diff --git a/.github/workflows/maven.yml b/.github/workflows/maven.yml
index 902077c..f4c8600 100644
--- a/.github/workflows/maven.yml
+++ b/.github/workflows/maven.yml
@@ -27,10 +27,10 @@ jobs:
 
     steps:
     - uses: actions/checkout@v4
-    - name: Set up JDK 17
+    - name: Set up JDK 21
       uses: actions/setup-java@v3
       with:
-        java-version: '17'
+        java-version: '21'
         distribution: 'temurin'
         cache: maven
     - name: Build with Maven
diff --git a/virage/.gitignore b/virage/.gitignore
index 5216f15..0f6ca3f 100644
--- a/virage/.gitignore
+++ b/virage/.gitignore
@@ -4,4 +4,9 @@
 /.classpath
 /.project
 /.settings/
+/.bloop/
+/.metals/
+/.vscode/
 /src/main/resources/generated/
+/.apt_generated/
+/.apt_generated_tests/
diff --git a/virage/pom.xml b/virage/pom.xml
index 8711097..8d49197 100644
--- a/virage/pom.xml
+++ b/virage/pom.xml
@@ -40,19 +40,24 @@
     <skipTests>false</skipTests>
     <project.build.sourceEncoding>UTF-8</project.build.sourceEncoding>
     <project.reporting.outputEncoding>UTF-8</project.reporting.outputEncoding>
-    <maven.version>3.8.1</maven.version>
-    <maven.clean.version>3.3.1</maven.clean.version>
-    <maven.install.version>3.1.1</maven.install.version>
-    <maven.jar.version>3.3.0</maven.jar.version>
-    <maven.compiler.release>17</maven.compiler.release>
-    <maven.compiler.source>17</maven.compiler.source>
-    <maven.compiler.target>17</maven.compiler.target>
-    <jdk.version>17</jdk.version>
-    <scala.version>2.13.12</scala.version>
-    <log4j.version>2.20.0</log4j.version>
-    <spotbugs.version>4.7.3</spotbugs.version>
-    <junit.jupiter.version>5.10.0</junit.jupiter.version>
-    <surefire.version>3.1.2</surefire.version>
+    <maven.version>3.6.3</maven.version>
+    <jdk.version>21</jdk.version>
+    <maven.clean.version>3.4.1</maven.clean.version>
+    <maven.install.version>3.1.4</maven.install.version>
+    <maven.exec.version>3.5.0</maven.exec.version>
+    <maven.jar.version>3.4.2</maven.jar.version>
+    <maven.resources.version>3.3.1</maven.resources.version>
+    <maven.checkstyle.version>3.6.0</maven.checkstyle.version>
+    <maven.buildhelper.version>3.6.0</maven.buildhelper.version>
+    <maven.reports.version>3.9.0</maven.reports.version>
+    <maven.compiler.release>${jdk.version}</maven.compiler.release>
+    <maven.compiler.source>${jdk.version}</maven.compiler.source>
+    <maven.compiler.target>${jdk.version}</maven.compiler.target>
+    <scala.version>2.13.16</scala.version>
+    <log4j.version>2.24.3</log4j.version>
+    <spotbugs.version>4.9.2.0</spotbugs.version>
+    <junit.jupiter.version>5.12.0</junit.jupiter.version>
+    <surefire.version>3.5.2</surefire.version>
     <swi.home.dir>/usr/lib/swi-prolog</swi.home.dir>
     <main.class>edu.kit.kastel.formal.virage.core.VirageMain</main.class>
   </properties>
@@ -108,7 +113,7 @@
     <dependency>
       <groupId>org.junit.platform</groupId>
       <artifactId>junit-platform-suite-engine</artifactId>
-      <version>1.10.0</version>
+      <version>1.12.0</version>
       <scope>test</scope>
     </dependency>
     <dependency>
@@ -135,22 +140,22 @@
     <dependency>
       <groupId>com.github.SWI-Prolog</groupId>
       <artifactId>packages-jpl</artifactId>
-      <version>V9.1.15</version>
+      <version>V9.3.21</version>
     </dependency>
     <dependency>
       <groupId>commons-cli</groupId>
       <artifactId>commons-cli</artifactId>
-      <version>1.5.0</version>
+      <version>1.9.0</version>
     </dependency>
     <dependency>
       <groupId>commons-io</groupId>
       <artifactId>commons-io</artifactId>
-      <version>2.14.0</version>
+      <version>2.18.0</version>
     </dependency>
     <dependency>
       <groupId>com.fasterxml.jackson.core</groupId>
       <artifactId>jackson-databind</artifactId>
-      <version>2.15.2</version>
+      <version>2.18.3</version>
     </dependency>
     <!-- <dependency> <groupId>com.github.VeriVote</groupId> <artifactId>ViRAGe</artifactId>
       <version>master-SNAPSHOT</version> </dependency> -->
@@ -158,22 +163,22 @@
     <dependency>
       <groupId>de.unruh</groupId>
       <artifactId>scala-isabelle_2.13</artifactId>
-      <version>0.4.2-RC1</version>
+      <version>0.4.2</version>
     </dependency>
     <dependency>
       <groupId>org.apache.commons</groupId>
       <artifactId>commons-configuration2</artifactId>
-      <version>2.9.0</version>
+      <version>2.11.0</version>
     </dependency>
     <dependency>
       <groupId>commons-beanutils</groupId>
       <artifactId>commons-beanutils</artifactId>
-      <version>1.9.4</version>
+      <version>1.10.1</version>
     </dependency>
     <dependency>
       <groupId>com.github.ferstl</groupId>
       <artifactId>depgraph-maven-plugin</artifactId>
-      <version>4.0.2</version>
+      <version>4.0.3</version>
     </dependency>
   </dependencies>
 
@@ -190,6 +195,7 @@
       <build>
         <plugins>
           <plugin>
+            <groupId>org.apache.maven.plugins</groupId>
             <artifactId>maven-compiler-plugin</artifactId>
             <executions>
               <execution>
@@ -207,6 +213,14 @@
             </executions>
           </plugin>
         </plugins>
+        <resources>
+          <resource>
+            <directory>${user.home}/.virage/</directory>
+            <includes>
+              <include>settings.properties</include>
+            </includes>
+          </resource>
+        </resources>
       </build>
     </profile>
   </profiles>
@@ -222,12 +236,12 @@
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-resources-plugin</artifactId>
-        <version>${maven.clean.version}</version>
+        <version>${maven.resources.version}</version>
       </plugin>
       <plugin>
         <groupId>net.alchim31.maven</groupId>
         <artifactId>scala-maven-plugin</artifactId>
-        <version>4.8.1</version>
+        <version>4.9.2</version>
         <configuration>
           <scalaVersion>${scala.version}</scalaVersion>
         </configuration>
@@ -251,7 +265,12 @@
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-compiler-plugin</artifactId>
-        <version>3.11.0</version>
+        <version>3.14.0</version>
+        <configuration>
+          <source>${jdk.version}</source>
+          <target>${jdk.version}</target>
+          <compilerArgument>-Xlint:deprecation</compilerArgument>
+        </configuration>
       </plugin>
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
@@ -278,7 +297,7 @@
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-site-plugin</artifactId>
-        <version>3.12.1</version>
+        <version>3.21.0</version>
         <configuration>
           <locales>en</locales>
         </configuration>
@@ -286,7 +305,7 @@
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-checkstyle-plugin</artifactId>
-        <version>${maven.jar.version}</version>
+        <version>${maven.checkstyle.version}</version>
         <configuration>
           <configLocation>checks.xml</configLocation>
           <encoding>UTF-8</encoding>
@@ -309,14 +328,14 @@
           <dependency>
             <groupId>com.puppycrawl.tools</groupId>
             <artifactId>checkstyle</artifactId>
-            <version>10.12.3</version>
+            <version>10.21.4</version>
           </dependency>
         </dependencies>
       </plugin>
       <plugin>
         <groupId>com.github.spotbugs</groupId>
         <artifactId>spotbugs-maven-plugin</artifactId>
-        <version>${spotbugs.version}.5</version>
+        <version>${spotbugs.version}</version>
         <configuration>
           <xmlOutput>true</xmlOutput>
           <failOnError>false</failOnError>
@@ -336,7 +355,7 @@
         <dependencies>
           <dependency>
             <groupId>com.github.spotbugs</groupId>
-            <artifactId>spotbugs</artifactId>
+            <artifactId>spotbugs-maven-plugin</artifactId>
             <version>${spotbugs.version}</version>
           </dependency>
         </dependencies>
@@ -344,7 +363,7 @@
       <plugin>
         <groupId>org.codehaus.mojo</groupId>
         <artifactId>build-helper-maven-plugin</artifactId>
-        <version>3.4.0</version>
+        <version>${maven.buildhelper.version}</version>
         <executions>
           <execution>
             <id>add-source</id>
@@ -364,7 +383,7 @@
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-enforcer-plugin</artifactId>
-        <version>3.4.1</version>
+        <version>3.5.0</version>
         <executions>
           <execution>
             <id>enforce-versions</id>
@@ -374,7 +393,7 @@
             <configuration>
               <rules>
                 <requireMavenVersion>
-                  <version>3.3.9</version>
+                  <version>3.6.3</version>
                 </requireMavenVersion>
                 <requireJavaVersion>
                   <version>${jdk.version}</version>
@@ -387,7 +406,7 @@
       <plugin>
         <groupId>org.codehaus.mojo</groupId>
         <artifactId>versions-maven-plugin</artifactId>
-        <version>2.16.0</version>
+        <version>2.18.0</version>
         <configuration>
           <rulesUri>file:///${basedir}/rules.xml</rulesUri>
           <allowSnapshots>false</allowSnapshots>
@@ -407,7 +426,7 @@
       <plugin>
         <groupId>org.codehaus.mojo</groupId>
         <artifactId>properties-maven-plugin</artifactId>
-        <version>1.2.0</version>
+        <version>1.2.1</version>
         <executions>
           <execution>
             <id>read-settings</id>
@@ -417,7 +436,7 @@
             </goals>
             <configuration>
               <files>
-                <file>${user.home}/.virage/settings</file>
+                <file>${user.home}/.virage/settings.properties</file>
               </files>
               <!-- settings will not be there on first startup. -->
               <!-- This is fine, as ViRAGe will copy it and enforce a restart if 
@@ -455,7 +474,7 @@
       <plugin>
         <groupId>org.codehaus.mojo</groupId>
         <artifactId>exec-maven-plugin</artifactId>
-        <version>3.1.0</version>
+        <version>${maven.exec.version}</version>
         <executions>
           <execution>
             <id>execute</id>
@@ -487,7 +506,7 @@
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-jxr-plugin</artifactId>
-        <version>${maven.jar.version}</version>
+        <version>${maven.checkstyle.version}</version>
       </plugin>
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
@@ -497,12 +516,12 @@
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-javadoc-plugin</artifactId>
-        <version>3.6.0</version>
+        <version>3.11.2</version>
       </plugin>
       <plugin>
         <groupId>org.apache.maven.plugins</groupId>
         <artifactId>maven-project-info-reports-plugin</artifactId>
-        <version>3.4.5</version>
+        <version>${maven.reports.version}</version>
         <reportSets>
           <reportSet>
             <reports>
@@ -525,7 +544,7 @@
       <plugin>
         <groupId>com.github.spotbugs</groupId>
         <artifactId>spotbugs-maven-plugin</artifactId>
-        <version>4.7.2.1</version>
+        <version>${spotbugs.version}</version>
         <reportSets>
           <reportSet>
             <reports>
diff --git a/virage/src/main/java/edu/kit/kastel/formal/util/ProcessUtils.java b/virage/src/main/java/edu/kit/kastel/formal/util/ProcessUtils.java
index f43549a..748432d 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/util/ProcessUtils.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/util/ProcessUtils.java
@@ -3,6 +3,7 @@
 import java.io.File;
 import java.io.IOException;
 import java.nio.charset.StandardCharsets;
+import java.util.StringTokenizer;
 
 import org.apache.logging.log4j.LogManager;
 import org.apache.logging.log4j.Logger;
@@ -40,6 +41,24 @@ public static void start(final String editorExecutable, final File path) {
         }
     }
 
+    /**
+     * Wrapper method for process execution, either strip and escape or simply format.
+     *
+     * @param cmd the command string, multiple ones separated by blanks
+     * @param escape flag for determining whether to strip and escape or not
+     * @return the resulting process
+     * @throws IOException if an I/O error occurs
+     */
+    public static Process exec(final String cmd, final boolean escape) throws IOException {
+        final String command = escape ? StringUtils.stripAndEscape(cmd) : String.format(cmd);
+        final StringTokenizer st = new StringTokenizer(command);
+        final String[] cmdarray = new String[st.countTokens()];
+        for (int i = 0; st.hasMoreTokens(); i++) {
+            cmdarray[i] = st.nextToken();
+        }
+        return new ProcessBuilder(cmdarray).start();
+    }
+
     /**
      * Executes a terminating command and prints its output to System.out/System.err, respectively.
      * <b>Does not return if the command is non-terminating!</b>
@@ -52,8 +71,7 @@ public static void start(final String editorExecutable, final File path) {
      */
     public static Output runTerminatingProcess(final String command)
             throws IOException, InterruptedException {
-        final Runtime rt = Runtime.getRuntime();
-        final Process p = rt.exec(StringUtils.stripAndEscape(command));
+        final Process p = exec(command, true);
         final int status = p.waitFor();
         final String stdErr = new String(p.getErrorStream().readAllBytes(), StandardCharsets.UTF_8);
         final String stdOut = new String(p.getInputStream().readAllBytes(), StandardCharsets.UTF_8);
diff --git a/virage/src/main/java/edu/kit/kastel/formal/util/StringUtils.java b/virage/src/main/java/edu/kit/kastel/formal/util/StringUtils.java
index 2e4e9fd..b017afe 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/util/StringUtils.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/util/StringUtils.java
@@ -37,6 +37,11 @@ public final class StringUtils {
      */
     public static final String HASH = "#";
 
+    /**
+     * A star sign.
+     */
+    public static final String STAR = "*";
+
     /**
      * An opening parenthesis sign.
      */
@@ -122,6 +127,16 @@ public final class StringUtils {
      */
     public static final int TEN = 10;
 
+    /**
+     * An opening bracket sign.
+     */
+    private static final String OPENING_BRACKET = "]";
+
+    /**
+     * A closing bracket sign.
+     */
+    private static final String CLOSING_BRACKET = "[";
+
     /**
      * The disjunction (or) sign.
      */
@@ -195,9 +210,9 @@ public static String printCollection(final Collection<?> c) {
         if (c.isEmpty()) {
             return EMPTY;
         }
-        final StringBuilder res = new StringBuilder(EMPTY);
+        final StringBuilder res = new StringBuilder();
         for (final Object obj: c) {
-            res.append(obj).append(COMMA);
+            res.append(obj).append(COMMA.charAt(0));
         }
         return res.deleteCharAt(res.length() - 1).toString();
     }
@@ -222,9 +237,9 @@ public static String printCollection2(final Collection<?> c) {
         if (c.isEmpty()) {
             return EMPTY;
         }
-        final StringBuilder res = new StringBuilder(EMPTY);
+        final StringBuilder res = new StringBuilder();
         for (final Object obj: c) {
-            res.append(obj).append(SPACE);
+            res.append(obj).append(SPACE.charAt(0));
         }
         return res.deleteCharAt(res.length() - 1).toString();
     }
@@ -279,6 +294,62 @@ public static String surroundWithSpaces(final String s) {
         return SPACE + s + SPACE;
     }
 
+    /**
+     * Adds the separator and a space before and a space and the separator after the given String.
+     *
+     * @param s the given String.
+     * @param separator the separator.
+     * @return new String with added separator and spaces before and after
+     */
+    public static String surroundWithSpacedSeparators(final String s, final String separator) {
+        return separator + surroundWithSpaces(s) + separator;
+    }
+
+    /**
+     * Put the given String in a raw Isabelle comment.
+     *
+     * @param s the given String.
+     * @return the resulting Isabelle comment containing the given String
+     */
+    public static String isabelleComment(final String s) {
+        return parenthesize(surroundWithSpacedSeparators(s, STAR));
+    }
+
+    /**
+     * Put the given String in a raw Isabelle comment line.
+     *
+     * @param s the given String.
+     * @return the resulting Isabelle comment line containing the given String
+     */
+    public static String isabelleCommentLine(final String s) {
+        return isabelleComment(s) + System.lineSeparator();
+    }
+
+    /**
+     * Put the given Strings in raw Isabelle comment lines.
+     *
+     * @param s the given Strings.
+     * @return the resulting Isabelle comment lines containing the given Strings
+     */
+    public static String isabelleCommentLines(final String... s) {
+        final StringBuilder sb = new StringBuilder();
+        for (final String s1: s) {
+            sb.append(isabelleCommentLine(s1));
+        }
+        return sb.toString();
+    }
+
+    /**
+     * Put the given Strings in raw Isabelle comment block.
+     *
+     * @param s the given Strings.
+     * @return the resulting Isabelle comment block containing the given Strings
+     */
+    public static String isabelleCommentBlock(final String... s) {
+        final String starLine = parenthesize(repeat(22, addSpace(STAR)) + STAR);
+        return starLine + System.lineSeparator() + isabelleCommentLines(s) + starLine;
+    }
+
     /**
      * Returns a disjunction of the two string arguments.
      *
@@ -372,6 +443,39 @@ public static String parenthesize2(final String... args) {
         return OPENING_PARENTHESIS + printCollection2(Arrays.asList(args)) + CLOSING_PARENTHESIS;
     }
 
+    /**
+     * Puts brackets around a comma-separated list of strings.
+     *
+     * @param args the string arguments to bracketize
+     * @return the bracketized string representation
+     */
+    public static String bracketize(final String... args) {
+        return OPENING_BRACKET + printCollection(Arrays.asList(args)) + CLOSING_BRACKET;
+    }
+
+    /**
+     * Puts parentheses around a comma-separated list of strings.
+     *
+     * @param args the string arguments to bracketize
+     * @return the bracketized string representation
+     */
+    public static String bracketize(final Collection<String> args) {
+        if (args == null || args.isEmpty()) {
+            return EMPTY;
+        }
+        return bracketize(args.toArray(new String[args.size()]));
+    }
+
+    /**
+     * Puts brackets around a space-separated list of strings.
+     *
+     * @param args the string arguments to bracketize
+     * @return the bracketized string representation
+     */
+    public static String bracketize2(final String... args) {
+        return OPENING_BRACKET + printCollection2(Arrays.asList(args)) + CLOSING_BRACKET;
+    }
+
     /**
      * Returns function string with a variable amount of argument.
      *
@@ -410,12 +514,12 @@ public static String map(final String fun, final String arg1, final String arg2)
     /**
      * Returns map function string with two separately parenthesized arguments.
      *
-     * @param arg1 the function which is to be applied by the map
-     * @param arg2 the collection argument to which the function is applied
+     * @param fun the function which is to be applied by the map
+     * @param app the collection argument to which the function is applied
      * @return the function string representation
      */
-    public static String map(final String arg1, final String arg2) {
-        return map(MAP, arg1, arg2);
+    public static String map(final String fun, final String app) {
+        return map(MAP, fun, app);
     }
 
     /**
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/analyzer/SimplePrologCompositionAnalyzer.java b/virage/src/main/java/edu/kit/kastel/formal/virage/analyzer/SimplePrologCompositionAnalyzer.java
index 4804a83..10381ee 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/analyzer/SimplePrologCompositionAnalyzer.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/analyzer/SimplePrologCompositionAnalyzer.java
@@ -194,10 +194,10 @@ public List<CompositionProof> proveClaims(final DecompositionTree composition,
             this.facade.setTimeout(oldTimeout);
             if (result.hasValue()) {
                 try {
-                    final Map<String, String> map = result.getValue();
-                    if (map.containsKey(proofVariable)) {
-                        LOGGER.debug(map.get(proofVariable));
-                        proofs.add(PrologProof.createProofFromString(map.get(proofVariable)));
+                    final String val = result.getValue().get(proofVariable);
+                    if (val != null) {
+                        LOGGER.debug(val);
+                        proofs.add(PrologProof.createProofFromString(val));
                     } else {
                         throw new IllegalArgumentException();
                     }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/beast/CCodeGenerator.java b/virage/src/main/java/edu/kit/kastel/formal/virage/beast/CCodeGenerator.java
index 4c5059c..def743d 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/beast/CCodeGenerator.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/beast/CCodeGenerator.java
@@ -99,11 +99,6 @@ public final class CCodeGenerator {
      */
     private final String codeFileTemplate;
 
-    /**
-     * Templates for the components of the framework.
-     */
-    private final String compositionsTemplate;
-
     /**
      * Templates for each component.
      */
@@ -144,7 +139,6 @@ public CCodeGenerator(final FrameworkRepresentation framework) {
         } catch (final IOException e) {
             e.printStackTrace();
         }
-        this.compositionsTemplate = writer.toString();
         this.templates = new HashMap<String, String>();
         this.signatures = new HashMap<String, List<String>>();
         final String componentNameGroup = "componentName";
@@ -155,7 +149,7 @@ public CCodeGenerator(final FrameworkRepresentation framework) {
                 + signatureGroup + ">[^\\)]*\\)).*)\\/\\/[ ]*\\k<" + componentNameGroup + ">"
                 + System.lineSeparator(),
                 Pattern.DOTALL);
-        final Matcher matcher = pattern.matcher(this.compositionsTemplate);
+        final Matcher matcher = pattern.matcher(writer.toString());
         while (matcher.find()) {
             this.templates.put(matcher.group(componentNameGroup),
                     matcher.group(implementationGroup));
@@ -210,8 +204,8 @@ private Pair<Pair<String, String>, Integer> getCCodeFromComposition(
                 this.frameworkField.getComponent(composition.getLabel());
         if (currentComponent != null) {
             final String componentName = composition.getLabel();
-            if (this.templates.containsKey(componentName)) {
-                String componentTemplate = this.templates.get(componentName);
+            String componentTemplate = this.templates.get(componentName);
+            if (componentTemplate != null) {
                 componentTemplate =
                         componentTemplate.replace(COMPONENT_INDEX_VARIABLE, String.valueOf(ctr));
                 int moduleCounter = 1;
@@ -256,7 +250,7 @@ private List<String> findMissingTemplates(final DecompositionTree composition) {
 
     private List<String> findMissingTemplates(final DecompositionTree composition,
                                               final List<String> found) {
-        if (!this.templates.keySet().contains(composition.getLabel())) {
+        if (!this.templates.containsKey(composition.getLabel())) {
             found.add(composition.getLabel());
         }
         for (final DecompositionTree child: composition.getChildren()) {
@@ -267,7 +261,7 @@ private List<String> findMissingTemplates(final DecompositionTree composition,
 
     /*public File generateAndCompileCCodeFromComposition(final String compositionString,
                                                          final int numVoters,
-                                                         final int numCandidates)
+                                                         final int numAlternatives)
                 throws IOException, InterruptedException, CompilationFailedException {
         final File votingRuleFile = this.getCCodeFromComposition(compositionString);
         String cFiles = StringUtils.EMPTY;
@@ -293,7 +287,7 @@ private List<String> findMissingTemplates(final DecompositionTree composition,
         final String gccCommand = ConfigReader.getInstance().getGccExecutable()
                 + StringUtils.prefixSpace(tmpVotingRuleFile.getAbsolutePath())
                 + cFiles + " -o " + exeFile.getAbsolutePath() + " -D V=" + numVoters
-                + " -D C=" + numCandidates;
+                + " -D C=" + numAlternatives;
         final Pair<String, String> output = ProcessUtils.runTerminatingProcess(gccCommand);
         if (!output.getFirstValue().isEmpty()) {
             LOGGER.warn(output.getFirstValue());
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/core/ConfigReader.java b/virage/src/main/java/edu/kit/kastel/formal/virage/core/ConfigReader.java
index 5f23779..8833481 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/core/ConfigReader.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/core/ConfigReader.java
@@ -1,13 +1,11 @@
 package edu.kit.kastel.formal.virage.core;
 
 import java.io.File;
-import java.io.FileInputStream;
 import java.io.IOException;
 import java.io.InputStream;
 import java.io.StringWriter;
 import java.nio.charset.StandardCharsets;
 import java.nio.file.Files;
-import java.util.HashMap;
 import java.util.LinkedList;
 import java.util.List;
 import java.util.Map;
@@ -24,6 +22,8 @@
 import org.apache.logging.log4j.Logger;
 import org.jpl7.JPL;
 
+import com.google.common.collect.Maps;
+
 import edu.kit.kastel.formal.util.Pair;
 import edu.kit.kastel.formal.util.ProcessUtils;
 import edu.kit.kastel.formal.util.SimpleFileReader;
@@ -68,7 +68,7 @@ public final class ConfigReader {
     /**
      * Name of the settings file.
      */
-    private static final String SETTINGS = "settings";
+    private static final String SETTINGS = "settings.properties";
 
     /**
      * Prefix for the old settings file.
@@ -391,13 +391,12 @@ public String checkAvailabilityAndGetVersions() {
             this.jplAvailable = false;
             this.swiplAvailable = false;
         }
-        res += getVersionString("JPL", null, JPL.version_string(), StringUtils.THREE);
-        return res;
+        return res + getVersionString("JPL", null, JPL.version_string(), StringUtils.THREE);
     }
 
     private Pair<Boolean, List<String>> checkConfigCompatibility() throws IOException {
         final Properties oldProperties = new Properties();
-        try (FileInputStream fis = new FileInputStream(this.configFile)) {
+        try (InputStream fis = java.nio.file.Files.newInputStream(this.configFile.toPath())) {
             oldProperties.load(fis);
         } catch (IOException e) {
             e.printStackTrace();
@@ -450,7 +449,7 @@ public void copyConfigToExpectedPath() throws IOException {
         }
         writer.writeToFile(oldPropertiesFile.getAbsolutePath(), reader.readFile(this.configFile));
         final Properties oldProperties = new Properties();
-        try (FileInputStream fis = new FileInputStream(oldPropertiesFile)) {
+        try (InputStream fis = java.nio.file.Files.newInputStream(oldPropertiesFile.toPath())) {
             oldProperties.load(fis);
         } catch (IOException e) {
             e.printStackTrace();
@@ -461,15 +460,16 @@ public void copyConfigToExpectedPath() throws IOException {
         IOUtils.copy(configFromResourcesStream, stringWriter, StandardCharsets.UTF_8);
         writer.writeToFile(this.configFile.getAbsolutePath(), stringWriter.toString());
         final Properties newProperties = new Properties();
-        try (FileInputStream fis = new FileInputStream(this.configFile)) {
+        try (InputStream fis = java.nio.file.Files.newInputStream(this.configFile.toPath())) {
             newProperties.load(fis);
         }
         for (final Object o: oldProperties.keySet()) {
-            if (o.toString().equals(VIRAGE_CONFIG_VERSION)) {
+            final String propName = o.toString();
+            if (VIRAGE_CONFIG_VERSION.equals(propName)) {
                 continue;
             }
             if (newProperties.keySet().contains(o)) {
-                this.updateValue(o.toString(), oldProperties.getProperty(o.toString()));
+                this.updateValue(propName, oldProperties.getProperty(propName));
             }
         }
     }
@@ -489,9 +489,11 @@ public List<String> getAdditionalProperties() {
      * @return the properties
      */
     public Map<String, String> getAllProperties() {
-        final Map<String, String> result = new HashMap<String, String>();
-        for (final Object prop: this.properties.keySet()) {
-            result.put(prop.toString(), this.properties.getProperty(prop.toString()));
+        final Properties props = this.properties;
+        final Map<String, String> result = Maps.newHashMapWithExpectedSize(props.size());
+        for (final Object prop: props.keySet()) {
+            final String propString = prop.toString();
+            result.put(propString, props.getProperty(propString));
         }
         return result;
     }
@@ -612,9 +614,9 @@ public String getPathToRootFile() {
      * @return the component aliases
      */
     public Map<String, String> getComponentAliases() {
-        final Map<String, String> toReturn = new HashMap<String, String>();
         final List<String> pairStrings =
                 readAndSplitList(this.properties.getProperty(COMPONENT_ALIASES));
+        final Map<String, String> toReturn = Maps.newHashMapWithExpectedSize(pairStrings.size());
         for (final String pairString: pairStrings) {
             final String[] elements = pairString.split(PAIR_SEPARATOR);
             toReturn.put(elements[0], elements[1]);
@@ -777,7 +779,7 @@ public void openConfigFileForEditing() throws ExternalSoftwareUnavailableExcepti
         ProcessUtils.start(XDG_OPEN, this.configFile);
         final String editorExecutable;
         if (this.properties.containsKey(SYSTEM_EDITOR)
-                && !this.properties.getProperty(SYSTEM_EDITOR).toString().isEmpty()) {
+                && !this.properties.getProperty(SYSTEM_EDITOR).isEmpty()) {
             editorExecutable = this.properties.getProperty(SYSTEM_EDITOR);
         } else if (System.getenv().containsKey(EDITOR)
                 && !System.getenv().get(EDITOR).isEmpty()) {
@@ -818,7 +820,7 @@ public void readConfigFile(final boolean overwriteIfNecessary) throws IOExceptio
             this.copyConfigToExpectedPath();
         }
         final Properties proposedProperties = new Properties();
-        try (FileInputStream fis = new FileInputStream(this.configFile)) {
+        try (InputStream fis = java.nio.file.Files.newInputStream(this.configFile.toPath())) {
             proposedProperties.load(fis);
         } catch (IOException e) {
             e.printStackTrace();
@@ -835,7 +837,7 @@ public void readConfigFile(final boolean overwriteIfNecessary) throws IOExceptio
                         + StringUtils.printCollection(configCompatibility.getSecondValue()));
             }
         }
-        try (FileInputStream fis = new FileInputStream(this.configFile)) {
+        try (InputStream fis = java.nio.file.Files.newInputStream(this.configFile.toPath())) {
             this.properties.load(fis);
         } catch (IOException e) {
             e.printStackTrace();
@@ -849,7 +851,7 @@ private String replaceTypeAliases(final String s) {
         }
         final List<String> replacements =
                 readAndSplitList(this.properties.getProperty(SESSION_SPECIFIC_TYPE_ALIASES));
-        final Map<String, String> replMap = new HashMap<String, String>();
+        final Map<String, String> replMap = Maps.newHashMapWithExpectedSize(replacements.size());
         for (final String repl: replacements) {
             final String[] parts = repl.split(PAIR_SEPARATOR);
             if (parts.length != 2) {
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageCommandLineInterface.java b/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageCommandLineInterface.java
index e5fd323..f699cb2 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageCommandLineInterface.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageCommandLineInterface.java
@@ -13,6 +13,7 @@
 import java.util.LinkedList;
 import java.util.List;
 import java.util.Scanner;
+import java.util.Set;
 
 import org.apache.commons.io.IOUtils;
 import org.apache.logging.log4j.LogManager;
@@ -37,15 +38,13 @@
 import edu.kit.kastel.formal.virage.jobs.VirageProveJob;
 import edu.kit.kastel.formal.virage.prolog.JplFacade;
 import edu.kit.kastel.formal.virage.prolog.PrologParser;
-import edu.kit.kastel.formal.virage.types.Component;
-import edu.kit.kastel.formal.virage.types.ComposableModule;
 import edu.kit.kastel.formal.virage.types.CompositionProof;
-import edu.kit.kastel.formal.virage.types.CompositionalStructure;
 import edu.kit.kastel.formal.virage.types.DecompositionTree;
 import edu.kit.kastel.formal.virage.types.ExternalSoftwareUnavailableException;
 import edu.kit.kastel.formal.virage.types.FrameworkRepresentation;
 import edu.kit.kastel.formal.virage.types.InvalidConfigVersionException;
 import edu.kit.kastel.formal.virage.types.Property;
+import edu.kit.kastel.formal.virage.types.TypedAndParameterized;
 
 /**
  * A simple command line interface for ViRAGe.
@@ -239,11 +238,6 @@ public final class VirageCommandLineInterface implements VirageUserInterface {
      */
     private final Writer outputWriter;
 
-    /**
-     * This UI's thread.
-     */
-    private Thread thread;
-
     /**
      * Simple constructor.
      *
@@ -551,12 +545,10 @@ public String requestString(final String message) {
             this.displayInputMarker();
             final String input = this.scanner.nextLine();
             switch (input) {
-            case StringUtils.QUESTION:
-            case "h":
-            case HELP:
+            case StringUtils.QUESTION, "h", "H", HELP:
                 this.displayHelp();
                 break;
-            case EXIT:
+            case "x", "X", "q", "Q", EXIT:
                 this.core.submit(new VirageExitJob(this, 0));
                 break;
             default:
@@ -641,6 +633,14 @@ private String requestPropertyString() {
         return propertiesString;
     }
 
+    private static List<String> convertNames(final Set<? extends TypedAndParameterized> cs) {
+        final List<String> names = new ArrayList<String>(cs.size());
+        for (final TypedAndParameterized c: cs) {
+            names.add(StringUtils.indentWithTab(c.toString()));
+        }
+        return names;
+    }
+
     private String requestCompositionString() {
         boolean invalid = false;
         String compositionString = this
@@ -661,18 +661,16 @@ private String requestCompositionString() {
             }
         }
         if (compositionString.isEmpty() || invalid) {
-            final List<String> sortedStrings = new ArrayList<String>();
-            for (final Component c: this.core.getFrameworkRepresentation().getComponents()) {
-                sortedStrings.add(StringUtils.indentWithTab(c.toString()));
-            }
-            for (final ComposableModule c: this.core.getFrameworkRepresentation()
-                    .getComposableModules()) {
-                sortedStrings.add(StringUtils.indentWithTab(c.toString()));
-            }
-            for (final CompositionalStructure c: this.core.getFrameworkRepresentation()
-                    .getCompositionalStructures()) {
-                sortedStrings.add(StringUtils.indentWithTab(c.toString()));
-            }
+            final FrameworkRepresentation fr = this.core.getFrameworkRepresentation();
+            final List<String> compNames = convertNames(fr.getComponents());
+            final List<String> modNames = convertNames(fr.getComposableModules());
+            final List<String> structNames = convertNames(fr.getCompositionalStructures());
+            final List<String> sortedStrings =
+                    new ArrayList<String>(compNames.size() + modNames.size() + structNames.size());
+
+            sortedStrings.addAll(compNames);
+            sortedStrings.addAll(modNames);
+            sortedStrings.addAll(structNames);
             Collections.sort(sortedStrings);
             for (final String s: sortedStrings) {
                 this.displayMessage(s);
@@ -707,8 +705,7 @@ private VirageGenerateJob createGenerationQuery() {
 
     private VirageProveJob createProofQuery(final String composition, final String propertyString) {
         final List<String> properties = StringUtils.separate(StringUtils.COMMA, propertyString);
-        final VirageProveJob res = new VirageProveJob(this, composition, properties);
-        return res;
+        return new VirageProveJob(this, composition, properties);
     }
 
     private VirageJob<?> createIsabelleQuery() {
@@ -753,9 +750,7 @@ private VirageJob<?> createIsabelleQuery() {
         }
         this.core.submit(generateJob);
         generateJob.waitFor();
-        final VirageIsabelleVerifyJob verifyJob =
-                new VirageIsabelleVerifyJob(this, generateJob.getResult());
-        return verifyJob;
+        return new VirageIsabelleVerifyJob(this, generateJob.getResult());
     }
 
     /**
@@ -937,8 +932,7 @@ private void extractOrParseInputs() {
      */
     @Override
     public void launch() {
-        this.thread = new Thread(this, "vcli");
-        this.thread.start();
+        new Thread(this, "vcli").start();
     }
 
     @Override
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageSearchManager.java b/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageSearchManager.java
index 03a5fa5..3fd6c3c 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageSearchManager.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/core/VirageSearchManager.java
@@ -59,10 +59,8 @@ public List<List<SearchResult<BooleanWithUncertainty>>> analyzeComposition(
         // TODO Parallelize.
         final List<List<SearchResult<BooleanWithUncertainty>>> results =
                 new LinkedList<List<SearchResult<BooleanWithUncertainty>>>();
-        for (int i = 0; i < this.analyzers.size(); i++) {
-            final List<SearchResult<BooleanWithUncertainty>> result =
-                    this.analyzers.get(i).analyzeComposition(composition, properties);
-            results.add(result);
+        for (final CompositionAnalyzer analyzer: this.analyzers) {
+            results.add(analyzer.analyzeComposition(composition, properties));
         }
         return results;
     }
@@ -82,10 +80,8 @@ public List<SearchResult<DecompositionTree>> generateComposition(
         // TODO Parallelize.
         final List<SearchResult<DecompositionTree>> results =
                 new LinkedList<SearchResult<DecompositionTree>>();
-        for (int i = 0; i < this.analyzers.size(); i++) {
-            final SearchResult<DecompositionTree> result =
-                    this.analyzers.get(i).generateComposition(properties);
-            results.add(result);
+        for (final CompositionAnalyzer analyzer: this.analyzers) {
+            results.add(analyzer.generateComposition(properties));
         }
         return results;
     }
@@ -106,10 +102,8 @@ public List<List<CompositionProof>> proveClaims(final DecompositionTree composit
         }
         // TODO Parallelize.
         final List<List<CompositionProof>> results = new LinkedList<List<CompositionProof>>();
-        for (int i = 0; i < this.analyzers.size(); i++) {
-            final List<CompositionProof> result =
-                    this.analyzers.get(i).proveClaims(composition, properties);
-            results.add(result);
+        for (final CompositionAnalyzer analyzer: this.analyzers) {
+            results.add(analyzer.proveClaims(composition, properties));
         }
         return results;
     }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleClientObserver.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleClientObserver.java
index 98249e0..cb7b498 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleClientObserver.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleClientObserver.java
@@ -26,11 +26,6 @@ public final class IsabelleClientObserver implements Runnable {
      */
     private final IsabelleProofChecker listener;
 
-    /**
-     * The Isabelle client process.
-     */
-    private final Process isabelleClient;
-
     /**
      * The Isabelle event factory.
      */
@@ -49,13 +44,12 @@ public final class IsabelleClientObserver implements Runnable {
     private IsabelleClientObserver(final IsabelleProofChecker listenerValue,
             final Process isabelleClientValue) {
         this.listener = listenerValue;
-        this.isabelleClient = isabelleClientValue;
         this.factory = new IsabelleEventFactory();
         this.stdoutReader = new BufferedReader(
-                new InputStreamReader(this.isabelleClient.getInputStream(),
+                new InputStreamReader(isabelleClientValue.getInputStream(),
                                       StandardCharsets.UTF_8));
         this.stderrReader = new BufferedReader(
-                new InputStreamReader(this.isabelleClient.getErrorStream(),
+                new InputStreamReader(isabelleClientValue.getErrorStream(),
                                       StandardCharsets.UTF_8));
         final Thread thread = new Thread(this, "Isabelle Console Output");
         thread.start();
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleCodeGenerator.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleCodeGenerator.java
index 0b24c14..dfa09a4 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleCodeGenerator.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleCodeGenerator.java
@@ -131,11 +131,6 @@ public final class IsabelleCodeGenerator {
      */
     private static final String PARENT_NAME_VAR = "$PARENT_NAME";
 
-    /**
-     * String for enumerable.
-     */
-    private static final String ENUM = "Enum";
-
     /**
      * Comment for enumerable.
      */
@@ -158,9 +153,11 @@ public final class IsabelleCodeGenerator {
     private static final String RELATION =
             StringUtils.parenthesize2(
                     ScalaIsabelleFacade.DEFAULT_VARIABLE + StringUtils.COLON,
-                    IsabelleUtils.SET + IsabelleUtils.THEORY_NAME_SEPARATOR + "set["
-                    + StringUtils.parenthesize(IsabelleUtils.DEFAULT_SET, IsabelleUtils.DEFAULT_SET)
-                    + "]") + StringUtils.COLON;
+                    IsabelleUtils.SET + IsabelleUtils.THEORY_NAME_SEPARATOR + IsabelleUtils.SET_FUNC
+                    + StringUtils.bracketize(
+                            StringUtils.parenthesize(
+                                    IsabelleUtils.DEFAULT_SET, IsabelleUtils.DEFAULT_SET)))
+            + StringUtils.COLON;
 
     /**
      * Option1 comment.
@@ -338,7 +335,7 @@ private String prepareTheoryFile(final File theory, final String language) throw
         String newName = StringUtils.EMPTY;
         final Map<String, String> map =
                 IsabelleTheoryParser.getAllFunctionsAndDefinitions(theory.getCanonicalPath());
-        if (map.keySet().size() != 1) {
+        if (map.size() != 1) {
             throw new IllegalArgumentException();
         }
         for (final String definition: map.keySet()) {
@@ -362,7 +359,7 @@ private String prepareTheoryFile(final File theory, final String language) throw
         final List<String> lines = this.reader.readFileByLine(theory);
         for (int i = 0; i < lines.size(); i++) {
             final String line = lines.get(i);
-            if (StringUtils.removeWhitespace(line).equals(END)) {
+            if (END.equals(StringUtils.removeWhitespace(line))) {
                 lines.add(i, result);
                 break;
             }
@@ -410,7 +407,7 @@ private File prepareVotingContext(final String theoryName, final String moduleNa
         String result =
                 IsabelleCodeGenerator.votingContextTemplate
                 .replace(THEORY_NAME_VAR, theoryName).replace(MODULE_NAME_VAR, moduleName);
-        final boolean containsEnum = code.contains(ENUM);
+        final boolean containsEnum = code.contains(IsabelleUtils.ENUM);
         final boolean containsEquality = code.contains(EQUALITY);
         final boolean requiresRelation = code.contains(RELATION);
         final List<String> parameters = new LinkedList<String>();
@@ -423,7 +420,7 @@ private File prepareVotingContext(final String theoryName, final String moduleNa
         if (containsEquality) {
             result = result.replace(COMMENT_OPEN + EQUALITY_COMMENT, StringUtils.EMPTY)
                     .replace(EQUALITY_COMMENT + COMMENT_CLOSE, StringUtils.EMPTY);
-            parameters.add(EQ_PARAMETER);
+            parameters.add(StringUtils.prefixSpace(EQ_PARAMETER));
         }
         if (requiresRelation) {
             result = result.replace(COMMENT_OPEN + OPTION2_COMMENT, StringUtils.EMPTY)
@@ -618,8 +615,10 @@ public File generateScalaCodeAndCompile(final File theory)
                         theory.getName().length() - (IsabelleUtils.DOT_THY.length()));
         final String sessionName = this.buildSessionRoot(theoryName, theory);
         final File codeFile = invokeScalaCodeFromIsabelleTheory(theory, sessionName, theoryName);
-        // First, try using implicit values only
-        File votingContext = generateVotingContextFile(theoryName, moduleName, codeFile, false);
+        // First, try using implicit values only.
+        final File cxtImplVals = generateVotingContextFile(theoryName, moduleName, codeFile, false);
+        // If implicit values did not work, try setting them explicitly.
+        final File cxtExplVals = generateVotingContextFile(theoryName, moduleName, codeFile, true);
         final String jarPath = codeFile.getParent() + File.separator + moduleName + DOT_JAR;
         final String isaExec;
         try {
@@ -627,10 +626,15 @@ public File generateScalaCodeAndCompile(final File theory)
         } catch (ExternalSoftwareUnavailableException e) {
             throw new CodeGenerationFailedException(e);
         }
-        int status = runScalaCompileProcess(codeFile, votingContext, isaExec, jarPath);
-        if (status != 0) { // Implicit values did not work, try setting them explicitly.
-            votingContext = generateVotingContextFile(theoryName, moduleName, codeFile, true);
-            status = runScalaCompileProcess(codeFile, votingContext, isaExec, jarPath);
+        int status = 1;
+        try {
+            status = runScalaCompileProcess(codeFile, cxtImplVals, isaExec, jarPath);
+        } catch (CodeGenerationFailedException e) {
+            LOGGER.info("Scala compilation with implicit values did not work,"
+                        + "now trying with explicit values.");
+        }
+        if (status != 0) {
+            status = runScalaCompileProcess(codeFile, cxtExplVals, isaExec, jarPath);
             if (status != 0) {
                 throw new CodeGenerationFailedException(
                         new CompilationFailedException(
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleFrameworkExtractor.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleFrameworkExtractor.java
index ce4eecc..3ed7462 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleFrameworkExtractor.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleFrameworkExtractor.java
@@ -12,6 +12,8 @@
 import org.apache.logging.log4j.LogManager;
 import org.apache.logging.log4j.Logger;
 
+import com.google.common.collect.Maps;
+
 import edu.kit.kastel.formal.util.Pair;
 import edu.kit.kastel.formal.util.StringUtils;
 import edu.kit.kastel.formal.util.SystemUtils;
@@ -80,17 +82,11 @@ public final class IsabelleFrameworkExtractor {
      */
     private final PrologParser parser;
 
-    /**
-     * List of Prolog Strings.
-     */
-    private final List<String> prologStrings;
-
     /**
      * Simple constructor.
      */
     public IsabelleFrameworkExtractor() {
         this.parser = new SimplePrologParser();
-        this.prologStrings = new LinkedList<String>();
     }
 
     private static String buildPrologClauseString(final String succedent,
@@ -185,18 +181,13 @@ private void convertCompositionRules(final FrameworkRepresentation framework,
                         continue thmLoop;
                     }
                 }
-                final List<String> prologStringList = new LinkedList<String>();
-                prologStringList.add(buildPrologClauseString(succedent, antecedents));
+                final String prologString = buildPrologClauseString(succedent, antecedents);
                 try {
-                    for (final String prologString: prologStringList) {
-                        final CompositionRule rule =
-                                new CompositionRule(thm.getKey(),
-                                                    thy.getKey().split(DOT_REGEX)[1]
-                                                            + IsabelleUtils.DOT_THY,
-                                                    this.parser.parseSingleClause(prologString));
-                        framework.add(rule);
-                        this.prologStrings.add(prologString);
-                    }
+                    framework.add(
+                            new CompositionRule(thm.getKey(),
+                                                thy.getKey().split(DOT_REGEX)[1]
+                                                        + IsabelleUtils.DOT_THY,
+                                                this.parser.parseSingleClause(prologString)));
                 } catch (final IllegalArgumentException e) {
                     this.logger.warn(e);
                 }
@@ -244,8 +235,7 @@ private static String convertIsabelleToProlog(final String s) {
                 break;
             }
         }
-        res += StringUtils.RIGHT_PAREN;
-        return res;
+        return res + StringUtils.RIGHT_PAREN;
     }
 
     /**
@@ -308,7 +298,7 @@ public FrameworkRepresentation extract(final String sessionDir, final String ses
                 computeTransitiveClosureOfComponentAliases() {
         final Map<String, String> input = ConfigReader.getInstance().getComponentAliases();
         final Map<PrologPredicate, List<PrologPredicate>> res =
-                    new HashMap<PrologPredicate, List<PrologPredicate>>();
+                Maps.newHashMapWithExpectedSize(input.size());
         for (final Map.Entry<String, String> in: input.entrySet()) {
             res.put(this.parser.parsePredicate(in.getKey()),
                     List.of(this.parser.parsePredicate(in.getValue())));
@@ -334,16 +324,16 @@ public FrameworkRepresentation extract(final String sessionDir, final String ses
             changesMade = false;
             for (final PrologPredicate alias: toReturn.keySet()) {
                 final List<PrologPredicate> aliasExpansions = toReturn.get(alias);
-                for (final PrologPredicate candidate: toReturn.keySet()) {
-                    if (candidate == alias) {
+                for (final PrologPredicate predicate: toReturn.keySet()) {
+                    if (predicate == alias) {
                         continue;
                     }
-                    final List<PrologPredicate> candidateExpansions = toReturn.get(candidate);
+                    final List<PrologPredicate> predicateExpansions = toReturn.get(predicate);
                     final List<PrologPredicate> newValues =
-                        computeAllExpansions(alias, aliasExpansions, candidateExpansions);
+                        computeAllExpansions(alias, aliasExpansions, predicateExpansions);
                     for (final PrologPredicate newValue: newValues) {
-                        if (!candidateExpansions.contains(newValue)) {
-                            candidateExpansions.add(newValue);
+                        if (!predicateExpansions.contains(newValue)) {
+                            predicateExpansions.add(newValue);
                             changesMade = true;
                         }
                     }
@@ -356,15 +346,15 @@ public FrameworkRepresentation extract(final String sessionDir, final String ses
     private static List<PrologPredicate>
                 computeAllExpansions(final PrologPredicate alias,
                                      final List<PrologPredicate> aliasExpansions,
-                                     final List<PrologPredicate> candidateExpansions)
+                                     final List<PrologPredicate> predicateExpansions)
                     throws ExternalSoftwareUnavailableException {
         final List<PrologPredicate> toReturn = new LinkedList<PrologPredicate>();
-        for (final PrologPredicate candidateExpansion: candidateExpansions) {
-            final List<PrologPredicate> children = candidateExpansion.getAllChildren();
+        for (final PrologPredicate predicateExpansion: predicateExpansions) {
+            final List<PrologPredicate> children = predicateExpansion.getAllChildren();
             for (final PrologPredicate child: children) {
                 if (child.getName().equals(alias.getName())) {
                     for (final PrologPredicate aliasExpansion: aliasExpansions) {
-                        toReturn.add(replaceAlias(candidateExpansion, child, aliasExpansion));
+                        toReturn.add(replaceAlias(predicateExpansion, child, aliasExpansion));
                     }
                 }
             }
@@ -384,7 +374,7 @@ private static PrologPredicate replaceAlias(final PrologPredicate original,
         } catch (final IllegalArgumentException e) {
             unifiable = false;
         }
-        if (replacements.keySet().contains(PrologPredicate.ANONYMOUS)) {
+        if (replacements.containsKey(PrologPredicate.ANONYMOUS)) {
             replacements.remove(PrologPredicate.ANONYMOUS);
         }
         final boolean onlyOneVariable =
@@ -527,7 +517,6 @@ private static String replaceVariables(final String isaString) {
                                                    varName.toUpperCase());
             matcher = pattern.matcher(prologString);
         }
-        prologString = prologString.replace("?", StringUtils.EMPTY);
-        return prologString;
+        return prologString.replace("?", StringUtils.EMPTY);
     }
 }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofChecker.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofChecker.java
index e25b05f..42363b7 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofChecker.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofChecker.java
@@ -148,6 +148,16 @@ public final class IsabelleProofChecker {
      */
     private static final String PWD_CMD = "`pwd`";
 
+    /**
+     * The prefix for ad-hoc Isabelle sessions.
+     */
+    private static final String AD_HOC_SESSION_PREFIX = "ad_hoc_session_";
+
+    /**
+     * The pattern for simple date formats, intended for creating quasi unique file names.
+     */
+    private static final String DATE_FORMAT_PATTERN = "yyyyMMddHHmmss";
+
     /**
      * The session name variable.
      */
@@ -163,11 +173,6 @@ public final class IsabelleProofChecker {
      */
     private static final String PARENT_NAME_VAR = "$PARENT_NAME";
 
-    /**
-     * The runtime.
-     */
-    private final Runtime runtime;
-
     /**
      * The server process.
      */
@@ -231,7 +236,6 @@ public final class IsabelleProofChecker {
     private IsabelleProofChecker(final String sessionNameValue, final String theoryPathValue)
             throws ExternalSoftwareUnavailableException, IsabelleBuildFailedException {
         this.theoryPath = theoryPathValue;
-        this.runtime = Runtime.getRuntime();
         this.solvers = ConfigReader.getInstance().getIsabelleTactics();
 
         // Use scala-isabelle to build the session, as my own solution
@@ -244,7 +248,7 @@ private IsabelleProofChecker(final String sessionNameValue, final String theoryP
                             ConfigReader.getInstance().getIsabelleExecutable(),
                             BUILD_TOOL, SYS_OPT, BROWSER_INFO,
                             BUILD_OPT, INCL_SEL_SESS_DIR, PWD_CMD);
-            final Process process = Runtime.getRuntime().exec(String.format(clString));
+            final Process process = ProcessUtils.exec(clString, false);
             process.waitFor();
 
             this.initServer();
@@ -321,7 +325,8 @@ private String buildSessionRoot(final String theoryName, final File theory) {
         // rebuilding single sessions without triggering full rebuilds.
         // TODO: Is there a way to do it?
         final String localSessionName =
-                "ad_hoc_session_" + new SimpleDateFormat("yyyyMMddHHmmss").format(new Date());
+                AD_HOC_SESSION_PREFIX
+                + new SimpleDateFormat(DATE_FORMAT_PATTERN).format(new Date());
         String result = this.rootTemplate
                 .replace(SESSION_NAME_VAR, localSessionName).replace(THEORY_NAME_VAR, theoryName);
         result = result.replace(PARENT_NAME_VAR, this.parentName);
@@ -380,7 +385,7 @@ private void initClient(final String localSessionName, final String localTheoryP
                         ConfigReader.getInstance().getIsabelleExecutable(),
                         CLIENT_TOOL, NAME_OPT, SERVER_NAME);
         this.sessionName = localSessionName;
-        this.client = this.runtime.exec(String.format(clString));
+        this.client = ProcessUtils.exec(clString, false);
         this.clientInput = this.client.getOutputStream();
 
         IsabelleClientObserver.start(this, this.client);
@@ -397,7 +402,7 @@ private void initServer() throws IOException, ExternalSoftwareUnavailableExcepti
                 StringUtils.printCollection2(
                         ConfigReader.getInstance().getIsabelleExecutable(),
                         SERVER_TOOL, NAME_OPT, SERVER_NAME);
-        this.server = this.runtime.exec(String.format(clString));
+        this.server = ProcessUtils.exec(clString, false);
         // The server will send a message when startup is finished.
         // Contents are irrelevant, just wait for it to appear.
         try (BufferedReader br = new BufferedReader(new InputStreamReader(
@@ -558,7 +563,7 @@ public Pair<Boolean, File> verifyTheoryFile(final File theory,
         String command = "use_theories " + sessionAndTheoriesCommand;
         this.sendCommandAndWaitForTermination(command);
 
-        final String result = this.lastEvent.getValue(IsabelleUtils.SUCCESS_STRING.toLowerCase());
+        final String result = this.lastEvent.getValue(IsabelleUtils.SUCCESS_STRING);
         if ("true".equals(result)) {
             LOGGER.info("Verification successful.");
             final String adHocSessionName = this.buildSessionRoot(
@@ -569,33 +574,35 @@ public Pair<Boolean, File> verifyTheoryFile(final File theory,
                     | ExternalSoftwareUnavailableException e) {
                 LOGGER.warn("No documentation could be generated.");
             }
-            return new Pair<Boolean, File>(true, theory);
+            return new Pair<Boolean, File>(Boolean.TRUE, theory);
         } else {
             LOGGER.info("Verification failed. Attempting to solve automatically "
                     + "by employing different solvers.");
             final String errors = this.lastEvent.getValue("errors");
 
-            command = "purge_theories "  + sessionAndTheoriesCommand;
-            this.sendCommandAndWaitForOk(command);
-            final Pattern pattern = Pattern.compile("line=[0-9]+");
-            final Matcher matcher = pattern.matcher(errors);
-
-            if (matcher.find()) {
-                final String line = errors.substring(matcher.start(), matcher.end());
-                // Isabelle starts counting at 1.
-                final int lineNum = Integer.parseInt(line.split("=")[1]) - 1;
-                final File newFile = this.replaceSolver(theory, lineNum);
-                if (newFile != null) {
-                    // The content of the file has changed, and this can
-                    // only happen IsabelleUtils.SOLVERS.length times,
-                    // so the recursive call is fine.
-                    return this.verifyTheoryFile(newFile, framework);
-                } else {
-                    LOGGER.info("Automatic verification failed. "
-                            + "You might be able to fix the errors manually within Isabelle.");
+            if (errors != null) {
+                command = "purge_theories "  + sessionAndTheoriesCommand;
+                this.sendCommandAndWaitForOk(command);
+                final Pattern pattern = Pattern.compile("line=[0-9]+");
+                final Matcher matcher = pattern.matcher(errors);
+
+                if (matcher.find()) {
+                    final String line = errors.substring(matcher.start(), matcher.end());
+                    // Isabelle starts counting at 1.
+                    final int lineNum = Integer.parseInt(line.split("=")[1]) - 1;
+                    final File newFile = this.replaceSolver(theory, lineNum);
+                    if (newFile != null) {
+                        // The content of the file has changed, and this can
+                        // only happen IsabelleUtils.SOLVERS.length times,
+                        // so the recursive call is fine.
+                        return this.verifyTheoryFile(newFile, framework);
+                    } else {
+                        LOGGER.info("Automatic verification failed. "
+                                + "You might be able to fix the errors manually within Isabelle.");
+                    }
                 }
             }
-            return new Pair<Boolean, File>(false, null);
+            return new Pair<Boolean, File>(Boolean.FALSE, null);
         }
     }
 
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofGenerator.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofGenerator.java
index 95c83f0..b8737e2 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofGenerator.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofGenerator.java
@@ -80,11 +80,6 @@ public class IsabelleProofGenerator {
      */
     private final IsabelleProofStepGenerator generator;
 
-    /**
-     * Functions and definitions from an Isabelle session.
-     */
-    private final Map<String, String> functionsAndDefinitions;
-
     /**
      * The parent theory generator.
      */
@@ -111,8 +106,7 @@ public IsabelleProofGenerator(final IsabelleTheoryGenerator parentValue,
             }
             setProofTemplate(writer.toString());
         }
-        this.functionsAndDefinitions = functionsAndDefinitionsValue;
-        this.generator = new IsabelleProofStepGenerator(this, this.functionsAndDefinitions);
+        this.generator = new IsabelleProofStepGenerator(this, functionsAndDefinitionsValue);
         this.parent = parentValue;
     }
 
@@ -123,13 +117,12 @@ private static synchronized void setProofTemplate(final String newTemplate) {
     private static String replaceVariables(final String theoremName, final String goal,
                                            final String proofSteps, final String subgoalIds,
                                            final String assumptions) {
-        String res = IsabelleProofGenerator.proofTemplate;
-        res = res.replace(VAR_THEOREM_NAME, theoremName);
-        res = res.replace(VAR_GOAL, goal);
-        res = res.replace(VAR_PROOF_STEPS, proofSteps);
-        res = res.replace(VAR_SUBGOAL_ID, subgoalIds);
-        res = res.replace(VAR_ASSUMPTIONS, assumptions);
-        return res;
+        return IsabelleProofGenerator.proofTemplate
+                .replace(VAR_THEOREM_NAME, theoremName)
+                .replace(VAR_GOAL, goal)
+                .replace(VAR_PROOF_STEPS, proofSteps)
+                .replace(VAR_SUBGOAL_ID, subgoalIds)
+                .replace(VAR_ASSUMPTIONS, assumptions);
     }
 
     /**
@@ -165,26 +158,24 @@ public String generateIsabelleProof(final CompositionProof proof) {
         final Set<String> assumptions = new HashSet<String>();
         final StringBuilder proofSteps = new StringBuilder();
         for (final CompositionProof subgoal: proof.getAllStepsDepthFirst()) {
-            if (subgoal.getAllCompositionRules().size() == 1) {
-                final CompositionRule rule = subgoal.getAllCompositionRules().iterator().next();
-                if (rule.getOrigin().equals(ExtendedPrologStrings.ASSUMPTION)) {
-                    final Property p =
-                            this.parent.getFramework().getProperty(rule.getSuccedent().getName());
-                    String newAssumptions =
-                            StringUtils.addSpace(StringUtils.QUOTATION
-                                                    + rule.getSuccedent().getName());
+            final Set<CompositionRule> rules = subgoal.getAllCompositionRules();
+            if (rules.size() == 1) {
+                final CompositionRule rule = rules.iterator().next();
+                if (ExtendedPrologStrings.ASSUMPTION.equals(rule.getOrigin())) {
+                    final String succName = rule.getSuccedent().getName();
+                    final Property p = this.parent.getFramework().getProperty(succName);
+                    String newAssumptions = StringUtils.addSpace(StringUtils.QUOTATION + succName);
                     for (final ComponentType child: p.getParameters()) {
+                        final String childVar =
+                                this.getParent().getTypedVariables().get(child.getName());
                         // Only parameters defined within the module definition can be referenced,
                         // so if
                         // any others
                         // occur, we skip.
-                        if (!this.getParent().getTypedVariables().containsKey(child.getName())) {
+                        if (childVar == null) {
                             break;
                         }
-                        newAssumptions +=
-                                StringUtils.addSpace(
-                                        this.getParent().getTypedVariables().get(child.getName()))
-                                    + StringUtils.QUOTATION;
+                        newAssumptions += StringUtils.addSpace(childVar) + StringUtils.QUOTATION;
                         assumptions.add(newAssumptions);
                     }
                     // This is needed to not add subgoal identities for these "virtual" goals.
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofStepGenerator.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofStepGenerator.java
index 3d5cf58..332e912 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofStepGenerator.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleProofStepGenerator.java
@@ -118,13 +118,12 @@ private static synchronized void setProotStepTemplate(final String newTemplate)
     private static String replaceVariables(final String goalId, final String goal,
                                            final String subgoalIds, final String rule,
                                            final String solver) {
-        String res = IsabelleProofStepGenerator.proofStepTemplate;
-        res = res.replace(VAR_GOAL_ID, goalId);
-        res = res.replace(VAR_GOAL, goal);
-        res = res.replace(VAR_SUBGOAL_IDS, subgoalIds);
-        res = res.replace(VAR_RULE, rule);
-        res = res.replace(VAR_SOLVER, solver);
-        return res;
+        return IsabelleProofStepGenerator.proofStepTemplate
+                .replace(VAR_GOAL_ID, goalId)
+                .replace(VAR_GOAL, goal)
+                .replace(VAR_SUBGOAL_IDS, subgoalIds)
+                .replace(VAR_RULE, rule)
+                .replace(VAR_SOLVER, solver);
     }
 
     /**
@@ -141,7 +140,7 @@ public String generateIsabelleProofStep(final CompositionProof step) {
         final Map<String, Set<String>> goalMap =
                 IsabelleUtils
                 .translatePrologToIsabelle(this.functionsAndDefinitions, predicate.toString());
-        if (goalMap.keySet().size() != 1) {
+        if (goalMap.size() != 1) {
             throw new IllegalArgumentException();
         }
         String goal = StringUtils.EMPTY;
@@ -151,9 +150,9 @@ public String generateIsabelleProofStep(final CompositionProof step) {
 
         final StringBuilder subgoalIds = new StringBuilder();
         for (final CompositionProof subgoal: step.getSubgoals()) {
-            if (subgoal.getAllCompositionRules().size() == 1) {
-                final CompositionRule rule = subgoal.getAllCompositionRules().iterator().next();
-                if (rule.getOrigin().equals(ExtendedPrologStrings.ASSUMPTION)) {
+            final Set<CompositionRule> rules = subgoal.getAllCompositionRules();
+            if (rules.size() == 1) {
+                if (ExtendedPrologStrings.ASSUMPTION.equals(rules.iterator().next().getOrigin())) {
                     continue;
                 }
             }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryGenerator.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryGenerator.java
index 5567d87..6ac866c 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryGenerator.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryGenerator.java
@@ -5,9 +5,11 @@
 import java.io.InputStream;
 import java.io.StringWriter;
 import java.nio.charset.StandardCharsets;
+import java.util.ArrayList;
 import java.util.Collections;
 import java.util.HashMap;
 import java.util.HashSet;
+import java.util.LinkedHashSet;
 import java.util.LinkedList;
 import java.util.List;
 import java.util.Map;
@@ -194,7 +196,7 @@ private static String findImportsFromCompositionRules(final List<CompositionProo
                 // Isabelle expects imports without suffix.
                 originStrings.add(origin.replace(IsabelleUtils.DOT_THY, StringUtils.EMPTY));
             } else {
-                if (origin.equals(ExtendedPrologStrings.UNPROVEN)) {
+                if (ExtendedPrologStrings.UNPROVEN.equals(origin)) {
                     // Proof relies on unproven facts, add a comment explaining this.
                     usingUnprovenFacts = true;
                 }
@@ -207,23 +209,11 @@ private static String findImportsFromCompositionRules(final List<CompositionProo
         }
         String res = r.toString();
         if (usingUnprovenFacts) {
-            final String star = "*";
-            final String separatorStart = StringUtils.addSpace(star);
-            final String separatorEnd = StringUtils.prefixSpace(star);
-            final String separatorLine =
-                    StringUtils.parenthesize(
-                            StringUtils.repeat(22, StringUtils.addSpace(star)) + star);
             res += System.lineSeparator() + System.lineSeparator()
-                    + separatorLine + System.lineSeparator()
-                    + StringUtils.parenthesize(separatorStart
-                            + "Some proofs appear to rely on facts not yet" + separatorEnd)
-                     + System.lineSeparator()
-                    + StringUtils.parenthesize(separatorStart
-                            + " proven within Isabelle/HOL. Check Isabelle" + separatorEnd)
-                     + System.lineSeparator()
-                    + StringUtils.parenthesize(separatorStart
-                            + "    error messages for more information.   " + separatorEnd)
-                     + System.lineSeparator() + separatorLine;
+                    + StringUtils.isabelleCommentBlock(
+                            "Some proofs appear to rely on facts not yet",
+                            " proven within Isabelle/HOL. Check Isabelle",
+                            "    error messages for more information.   ");
         }
         return res;
     }
@@ -262,48 +252,51 @@ private static String generateModuleParameterTypes(final List<String> modulePara
         for (final String type: moduleParamTypesList) {
             moduleParamTypes.append(type);
         }
-        final String moduleParameterTypeString = moduleParamTypes.toString();
-        return moduleParameterTypeString;
+        return moduleParamTypes.toString();
     }
 
     private static Pair<String, String> buildModuleDef(final Map<String, Set<String>> moduleDefMap,
                                                        final String importsParam,
                                                        final String sessionName) {
-        String moduleDef = StringUtils.EMPTY;
-        final StringBuilder imports = new StringBuilder(importsParam);
-        // There will be only one iteration, but it is a bit tedious to extract
-        // single elements from sets.
-        for (final Map.Entry<String, Set<String>> entry: moduleDefMap.entrySet()) {
-            moduleDef = entry.getKey();
-            // Additional imports might have occurred, which are only relevant
-            // for the definition of the module, but not used within the proofs.
-            // Add those.
-            for (final String importString: entry.getValue()) {
-                final String importStringWithoutSuffix =
-                        importString.replace(IsabelleUtils.DOT_THY, StringUtils.EMPTY);
-                if (!imports.toString().contains(importStringWithoutSuffix)) {
-                    final String importWithSuffix =
-                            sessionName + StringUtils.PERIOD + importStringWithoutSuffix;
-                    imports.append(StringUtils.surroundWithSpaces(importWithSuffix));
-                }
+        final Set<String> imports = new LinkedHashSet<String>();
+        imports.add(importsParam);
+        // There exists only one element, otherwise there is an exception thrown beforehand.
+        final String moduleDef = new ArrayList<String>(moduleDefMap.keySet()).get(0);
+        // imports.add(sessionName + StringUtils.PERIOD + moduleDef);
+        // Additional imports might have occurred, which are only relevant
+        // for the definition of the module, but not used within the proofs.
+        // Add those.
+        for (final String importString: new ArrayList<Set<String>>(moduleDefMap.values()).get(0)) {
+            final String importStringWithoutSuffix =
+                    importString.replace(IsabelleUtils.DOT_THY, StringUtils.EMPTY);
+            if (!imports.contains(importStringWithoutSuffix)) {
+                final String importWithSuffix =
+                        sessionName + StringUtils.PERIOD + importStringWithoutSuffix;
+                imports.add(importWithSuffix);
             }
         }
-        return new Pair<String, String>(moduleDef, imports.toString());
+        // Build string from chain of imports
+        final StringBuilder importBuild = new StringBuilder(importsParam);
+        for (final String entry: imports) {
+            importBuild.append(StringUtils.surroundWithSpaces(entry));
+        }
+        final String importString =
+                importBuild.toString().isBlank() ? IsabelleUtils.MAIN : importBuild.toString();
+        return new Pair<String, String>(moduleDef, importString);
     }
 
     private static String replaceVariables(final String theoryName, final String imports,
                                            final String moduleParamTypes, final String moduleName,
                                            final String moduleParameters, final String moduleDef,
                                            final String proofs) {
-        String res = IsabelleTheoryGenerator.theoryTemplate;
-        res = res.replace(VAR_PROOFS, proofs);
-        res = res.replace(VAR_THEORY_NAME, theoryName);
-        res = res.replace(VAR_IMPORTS, imports);
-        res = res.replace(VAR_MODULE_PARAM_TYPES, moduleParamTypes);
-        res = res.replace(VAR_MODULE_DEF, moduleDef);
-        res = res.replace(VAR_MODULE_NAME, moduleName);
-        res = res.replace(VAR_MODULE_PARAMETERS, moduleParameters);
-        return res;
+        return IsabelleTheoryGenerator.theoryTemplate
+                .replace(VAR_PROOFS, proofs)
+                .replace(VAR_THEORY_NAME, theoryName)
+                .replace(VAR_IMPORTS, imports)
+                .replace(VAR_MODULE_PARAM_TYPES, moduleParamTypes)
+                .replace(VAR_MODULE_DEF, moduleDef)
+                .replace(VAR_MODULE_NAME, moduleName)
+                .replace(VAR_MODULE_PARAMETERS, moduleParameters);
     }
 
     /**
@@ -351,21 +344,23 @@ public File generateTheoryFile(final String passedComposition,
         final Map<String, Set<String>> moduleDefMap =
                 IsabelleUtils.translatePrologToIsabelle(this.functionsAndDefinitions,
                                                         proofPredicate);
-        if (moduleDefMap.keySet().size() != 1) {
+        if (moduleDefMap.size() != 1) {
             throw new IllegalArgumentException();
         }
         final Pair<String, String> moduleDefResult =
                 buildModuleDef(moduleDefMap, imports, this.framework.getSessionName());
+        if (moduleDefResult.getSecondValue().isBlank()) {
+            throw new IllegalArgumentException();
+        }
         final StringBuilder proofsString = new StringBuilder();
         for (final CompositionProof proof: proofs) {
             proofsString.append(this.generator.generateIsabelleProof(proof)
                                 + System.lineSeparator() + System.lineSeparator());
         }
-        final String prfString = proofsString.toString();
         final String fileContents =
                 replaceVariables(theoryName, moduleDefResult.getSecondValue(),
                                  moduleParameterTypeString, moduleName, moduleParameterString,
-                                 moduleDefResult.getFirstValue(), prfString);
+                                 moduleDefResult.getFirstValue(), proofsString.toString());
         final SimpleFileWriter writer = new SimpleFileWriter();
         final String outputPath = buildOutputPath(passedOutputPath, theoryName);
         writer.writeToFile(outputPath, fileContents);
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryParser.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryParser.java
index f3b48b9..2d4e0e6 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryParser.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleTheoryParser.java
@@ -112,7 +112,7 @@ public static List<String> getImportsFromTheoryFile(final File theory) throws IO
             if (line.contains(IsabelleUtils.IMPORTS)) {
                 final String[] splits = line.split(StringUtils.SPACE);
                 for (final String split: splits) {
-                    if (split.isBlank() || split.equals(IsabelleUtils.IMPORTS)) {
+                    if (split.isBlank() || IsabelleUtils.IMPORTS.equals(split)) {
                         continue;
                     }
                     res.add(split);
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleUtils.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleUtils.java
index 52ade5f..78c94cc 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleUtils.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/IsabelleUtils.java
@@ -39,7 +39,7 @@ public final class IsabelleUtils {
     /**
      * String used by Isabelle to denote successful termination.
      */
-    public static final String SUCCESS_STRING = "OK ";
+    public static final String SUCCESS_STRING = "ok ";
 
     /**
      * String used by Isabelle to denote the line stating imported theories.
@@ -81,6 +81,36 @@ public final class IsabelleUtils {
      */
     public static final String THEORY_NAME_SEPARATOR = StringUtils.PERIOD;
 
+    /**
+     * Suffix for definition-unfolding rules.
+     */
+    public static final String DEFINITION_SUFFIX = "_def";
+
+    /**
+     * Suffix for relation rules.
+     */
+    public static final String RELATION_SUFFIX = "_rel";
+
+    /**
+     * Suffix for raw rules.
+     */
+    public static final String RAW_SUFFIX = "_raw";
+
+    /**
+     * Suffix for graph rules.
+     */
+    public static final String GRAPH_SUFFIX = "_graph";
+
+    /**
+     * Suffix for sum rules.
+     */
+    public static final String SUM_SUFFIX = "_sumC";
+
+    /**
+     * Main theory.
+     */
+    public static final String MAIN = "Main";
+
     /**
      * Higher-order logic theory.
      */
@@ -91,6 +121,16 @@ public final class IsabelleUtils {
      */
     public static final String SET = "Set";
 
+    /**
+     * Isabelle's enumerable theory.
+     */
+    public static final String ENUM = "Enum";
+
+    /**
+     * Isabelle's set function.
+     */
+    public static final String SET_FUNC = "set";
+
     /**
      * String to represent the HOL type for natural numbers.
      */
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java
index 5fb4579..eb5899e 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/isabelle/ScalaIsabelleFacade.java
@@ -18,6 +18,8 @@
 import org.apache.logging.log4j.LogManager;
 import org.apache.logging.log4j.Logger;
 
+import com.google.common.collect.Maps;
+
 import de.unruh.isabelle.control.Isabelle;
 import de.unruh.isabelle.control.Isabelle.Setup;
 import de.unruh.isabelle.control.IsabelleBuildException;
@@ -155,7 +157,7 @@ public final class ScalaIsabelleFacade {
      * The theory-name function.
      */
     private static final String THY_NAME =
-            "Context" + IsabelleUtils.THEORY_NAME_SEPARATOR + "theory_name";
+            "Context" + IsabelleUtils.THEORY_NAME_SEPARATOR + "theory_long_name";
 
     /**
      * The specifications-of function.
@@ -342,13 +344,12 @@ private static Setup setup(final String isabelleHome, final boolean verbose,
                                final String sessionDir,
                                final Iterable<String> sessionDirs) {
         Setup s = JIsabelle.setup(Path.of(isabelleHome));
-        s = JIsabelle.setupSetVerbose(true, s);
+        s = JIsabelle.setupSetVerbose(verbose, s);
         s = JIsabelle.setupSetUserDir(Path.of(isabelleDirectory), s);
         s = JIsabelle.setupSetWorkingDirectory(Path.of(sessionDir), s);
         s = JIsabelle.setupSetSessionRoots(toPaths(sessionDirs), s);
-        s = JIsabelleWrapper.setupSetLogic(sessionName, s);
-        s = JIsabelle.setupSetBuild(true, s);
-        return s;
+        s = JIsabelle.setupSetLogic(sessionName, s);
+        return JIsabelle.setupSetBuild(true, s);
     }
 
     private static synchronized void setIsabelleInstance(final Isabelle instance) {
@@ -458,7 +459,7 @@ public static void buildSession(final String sessionDir, final String sessionNam
      * Extracts functions and definitions from a given framework in Isabelle.
      */
     public void extractFunctionsAndDefinitions() {
-        this.functionsAndDefinitions = new HashMap<String, Map<String, String>>();
+        this.functionsAndDefinitions = Maps.newHashMapWithExpectedSize(this.theoryNames.size());
         final MLFunction<Theory, scala.collection.immutable.List<String>> mlFunToExtractAllNames =
                 MLValue.compileFunction(funToExtractAllNames(THY), ScalaIsabelleFacade.isabelle,
                                         Implicits.theoryConverter(), LIST_CONVERTER);
@@ -468,8 +469,6 @@ public void extractFunctionsAndDefinitions() {
                                     Implicits.theoryConverter(), PAIR_TO_LIST_CONVERTER);
         for (final String thy: this.theoryNames) {
             final String thyWithoutSession = thy.split("\\.")[1];
-            final Map<String, String> toBeFilled = new HashMap<String, String>();
-            this.functionsAndDefinitions.put(thy, toBeFilled);
             final Context ctxt = Context.apply(thy, ScalaIsabelleFacade.isabelle);
             final Theory theory = ctxt.theoryOf(ScalaIsabelleFacade.isabelle);
             final List<String> names =
@@ -480,41 +479,42 @@ public void extractFunctionsAndDefinitions() {
                     JavaConverters.asJava(mlFunToExtractSigns
                             .apply(theory.mlValue(), ScalaIsabelleFacade.isabelle)
                             .retrieveNow(PAIR_TO_LIST_CONVERTER, ScalaIsabelleFacade.isabelle));
-            for (int i = 0; i < names.size(); i++) {
-                String name = names.get(i);
+            final Map<String, String> toBeFilled = Maps.newHashMapWithExpectedSize(names.size());
+            this.functionsAndDefinitions.put(thy, toBeFilled);
+            for (final String name: names) {
                 if (!name.startsWith(thyWithoutSession + IsabelleUtils.THEORY_NAME_SEPARATOR)) {
                     continue;
                 }
-                final String defSfx = "_def";
-                final String relSfx = "_rel";
-                final String rawSfx = "_raw";
-                final String graphSfx = "_graph";
-                final String sumSfx = "_sumC";
-                final String defRawSfx = defSfx + rawSfx;
-                if (name.endsWith(defSfx) || name.endsWith(defRawSfx)) {
-                    name = name.replace(rawSfx, StringUtils.EMPTY);
-                    name = name.replace(defSfx, StringUtils.EMPTY);
-                    if (name.endsWith(relSfx) || name.endsWith(graphSfx) || name.endsWith(sumSfx)) {
+                if (name.endsWith(IsabelleUtils.DEFINITION_SUFFIX)
+                        || name.endsWith(IsabelleUtils.DEFINITION_SUFFIX
+                                            + IsabelleUtils.RAW_SUFFIX)) {
+                    final String def =
+                            name.replace(IsabelleUtils.RAW_SUFFIX, StringUtils.EMPTY)
+                                .replace(IsabelleUtils.DEFINITION_SUFFIX, StringUtils.EMPTY);
+                    if (def.endsWith(IsabelleUtils.RELATION_SUFFIX)
+                            || def.endsWith(IsabelleUtils.GRAPH_SUFFIX)
+                            || def.endsWith(IsabelleUtils.SUM_SUFFIX)) {
                         continue;
                     }
                     String type = StringUtils.EMPTY;
                     for (final Tuple2<String, String> tup: types) {
-                        if (tup._1.equals(name)) {
+                        if (tup._1.equals(def)) {
                             type = tup._2;
+                            break;
                         }
                     }
                     type = type.replace(
                             IsabelleUtils.SET + IsabelleUtils.THEORY_NAME_SEPARATOR
                             + StringUtils.func(SET_FUN, ALTERNATIVE),
                             ALTERNATIVES_NAME);
-                    toBeFilled.put(name, type);
+                    toBeFilled.put(def, type);
                 }
             }
         }
     }
 
     private void extractTheorems() {
-        this.theorems = new HashMap<String, Map<String, String>>();
+        this.theorems = Maps.newHashMapWithExpectedSize(this.theoryNames.size());
         final MLFunction<Theory, scala.collection.immutable.List<String>> mlFun =
                 MLValue.compileFunction(
                         addSpace(fnTo(THY,
@@ -531,13 +531,13 @@ private void extractTheorems() {
                         ScalaIsabelleFacade.isabelle,
                         Implicits.thmConverter(), STRING_CONVERTER);
         for (final String thyName: this.theoryNames) {
-            final Map<String, String> toBeFilled = new HashMap<String, String>();
-            this.theorems.put(thyName, toBeFilled);
             final Context ctxt = Context.apply(thyName, ScalaIsabelleFacade.isabelle);
             final Theory theory = ctxt.theoryOf(ScalaIsabelleFacade.isabelle);
             final scala.collection.immutable.List<String> defs =
                     mlFun.apply(theory, ScalaIsabelleFacade.isabelle, Implicits.theoryConverter())
                     .retrieveNow(LIST_CONVERTER, ScalaIsabelleFacade.isabelle);
+            final Map<String, String> toBeFilled = new HashMap<String, String>(defs.length());
+            this.theorems.put(thyName, toBeFilled);
             for (int i = 0; i < defs.length(); i++) {
                 final String thmName = defs.apply(i);
                 // This filters out all theorems generated by Isabelle for internal usage.
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeAllPropertiesJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeAllPropertiesJob.java
index 377d70a..4d948e6 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeAllPropertiesJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeAllPropertiesJob.java
@@ -55,7 +55,7 @@ public final String getDescription() {
     public final String presentConcreteResult() {
         final List<Boolean> hasProperties = new LinkedList<Boolean>();
         for (int i = 0; i < this.unaryProperties.size(); i++) {
-            hasProperties.add(false);
+            hasProperties.add(Boolean.FALSE);
         }
         for (final List<SearchResult<BooleanWithUncertainty>> resultList: this.getResult()) {
             for (int i = 0; i < resultList.size(); i++) {
@@ -80,9 +80,9 @@ public final String presentConcreteResult() {
     protected final void concreteExecute() {
         this.unaryProperties = new LinkedList<Property>();
         final VirageCore execCore = this.getExecutingCore();
-        for (final Property candidate: execCore.getFrameworkRepresentation().getProperties()) {
-            if (!candidate.isAtomic() && candidate.getArity() == 1) {
-                this.unaryProperties.add(candidate);
+        for (final Property property: execCore.getFrameworkRepresentation().getProperties()) {
+            if (!property.isAtomic() && property.getArity() == 1) {
+                this.unaryProperties.add(property);
             }
         }
         Collections.sort(this.unaryProperties,
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeJob.java
index fbffd5d..6be49dd 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageAnalyzeJob.java
@@ -35,16 +35,6 @@ public final class VirageAnalyzeJob
      */
     private final DecompositionTree tree;
 
-    /**
-     * The compositional framework.
-     */
-    private FrameworkRepresentation framework;
-
-    /**
-     * The search manager.
-     */
-    private VirageSearchManager manager;
-
     /**
      * Simple constructor.
      *
@@ -61,13 +51,14 @@ public VirageAnalyzeJob(final VirageUserInterface issuer, final String treeValue
 
     @Override
     public void concreteExecute() {
-        this.framework = this.getExecutingCore().getFrameworkRepresentation();
-        this.manager = this.getExecutingCore().getSearchManager();
+        final FrameworkRepresentation framework =
+                this.getExecutingCore().getFrameworkRepresentation();
+        final VirageSearchManager manager = this.getExecutingCore().getSearchManager();
         this.properties = new LinkedList<Property>();
         for (final String s: this.propertyStrings) {
-            this.properties.add(this.framework.getProperty(s));
+            this.properties.add(framework.getProperty(s));
         }
-        this.setResult(this.manager.analyzeComposition(this.tree, this.properties));
+        this.setResult(manager.analyzeComposition(this.tree, this.properties));
     }
 
     @Override
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateCCodeJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateCCodeJob.java
index d218d95..61fd06b 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateCCodeJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateCCodeJob.java
@@ -11,11 +11,6 @@
  * @author VeriVote
  */
 public final class VirageGenerateCCodeJob extends VirageJobWithExplicitResult<File> {
-    /**
-     * The C code generator.
-     */
-    private CCodeGenerator generator;
-
     /**
      * The generation from which code shall be generated.
      */
@@ -34,8 +29,8 @@ public VirageGenerateCCodeJob(final VirageUserInterface issuer, final String com
 
     @Override
     protected void concreteExecute() {
-        this.generator = this.getExecutingCore().getCCodeGenerator();
-        this.setResult(this.generator.getCCodeFromComposition(this.composition));
+        final CCodeGenerator generator = this.getExecutingCore().getCCodeGenerator();
+        this.setResult(generator.getCCodeFromComposition(this.composition));
     }
 
     @Override
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateJob.java
index ba96d9d..2235aec 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageGenerateJob.java
@@ -29,11 +29,6 @@ public final class VirageGenerateJob
      */
     private List<Property> properties;
 
-    /**
-     * The search manager to be used.
-     */
-    private VirageSearchManager manager;
-
     /**
      * Simple constructor.
      *
@@ -48,12 +43,12 @@ public VirageGenerateJob(final VirageUserInterface issuer, final List<String> pr
     @Override
     public void concreteExecute() {
         final VirageCore execCore = this.getExecutingCore();
-        this.manager = execCore.getSearchManager();
+        final VirageSearchManager manager = execCore.getSearchManager();
         this.properties = new LinkedList<Property>();
         for (final String s: this.propertyStrings) {
             this.properties.add(execCore.getFrameworkRepresentation().getProperty(s));
         }
-        this.setResult(this.manager.generateComposition(this.properties));
+        this.setResult(manager.generateComposition(this.properties));
     }
 
     @Override
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateJob.java
index 3169411..9d795f9 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateJob.java
@@ -26,11 +26,6 @@ public final class VirageIsabelleGenerateJob extends VirageJobWithExplicitResult
      */
     private final String outputPath;
 
-    /**
-     * The Isabelle theory generator to be used.
-     */
-    private IsabelleTheoryGenerator generator;
-
     /**
      * Simple constructor.
      *
@@ -65,8 +60,9 @@ public String presentConcreteResult() {
 
     @Override
     protected void concreteExecute() {
-        this.generator = this.getExecutingCore().getIsabelleTheoryGenerator();
-        this.setResult(this.generator
+        final IsabelleTheoryGenerator generator =
+                this.getExecutingCore().getIsabelleTheoryGenerator();
+        this.setResult(generator
                 .generateTheoryFile(this.composition, this.proofs, this.outputPath));
     }
 }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateScalaJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateScalaJob.java
index b9e72e7..c777bd4 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateScalaJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleGenerateScalaJob.java
@@ -13,11 +13,6 @@
  * @author VeriVote
  */
 public final class VirageIsabelleGenerateScalaJob extends VirageJobWithExplicitResult<File> {
-    /**
-     * The Isabelle Scala generator.
-     */
-    private IsabelleCodeGenerator generator;
-
     /**
      * The composition from which an implementation shall be generated.
      */
@@ -30,9 +25,8 @@ public final class VirageIsabelleGenerateScalaJob extends VirageJobWithExplicitR
      * @param compositionValue the composition
      */
     public VirageIsabelleGenerateScalaJob(final VirageUserInterface issuer,
-            final String compositionValue) {
+                                          final String compositionValue) {
         super(issuer);
-
         this.composition = compositionValue;
     }
 
@@ -53,7 +47,7 @@ public String presentConcreteResult() {
 
     @Override
     protected void concreteExecute() throws CodeGenerationFailedException {
-        this.generator = this.getExecutingCore().getIsabelleCodeGenerator();
-        this.setResult(this.generator.generateScalaCodeAndCompile(this.composition));
+        final IsabelleCodeGenerator generator = this.getExecutingCore().getIsabelleCodeGenerator();
+        this.setResult(generator.generateScalaCodeAndCompile(this.composition));
     }
 }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleVerifyJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleVerifyJob.java
index 3caef28..c7c8254 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleVerifyJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageIsabelleVerifyJob.java
@@ -5,8 +5,8 @@
 
 import edu.kit.kastel.formal.util.Pair;
 import edu.kit.kastel.formal.virage.core.ConfigReader;
+import edu.kit.kastel.formal.virage.core.VirageCore;
 import edu.kit.kastel.formal.virage.core.VirageUserInterface;
-import edu.kit.kastel.formal.virage.isabelle.IsabelleProofChecker;
 
 /**
  * A {@link VirageJob} that invokes Isabelle to automatically attempt proof verification.
@@ -15,11 +15,6 @@
  */
 public final class VirageIsabelleVerifyJob
         extends VirageJobWithExplicitResult<Pair<Boolean, File>> {
-    /**
-     * The checker used to verify the proof.
-     */
-    private IsabelleProofChecker checker;
-
     /**
      * The theory file to be checked.
      */
@@ -59,10 +54,8 @@ public String presentConcreteResult() {
 
     @Override
     protected void concreteExecute() throws IOException, InterruptedException {
-        this.checker = this.getExecutingCore().getIsabelleProofChecker();
-
-        this.setResult(this.checker.verifyTheoryFile(this.file,
-                this.getExecutingCore().getFrameworkRepresentation()));
+        final VirageCore core = this.getExecutingCore();
+        this.setResult(core.getIsabelleProofChecker()
+                        .verifyTheoryFile(this.file, core.getFrameworkRepresentation()));
     }
-
 }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageJob.java
index bf011ae..08f6f02 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageJob.java
@@ -155,8 +155,7 @@ public String presentResult() {
         } else {
             res += StringUtils.sentence("Something went wrong while executing this job");
         }
-        res += StringUtils.repeat(StringUtils.TEN, StringUtils.DASH);
-        return res;
+        return res + StringUtils.repeat(StringUtils.TEN, StringUtils.DASH);
     }
 
     /**
@@ -198,8 +197,7 @@ public String toString() {
         res += printPropertyLine(finishedString, SystemUtils.getTime(lastMeasurement));
         res += printPropertyLine("Elapsed time:", elapsedTime);
         res += printPropertyLine(StringUtils.repeat(StringUtils.FIVE, StringUtils.DASH));
-        res += printPropertyLine("State:", this.state.toString());
-        return res;
+        return res + printPropertyLine("State:", this.state.toString());
     }
 
     /**
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageProveJob.java b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageProveJob.java
index a156ac9..0632b7b 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageProveJob.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/jobs/VirageProveJob.java
@@ -33,11 +33,6 @@ public final class VirageProveJob
      */
     private final DecompositionTree tree;
 
-    /**
-     * The framework representation.
-     */
-    private FrameworkRepresentation framework;
-
     /**
      * Simple constructor.
      *
@@ -54,10 +49,11 @@ public VirageProveJob(final VirageUserInterface issuer, final String treeValue,
 
     @Override
     public void concreteExecute() {
-        this.framework = this.getExecutingCore().getFrameworkRepresentation();
+        final FrameworkRepresentation framework =
+                this.getExecutingCore().getFrameworkRepresentation();
         this.properties = new LinkedList<Property>();
         for (final String s: this.propertyStrings) {
-            this.properties.add(this.framework.getProperty(s));
+            this.properties.add(framework.getProperty(s));
         }
         this.setResult(this.getExecutingCore().getSearchManager().proveClaims(this.tree,
                                                                               this.properties));
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/JplFacade.java b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/JplFacade.java
index 1b200d7..75adaa1 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/JplFacade.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/JplFacade.java
@@ -20,6 +20,8 @@
 import org.jpl7.Term;
 import org.jpl7.Variable;
 
+import com.google.common.collect.Maps;
+
 import edu.kit.kastel.formal.util.StringUtils;
 import edu.kit.kastel.formal.util.SystemUtils;
 import edu.kit.kastel.formal.virage.core.ConfigReader;
@@ -163,19 +165,22 @@ public static String findUnusedVariable(final String queryString) {
      */
     public static Map<String, String> parseReplacementMap(final String variable,
                                                           final Map<String, String> map) {
-        final Map<String, String> result = new HashMap<String, String>();
-        // Looks like this: ['='(_108, 1)]
-        // _NUMBER is an alias for a variable.
         String replacementString = map.get(variable);
         replacementString = StringUtils.removeWhitespace(replacementString);
         replacementString = replacementString.substring(1, replacementString.length() - 1);
+
+        final String[] replacements = replacementString.isEmpty()
+                ? new String[] {} : replacementString.split("\\'=\\'");
+        // replacements[0] is always empty, as the String starts with the splitting
+        // regular expression.
+
+        final Map<String, String> result = new HashMap<String, String>(replacements.length);
+        // Looks like this: ['='(_108, 1)]
+        // _NUMBER is an alias for a variable.
         if (replacementString.isEmpty()) {
             // List of replacements is empty.
             return result;
         }
-        final String[] replacements = replacementString.split("\\'=\\'");
-        // replacements[0] is always empty, as the String starts with the splitting
-        // regular expression.
         for (int i = 1; i < replacements.length; i++) {
             String replacement = replacements[i];
             if (replacement.endsWith(StringUtils.COMMA)) {
@@ -316,18 +321,18 @@ public SearchResult<Boolean> factQuery(final String queryString, final long time
             SearchResult<Boolean> toReturn = null;
             if (result == null) {
                 // No solution, query failed.
-                toReturn = new SearchResult<Boolean>(QueryState.FAILED, false);
+                toReturn = new SearchResult<Boolean>(QueryState.FAILED, Boolean.FALSE);
             } else if (!result.containsKey(unusedVariable)) {
                 // Empty Map was received, timeout.
                 toReturn = new SearchResult<Boolean>(QueryState.TIMEOUT, null);
-            } else if (result.get(unusedVariable).equals(DEPTH_LIMIT_EXCEEDED)) {
+            } else if (DEPTH_LIMIT_EXCEEDED.equals(result.get(unusedVariable))) {
                 // Depth limit exceeded, increase and try again.
                 maxDepth++;
                 continue;
             } else if (StringUtils.isNumeric(result.get(unusedVariable))) {
                 // Query succeeded.
                 result.remove(unusedVariable);
-                toReturn = new SearchResult<Boolean>(QueryState.SUCCESS, true);
+                toReturn = new SearchResult<Boolean>(QueryState.SUCCESS, Boolean.TRUE);
             }
             if (toReturn != null) {
                 return toReturn;
@@ -387,7 +392,7 @@ public SearchResult<Map<String, String>> iterativeDeepeningQuery(final String qu
                 // Empty Map was received, timeout.
                 toReturn = new SearchResult<Map<String, String>>(QueryState.TIMEOUT, null);
             } else {
-                if (result.get(unusedVariable).equals(DEPTH_LIMIT_EXCEEDED)) {
+                if (DEPTH_LIMIT_EXCEEDED.equals(result.get(unusedVariable))) {
                     // Depth limit exceeded, increase and try again.
                     maxDepth++;
                     continue;
@@ -452,7 +457,8 @@ public Map<String, String> simpleQueryWithTimeout(final String queryString,
             if (query.hasMoreSolutions()) {
                 try {
                     final Map<String, Term> solution = query.nextSolution();
-                    final Map<String, String> result = new HashMap<String, String>();
+                    final Map<String, String> result =
+                            Maps.newHashMapWithExpectedSize(solution.size());
                     for (final Map.Entry<String, Term> entry: solution.entrySet()) {
                         final String termString = entry.getValue().toString();
                         result.put(entry.getKey(), termString);
@@ -473,7 +479,7 @@ public Map<String, String> simpleQueryWithTimeout(final String queryString,
                 return null;
             }
         } catch (final PrologException e) {
-            if (!e.getMessage().equals("PrologException: time_limit_exceeded")) {
+            if (!"PrologException: time_limit_exceeded".equals(e.getMessage())) {
                 LOGGER.error("A Prolog error occured.");
                 throw e;
             }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologClause.java b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologClause.java
index ff5d891..b5dd558 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologClause.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologClause.java
@@ -143,8 +143,7 @@ public int hashCode() {
         final int prime = 31;
         int result = 1;
         result = prime * result + ((this.antecedents == null) ? 0 : this.antecedents.hashCode());
-        result = prime * result + ((this.succedent == null) ? 0 : this.succedent.hashCode());
-        return result;
+        return prime * result + ((this.succedent == null) ? 0 : this.succedent.hashCode());
     }
 
     /**
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologPredicate.java b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologPredicate.java
index ef368ee..3b899f9 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologPredicate.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/PrologPredicate.java
@@ -116,15 +116,6 @@ public List<PrologPredicate> getParameters() {
         return this.parameters;
     }
 
-    /**
-     * Checks whether a PrologPredicate is a variable.
-     *
-     * @return true if this is a variable, false otherwise
-     */
-    public boolean isVariable() {
-        return PrologPredicate.isVariable(this.name);
-    }
-
     /**
      * Checks whether a string represents a Prolog variable.
      *
@@ -135,6 +126,15 @@ public static boolean isVariable(final String s) {
         return s.matches("[A-Z_][a-zA-Z_0-9]*");
     }
 
+    /**
+     * Checks whether a PrologPredicate is a variable.
+     *
+     * @return true if this is a variable, false otherwise
+     */
+    public boolean isVariable() {
+        return isVariable(this.name);
+    }
+
     /**
      * Sets a new name for the Prolog predicate.
      *
@@ -164,8 +164,9 @@ public static PrologPredicate copy(final PrologPredicate pred) {
      * @param replacements the replacements
      */
     public void replaceVariables(final Map<String, String> replacements) {
-        if (replacements.keySet().contains(this.name)) {
-            this.name = replacements.get(this.name);
+        final String s = replacements.get(this.name);
+        if (s != null) {
+            this.name = s;
         } else {
             for (final PrologPredicate parameter: this.parameters) {
                 parameter.replaceVariables(replacements);
@@ -179,8 +180,7 @@ public int hashCode() {
         int result = 1;
         result = prime * result + this.arity;
         result = prime * result + ((this.name == null) ? 0 : this.name.hashCode());
-        result = prime * result + ((this.parameters == null) ? 0 : this.parameters.hashCode());
-        return result;
+        return prime * result + ((this.parameters == null) ? 0 : this.parameters.hashCode());
     }
 
     @Override
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/SimpleExtendedPrologParser.java b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/SimpleExtendedPrologParser.java
index 2ca86e9..b399f31 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/SimpleExtendedPrologParser.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/prolog/SimpleExtendedPrologParser.java
@@ -91,8 +91,7 @@ private FrameworkRepresentation parseFramework(final List<String> representation
         final List<String> propertySection = new LinkedList<String>();
         final List<String> compositionRuleSection = new LinkedList<String>();
 
-        for (int lineNumber = 0; lineNumber < representation.size(); lineNumber++) {
-            String currentLine = representation.get(lineNumber);
+        for (final String currentLine: representation) {
             // Skip comments
             if (currentLine.startsWith("%%")) {
                 continue;
@@ -101,29 +100,29 @@ private FrameworkRepresentation parseFramework(final List<String> representation
                 this.findTheoryPath(currentLine, framework);
             }
             // Remove Prolog comment markings, they are not necessary any more.
-            currentLine = currentLine.replace("% ", StringUtils.EMPTY);
-            currentLine = currentLine.replace("%", StringUtils.EMPTY);
+            final String line =
+                    currentLine.replace("% ", StringUtils.EMPTY).replace("%", StringUtils.EMPTY);
             // Skip empty lines. (Careful: currentLine is not actually sanitized after
             // this!)
-            if (this.sanitizeLine(currentLine).isEmpty()) {
+            if (this.sanitizeLine(line).isEmpty()) {
                 continue;
             }
-            state = this.newState(currentLine, state);
+            state = this.newState(line, state);
             switch (state) {
             case COMPOSITION_TYPE:
-                compositionTypeSection.add(currentLine);
+                compositionTypeSection.add(line);
                 break;
             case COMPOSABLE_MODULE:
-                composableModuleSection.add(currentLine);
+                composableModuleSection.add(line);
                 break;
             case COMPOSITIONAL_STRUCTURE:
-                compositionalStructureSection.add(currentLine);
+                compositionalStructureSection.add(line);
                 break;
             case PROPERTY:
-                propertySection.add(currentLine);
+                propertySection.add(line);
                 break;
             case COMPOSITION_RULE:
-                compositionRuleSection.add(currentLine);
+                compositionRuleSection.add(line);
                 break;
             default: // skip operation, invalid call.
             }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/types/ComponentType.java b/virage/src/main/java/edu/kit/kastel/formal/virage/types/ComponentType.java
index 0c626f2..1b8192d 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/types/ComponentType.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/types/ComponentType.java
@@ -60,9 +60,8 @@ public String getName() {
     @Override
     public int hashCode() {
         final int prime = 31;
-        int result = 1;
-        result = prime * result + ((this.name == null) ? 0 : this.name.hashCode());
-        return result;
+        final int result = 1;
+        return prime * result + ((this.name == null) ? 0 : this.name.hashCode());
     }
 
     /**
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionProof.java b/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionProof.java
index 9b24ba4..63a6719 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionProof.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionProof.java
@@ -5,6 +5,8 @@
 import java.util.List;
 import java.util.Set;
 
+import com.google.common.collect.Sets;
+
 import edu.kit.kastel.formal.util.StringUtils;
 import edu.kit.kastel.formal.virage.isabelle.IsabelleUtils;
 
@@ -36,29 +38,27 @@ public final class CompositionProof {
     private String id = StringUtils.EMPTY;
 
     /**
-     * Simple constructor for a terminal proof step.
+     * Simple constructor.
      *
      * @param goalValue the goal
+     * @param subgoalsValue the subgoals
      * @param ruleValue the rule
      */
-    public CompositionProof(final String goalValue, final CompositionRule ruleValue) {
+    public CompositionProof(final String goalValue, final List<CompositionProof> subgoalsValue,
+                            final CompositionRule ruleValue) {
         this.goal = goalValue;
-        this.subgoals = new LinkedList<CompositionProof>();
+        this.subgoals = subgoalsValue;
         this.rule = ruleValue;
     }
 
     /**
-     * Simple constructor.
+     * Simple constructor for a terminal proof step.
      *
      * @param goalValue the goal
-     * @param subgoalsValue the subgoals
      * @param ruleValue the rule
      */
-    public CompositionProof(final String goalValue, final List<CompositionProof> subgoalsValue,
-                            final CompositionRule ruleValue) {
-        this.goal = goalValue;
-        this.subgoals = subgoalsValue;
-        this.rule = ruleValue;
+    public CompositionProof(final String goalValue, final CompositionRule ruleValue) {
+        this(goalValue, new LinkedList<CompositionProof>(), ruleValue);
     }
 
     /**
@@ -81,8 +81,8 @@ public Set<CompositionRule> getAllCompositionRules() {
      * @return all origins
      */
     public Set<String> getAllOrigins() {
-        final Set<String> origins = new HashSet<String>();
         final Set<CompositionRule> allRules = this.getAllCompositionRules();
+        final Set<String> origins = Sets.newHashSetWithExpectedSize(allRules.size());
         for (final CompositionRule localRule: allRules) {
             origins.add(localRule.getOrigin());
         }
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionRule.java b/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionRule.java
index cb2d037..7cc73c3 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionRule.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/types/CompositionRule.java
@@ -149,8 +149,7 @@ public int hashCode() {
         int result = 1;
         result = prime * result + ((this.clause == null) ? 0 : this.clause.hashCode());
         result = prime * result + ((this.name == null) ? 0 : this.name.hashCode());
-        result = prime * result + ((this.origin == null) ? 0 : this.origin.hashCode());
-        return result;
+        return prime * result + ((this.origin == null) ? 0 : this.origin.hashCode());
     }
 
     /**
@@ -162,8 +161,7 @@ public String toEplString() {
         String res = StringUtils.EMPTY;
         res += StringUtils.printCollection2("% =", this.origin) + System.lineSeparator();
         res += StringUtils.printCollection2("%", this.name) + System.lineSeparator();
-        res += this.clause.toString() + System.lineSeparator();
-        return res;
+        return res + this.clause.toString() + System.lineSeparator();
     }
 
     @Override
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/types/DecompositionTree.java b/virage/src/main/java/edu/kit/kastel/formal/virage/types/DecompositionTree.java
index 8a529a0..17b9951 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/types/DecompositionTree.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/types/DecompositionTree.java
@@ -208,8 +208,7 @@ public int hashCode() {
         int result = 1;
         result = prime * result + this.arity;
         result = prime * result + ((this.children == null) ? 0 : this.children.hashCode());
-        result = prime * result + ((this.label == null) ? 0 : this.label.hashCode());
-        return result;
+        return prime * result + ((this.label == null) ? 0 : this.label.hashCode());
     }
 
     @Override
@@ -217,9 +216,8 @@ public String toString() {
         if (this.arity == 0) {
             return this.label;
         } else {
-            String res = this.label;
-            res += StringUtils.parenthesize(StringUtils.printCollection(this.children));
-            return res;
+            return this.label
+                    + StringUtils.parenthesize(StringUtils.printCollection(this.children));
         }
     }
 
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/types/FrameworkRepresentation.java b/virage/src/main/java/edu/kit/kastel/formal/virage/types/FrameworkRepresentation.java
index 2dcdf23..c946aee 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/types/FrameworkRepresentation.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/types/FrameworkRepresentation.java
@@ -5,6 +5,7 @@
 import java.time.format.DateTimeFormatter;
 import java.util.Collections;
 import java.util.HashSet;
+import java.util.LinkedHashSet;
 import java.util.LinkedList;
 import java.util.List;
 import java.util.Map;
@@ -93,9 +94,9 @@ public final class FrameworkRepresentation {
     private List<Pair<String, String>> typeSynonyms;
 
     /**
-     * List of atomic types.
+     * Set of atomic types.
      */
-    private List<ComponentType> atomicTypes;
+    private Set<ComponentType> atomicTypes;
 
     /**
      * Alias for modules that can be composed.
@@ -263,7 +264,7 @@ private void addAliasRules() {
      */
     private void addDummyRules() {
         final List<String> atomicTypeStrings = ConfigReader.getInstance().getAtomicTypes();
-        this.atomicTypes = new LinkedList<ComponentType>();
+        this.atomicTypes = new LinkedHashSet<ComponentType>(atomicTypeStrings.size());
         for (final String s: atomicTypeStrings) {
             final ComponentType type = new ComponentType(s);
             this.add(type);
@@ -340,8 +341,7 @@ private String createHeader() {
                 + StringUtils.sentence(
                         "%% % example_rule_without_meaning" + System.lineSeparator()
                                 + "%% property_a(X) :- property_b(X)");
-        res += SEPARATOR;
-        return res;
+        return res + SEPARATOR;
     }
 
     /**
diff --git a/virage/src/main/java/edu/kit/kastel/formal/virage/types/Property.java b/virage/src/main/java/edu/kit/kastel/formal/virage/types/Property.java
index d56f696..0852934 100644
--- a/virage/src/main/java/edu/kit/kastel/formal/virage/types/Property.java
+++ b/virage/src/main/java/edu/kit/kastel/formal/virage/types/Property.java
@@ -173,8 +173,7 @@ public int hashCode() {
         final int prime = 31;
         int result = 1;
         result = prime * result + this.arity;
-        result = prime * result + ((this.name == null) ? 0 : this.name.hashCode());
-        return result;
+        return prime * result + ((this.name == null) ? 0 : this.name.hashCode());
     }
 
     @Override
diff --git a/virage/src/main/resources/c_implementations/wrapper.c b/virage/src/main/resources/c_implementations/wrapper.c
index 869fb71..5662b26 100644
--- a/virage/src/main/resources/c_implementations/wrapper.c
+++ b/virage/src/main/resources/c_implementations/wrapper.c
@@ -25,9 +25,9 @@ char** split_string(char* str) {
   return res;
 }
 
-int find_order_index(char* candidate) {
+int find_order_index(char* alternative) {
   for (int i = 0; i < C; i++) {
-    if (!strcmp(candidate, order[i])) return i;
+    if (!strcmp(alternative, order[i])) return i;
   }
 
   return C;
diff --git a/virage/src/main/resources/code_file.template b/virage/src/main/resources/code_file.template
index 7289df2..c753081 100644
--- a/virage/src/main/resources/code_file.template
+++ b/virage/src/main/resources/code_file.template
@@ -7,7 +7,7 @@
 // via an additional header or via gcc option "-D DEFAULT_X="concrete implementation"",
 // where "concrete implementation" has to be supplied by the user.
 //
-// "V" and "C": These defines specify the number of Voters and Candidates. For SBMC
+// "V" and "C": These defines specify the number of Voters and Alternatives. For SBMC
 // performance, these have to be known at compile time. Default values are 3 and 3,
 // but they have to fit your test instances. If the numbers don't match, segmentation
 // faults and undefined behaviour will occur. Recompile and explicitly set the correct
diff --git a/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/BEAST.fxml b/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/BEAST.fxml
index dd22382..96ea06c 100644
--- a/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/BEAST.fxml
+++ b/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/BEAST.fxml
@@ -251,7 +251,7 @@
                </children>
             </VBox>
             <Separator orientation="VERTICAL" />
-            <Text strokeType="OUTSIDE" strokeWidth="0.0" text="Candidates:" />
+            <Text strokeType="OUTSIDE" strokeWidth="0.0" text="Alternatives:" />
             <VBox>
                <children>
                   <HBox>
@@ -261,7 +261,7 @@
                               <Text strokeType="OUTSIDE" strokeWidth="0.0" text="min" BorderPane.alignment="CENTER" />
                            </left>
                         </BorderPane>
-              <TextField fx:id="minCandidates" alignment="CENTER_RIGHT" onAction="#minCandidates" prefWidth="50.0" text="1" />
+              <TextField fx:id="minAlternatives" alignment="CENTER_RIGHT" onAction="#minAlternatives" prefWidth="50.0" text="1" />
                      </children>
                   </HBox>
                   <HBox>
@@ -271,7 +271,7 @@
                               <Text strokeType="OUTSIDE" strokeWidth="0.0" text="max" BorderPane.alignment="CENTER" />
                            </left>
                         </BorderPane>
-              <TextField fx:id="maxCandidates" alignment="CENTER_RIGHT" onAction="#maxCandidates" prefWidth="50.0" text="1" />
+              <TextField fx:id="maxAlternatives" alignment="CENTER_RIGHT" onAction="#maxAlternatives" prefWidth="50.0" text="1" />
                      </children>
                   </HBox>
                </children>
@@ -354,7 +354,7 @@
                                                                               </MenuItem>
                                                                               <MenuItem mnemonicParsing="false" onAction="#addSymbCand">
                                                                                  <graphic>
-                                                                                    <Label text="%key.addCandidate" />
+                                                                                    <Label text="%key.addAlternative" />
                                                                                  </graphic>
                                                                               </MenuItem>
                                                                               <MenuItem mnemonicParsing="false" onAction="#addSymbSeat">
@@ -458,9 +458,9 @@
                                                       </BorderPane>
                                                    </left>
                                                    <center>
-                                                      <ScrollPane fx:id="candidateScrollPane" fitToHeight="true" hbarPolicy="NEVER" vbarPolicy="NEVER" BorderPane.alignment="CENTER">
+                                                      <ScrollPane fx:id="alternativeScrollPane" fitToHeight="true" hbarPolicy="NEVER" vbarPolicy="NEVER" BorderPane.alignment="CENTER">
                                                          <content>
-                                                            <GridPane fx:id="candidateGridPane">
+                                                            <GridPane fx:id="alternativeGridPane">
                                                                <columnConstraints>
                                                                   <ColumnConstraints hgrow="SOMETIMES" minWidth="10.0" prefWidth="50.0" />
                                                                </columnConstraints>
@@ -480,8 +480,8 @@
                                             <items>
                                                 <Label text="Voters" />
                                                 <TextField fx:id="inputVoterField" maxWidth="50.0" text="0" />
-                                                <Label text="Candidates" />
-                                                <TextField fx:id="inputCandidateField" maxWidth="50.0" text="0" />
+                                                <Label text="Alternatives" />
+                                                <TextField fx:id="inputAlternativeField" maxWidth="50.0" text="0" />
                                                 <Label text="Seats" />
                                                 <TextField fx:id="inputSeatField" maxWidth="50.0" text="0" />
                                                 <Pane HBox.hgrow="ALWAYS" />
diff --git a/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_de.properties b/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_de.properties
index e4c6ea8..b424e54 100644
--- a/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_de.properties
+++ b/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_de.properties
@@ -10,7 +10,7 @@ key.code=Code
 key.property=Eigenschaft
 key.result=Ergebnis
 key.voters=Wähler
-key.candidates=Kandidaten
+key.alternatives=Alternativen
 key.seats=Sitze
 key.file=Datei
 key.edit=Bearbeiten
@@ -45,7 +45,7 @@ key.newProp=Property hinzufügen
 key.loadProp=Property laden
 key.loadPropList=Propertyliste laden
 key.addVoter=Wählen hinzufügen
-key.addCandidate=Kandidat hinzufügen
+key.addAlternative=Alternative hinzufügen
 key.addSeat=Sitz hinzufügen
 key.deleteItems=Temporäre Dateien löschen
 key.reset=Reset
diff --git a/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_en.properties b/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_en.properties
index 50f9b38..5e56e79 100644
--- a/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_en.properties
+++ b/virage/src/main/resources/edu/kit/kastel/formal/beast/highlevel/javafx/bundles/LangBundle_en.properties
@@ -10,7 +10,7 @@ key.code=Code
 key.property=Property
 key.result=Result
 key.voters=Voters
-key.candidates=Candidates
+key.alternatives=Alternatives
 key.seats=Seats
 key.file=File
 key.edit=Edit
@@ -46,7 +46,7 @@ key.newProp=Create New Property
 key.loadProp=Load Property
 key.loadPropList=Load List of Multiple Properties
 key.addVoter=Add Voter
-key.addCandidate=Add Candidate
+key.addAlternative=Add Alternative
 key.addSeat=Add Seat
 key.deleteItems=Delete Temporary Files
 key.reset=Reset
diff --git a/virage/src/main/resources/settings b/virage/src/main/resources/settings.properties
similarity index 99%
rename from virage/src/main/resources/settings
rename to virage/src/main/resources/settings.properties
index ba54cb3..7f4b3b8 100644
--- a/virage/src/main/resources/settings
+++ b/virage/src/main/resources/settings.properties
@@ -3,7 +3,6 @@ TIMESTAMP=2023-09-01 00:00:00 GMT+02:00
 # This value is used for internal consistency checks, please do not alter it!
 VIRAGE_CONFIG_VERSION=0.1.0
 
-
 ## External executables
 
 ISABELLE_EXECUTABLE=isabelle
@@ -19,7 +18,6 @@ SWI_PROLOG_LIBSWIPL_PATH=/usr/lib/swi-prolog/lib/x86_64-linux/libswipl.so
 # should contain libjpl.so, otherwise install swi-prolog-java)
 SWI_PROLOG_LIBRARIES_PATH=/usr/lib/swi-prolog/lib/x86_64-linux/
 
-
 ## System settings
 #verbosity_level
 
@@ -30,7 +28,6 @@ SYSTEM_DEFAULT_OUTPUT_PATH=./target/generated-sources/
 # variable $EDITOR is used).
 SYSTEM_TEXT_EDITOR=/usr/bin/kate
 
-
 ## Isabelle settings
 
 # Isabelle tactics to be used by the brute force tactics choice
@@ -41,7 +38,6 @@ ISABELLE_TACTICS=simp;blast;metis;fastforce
 # A compositional framework will be extracted and written to an (E)PL file.
 ISABELLE_PATH_TO_ROOT_FILE=src/test/resources/verifiedVotingRuleConstruction/theories/ROOT
 
-
 ## Voting-specific settings
 
 # Names for generic type variables. These can be used in all session-specific
@@ -84,6 +80,3 @@ SESSION_SPECIFIC_COMPONENT_ALIASES=elector(X)->sequential_composition(X,elect_mo
   maximum_parallel_composition(X,Y)->parallel_composition(X,Y,max_aggregator);\
   smc(X)->iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),\
   sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))
-
-
-
diff --git a/virage/src/main/resources/voting_context.template b/virage/src/main/resources/voting_context.template
index 41f1db2..5f7553e 100644
--- a/virage/src/main/resources/voting_context.template
+++ b/virage/src/main/resources/voting_context.template
@@ -1,17 +1,16 @@
-import scala.util.chaining.scalaUtilChainingOps
-import scala.language.postfixOps
-
 import scala.io.Source
+import scala.language.postfixOps
+import scala.util.chaining.scalaUtilChainingOps
 
 /* ENUM
-class Bounded[A](entities: List[A]) extends Enum.`enum`[A] {
+class Bounded[A](entities: List[A]) extends Enum.`enuma`[A] {
 
-  val `Enum.enum`: List[A] = entities
+  val `Enum.enuma`: List[A] = entities
 
   val `Enum.enum_all`: (A => Boolean) => Boolean = (p => entities.forall(p))
 
   val `Enum.enum_ex`: (A => Boolean) => Boolean = (p => entities.exists(p))
-} 
+}
 ENUM */
 
 /* EQUALITY
@@ -21,46 +20,46 @@ class Equality[A] extends HOL.equal[A] {
 EQUALITY */
 
 object VotingContext {
-  def readBallotsFromFile(file: String): (List[String], List[Set.seta[(String,String)]]) = {
+  def readBallotsFromFile(file: String): (List[String], List[Set.seta[(String, String)]]) = {
     var ballots: List[List[String]] = Nil
-    
-    var candidates: List[String] = Nil
+
+    var alternatives: List[String] = Nil
 
     for (line <- Source.fromFile(file).getLines()) {
       val strings: Array[String] = line split ';'
-      
+
       val stringList = List(strings.toList)
-      if (candidates == Nil) candidates = strings.toList
-      
+      if (alternatives == Nil) alternatives = strings.toList
+
       ballots = ballots ::: stringList
     }
-    
-    (candidates, transformProfileToRelations[String](ballots))
+
+    (alternatives, transformProfileToRelations[String](ballots))
   }
   
-  def transformProfileToRelations[A](profile: List[List[A]]): List[Set.seta[(A,A)]] = profile match {
+  def transformProfileToRelations[A](profile: List[List[A]]): List[Set.seta[(A, A)]] = profile match {
     case Nil => Nil
     case p :: ps =>
-      val list = List(Set.seta[(A,A)](buildLinearOrder(p)))
+      val list = List(Set.seta[(A, A)](buildLinearOrder(p)))
       list ::: transformProfileToRelations(ps)
   }
-  
-  def buildLinearOrder[A](ballot: List[A]): List[(A,A)] = ballot match {
+
+  def buildLinearOrder[A](ballot: List[A]): List[(A, A)] = ballot match {
     case Nil => Nil
-    case x :: xs => 
-      var newList = List((x,x))
+    case x :: xs =>
+      var newList = List((x, x))
       for (next <- xs) {
-        newList = (x,next) :: newList
+        newList = (x, next) :: newList
       }
-    
+
       newList ::: buildLinearOrder(xs)
   }
-  
+
   def prettyPrint(result: (Set.set[String], (Set.set[String], Set.set[String]))): Unit = {
     val elected = result._1
     val rejected = result._2._1
     val deferred = result._2._2
-    
+
     print("elected: "); println(elected)
     print("deferred: "); println(deferred)
     print("rejected: "); println(rejected)
@@ -69,12 +68,12 @@ object VotingContext {
 
 object Main {
   def main(args: Array[String]): Unit = {
-    val (candidates, ballots) = VotingContext.readBallotsFromFile(args(0))
+    val (alternatives, ballots) = VotingContext.readBallotsFromFile(args(0))
 
     println("Reading finished.")
-    
+
     /* ENUM
-    implicit val bounded = new Bounded[String](candidates) 
+    implicit val bounded = new Bounded[String](alternatives)
     ENUM */
     /* EQUALITY
     implicit val eq = new Equality[String]
@@ -83,11 +82,11 @@ object Main {
     val t1 = System.nanoTime
 
     /* OPTION1
-    val module = $THEORY_NAME.$MODULE_NAME[String]$PARAMS(Set.seta[String](candidates))
+    val module = $THEORY_NAME.$MODULE_NAME[String]$PARAMS(Set.seta[String](alternatives))
     OPTION1 */
     /* OPTION2
     val relation = $THEORY_NAME.$MODULE_NAME[String]$PARAMS(ballots(1))
-    val module = relation(Set.seta[String](candidates))
+    val module = relation(Set.seta[String](alternatives))
     OPTION2 */
 
     val result = module(ballots)
@@ -98,4 +97,3 @@ object Main {
     println("time in seconds: " + duration)
   }
 }
-
diff --git a/virage/src/main/scala/edu/kit/kastel/formal/util/JIsabelleWrapper.scala b/virage/src/main/scala/edu/kit/kastel/formal/util/JIsabelleWrapper.scala
index 6b3ebae..9178e1f 100644
--- a/virage/src/main/scala/edu/kit/kastel/formal/util/JIsabelleWrapper.scala
+++ b/virage/src/main/scala/edu/kit/kastel/formal/util/JIsabelleWrapper.scala
@@ -8,15 +8,6 @@ import de.unruh.isabelle.control.IsabelleBuildException
 import scala.jdk.CollectionConverters._
 
 object JIsabelleWrapper {
-  /** Sets the [[de.unruh.isabelle.control.Isabelle.Setup.logic logic]] directory in the
-   *  setup `setup`.
-   *
-   * @return `setup` with [[de.unruh.isabelle.control.Isabelle.Setup.logic logic]] set to `logic`
-   * @param logic the new value for `setup.`[[de.unruh.isabelle.control.Isabelle.Setup.logic logic]]
-   * */
-  def setupSetLogic(logic: String, setup: Isabelle.Setup): Isabelle.Setup =
-    setup.copy(logic = logic)
-
   /**
    * Invokes the constructor [[de.unruh.isabelle.control.Isabelle]]`(setup: SetupGeneral)` and
    * explicitly declares that exceptions of [[de.unruh.isabelle.control.IsabelleBuildException]]
diff --git a/virage/src/test/resources/c_implementations/wrapper.c b/virage/src/test/resources/c_implementations/wrapper.c
index 9b4342b..be35a1e 100644
--- a/virage/src/test/resources/c_implementations/wrapper.c
+++ b/virage/src/test/resources/c_implementations/wrapper.c
@@ -25,9 +25,9 @@ char** split_string(char* str) {
   return res;
 }
 
-int find_order_index(char* candidate) {
+int find_order_index(char* alternative) {
   for (int i = 0; i < C; i++) {
-    if (!strcmp(candidate, order[i])) return i;
+    if (!strcmp(alternative, order[i])) return i;
   }
 
   return C;
diff --git a/virage/src/test/resources/instances/README.md b/virage/src/test/resources/instances/README.md
index 0cd7234..86be839 100644
--- a/virage/src/test/resources/instances/README.md
+++ b/virage/src/test/resources/instances/README.md
@@ -2,5 +2,5 @@ These are randomly generated test instances for the voting rule implementations
 They can be used for both the Scala and C implementations.
 
 The format is as follows:
-The file name gives both the number of voters and candidates, the file XcYv contains a profile with X candidates and Y voters.
-Each line represents a ballot, with candidates separated by ; and ordered from lowest to highest preference (might be counterintuitive).
\ No newline at end of file
+The file name gives both the number of voters and alternatives, the file XcYv contains a profile with X alternatives and Y voters.
+Each line represents a ballot, with alternatives separated by ; and ordered from lowest to highest preference (might be counterintuitive).
diff --git a/virage/src/test/resources/old_theories/output/document/Aggregator.tex b/virage/src/test/resources/old_theories/output/document/Aggregator.tex
deleted file mode 100644
index a0a5617..0000000
--- a/virage/src/test/resources/old_theories/output/document/Aggregator.tex
+++ /dev/null
@@ -1,159 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Aggregator}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-\isanewline
-%
-\isamarkupsection{Aggregator%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Aggregator\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Result{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-An aggregator gets two partitions (results of electoral modules) as input and
-output another partition. They are used to aggregate results of parallel
-composed electoral modules.
-They are commutative, i.e., the order of the aggregated modules does not affect
-the resulting aggregation. Moreover, they are conservative in the sense that
-the resulting decisions are subsets of the two given partitions' decisions.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Aggregator\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ aggregator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}aggregator\ agg\ {\isasymequiv}\isanewline
-\ \ \ \ {\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}commutative\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}commutative\ agg\ {\isasymequiv}\isanewline
-\ \ \ \ aggregator\ agg\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}conservative\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ agg\ {\isasymequiv}\isanewline
-\ \ \ \ aggregator\ agg\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Component_Types/Aggregator.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%13=6%:%
-%:%15=9%:%
-%:%31=11%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%48=21%:%
-%:%57=23%:%
-%:%67=25%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%82=33%:%
-%:%92=35%:%
-%:%93=35%:%
-%:%94=36%:%
-%:%96=38%:%
-%:%97=39%:%
-%:%98=40%:%
-%:%99=40%:%
-%:%100=41%:%
-%:%106=47%:%
-%:%109=48%:%
-%:%114=49%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Blacks_Rule.tex b/virage/src/test/resources/old_theories/output/document/Blacks_Rule.tex
deleted file mode 100644
index 7833272..0000000
--- a/virage/src/test/resources/old_theories/output/document/Blacks_Rule.tex
+++ /dev/null
@@ -1,111 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Blacks{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Black's Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Blacks{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ Pairwise{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \ \ \ \ \ \ \ \ Borda{\isacharunderscore}{\kern0pt}Rule\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is Black's voting rule. It is composed of a function that determines
-the Condorcet winner, i.e., the Pairwise Majority rule, and the Borda rule.
-Whenever there exists no Condorcet winner, it elects the choice made by the
-Borda rule, otherwise the Condorcet winner is elected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ blacks{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}blacks{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule\ {\isasymtriangleright}\ borda{\isacharunderscore}{\kern0pt}rule{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ blacks{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}blacks{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isasymtriangleright}\ borda{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Blacks_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%54=20%:%
-%:%64=22%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=25%:%
-%:%70=26%:%
-%:%73=27%:%
-%:%78=28%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Borda_Module.tex b/virage/src/test/resources/old_theories/output/document/Borda_Module.tex
deleted file mode 100644
index 34d3f2e..0000000
--- a/virage/src/test/resources/old_theories/output/document/Borda_Module.tex
+++ /dev/null
@@ -1,131 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Borda{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Borda Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Borda{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Borda module used by the Borda rule. The Borda rule is a voting
-rule, where on each ballot, each alternative is assigned a score that depends
-on how many alternatives are ranked below. The sum of all such scores for an
-alternative is hence called their Borda score. The alternative with the highest
-Borda score is elected. The module implemented herein only rejects the
-alternatives not elected by the voting rule, and defers the alternatives that
-would be elected by the full voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ borda{\isacharunderscore}{\kern0pt}score\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ borda{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ x\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ borda\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda\ A\ p\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ borda{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Borda_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%41=14%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%56=22%:%
-%:%66=24%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%73=29%:%
-%:%74=30%:%
-%:%75=30%:%
-%:%76=31%:%
-%:%77=32%:%
-%:%78=33%:%
-%:%79=33%:%
-%:%80=34%:%
-%:%83=35%:%
-%:%88=36%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Borda_Rule.tex b/virage/src/test/resources/old_theories/output/document/Borda_Rule.tex
deleted file mode 100644
index b96ff8a..0000000
--- a/virage/src/test/resources/old_theories/output/document/Borda_Rule.tex
+++ /dev/null
@@ -1,116 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Borda{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupchapter{Voting Rules%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Borda Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Borda{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Borda{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Borda rule. On each ballot, each alternative is assigned a score
-that depends on how many alternatives are ranked below. The sum of all such
-scores for an alternative is hence called their Borda score. The alternative
-with the highest Borda score is elected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ borda{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ borda\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ borda{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ borda{\isacharunderscore}{\kern0pt}code\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Borda_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%17=9%:%
-%:%33=11%:%
-%:%34=11%:%
-%:%35=12%:%
-%:%36=13%:%
-%:%37=14%:%
-%:%46=17%:%
-%:%47=18%:%
-%:%48=19%:%
-%:%49=20%:%
-%:%58=22%:%
-%:%68=24%:%
-%:%69=24%:%
-%:%70=25%:%
-%:%71=26%:%
-%:%72=27%:%
-%:%73=27%:%
-%:%74=28%:%
-%:%77=29%:%
-%:%82=30%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Classic_Nanson_Rule.tex b/virage/src/test/resources/old_theories/output/document/Classic_Nanson_Rule.tex
deleted file mode 100644
index b84db79..0000000
--- a/virage/src/test/resources/old_theories/output/document/Classic_Nanson_Rule.tex
+++ /dev/null
@@ -1,117 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Classic{\isacharunderscore}{\kern0pt}Nanson{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Classic Nanson Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Classic{\isacharunderscore}{\kern0pt}Nanson{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Borda{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the classic Nanson's voting rule, i.e., the rule that was originally
-invented by Nanson, but not the Nanson-Baldwin rule. The idea is similar,
-however, as alternatives with a Borda score less or equal than the average
-Borda score are excluded. The Borda scores of the remaining alternatives
-are hence adjusted to the new set of (still) eligible alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ classic{\isacharunderscore}{\kern0pt}nanson{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}classic{\isacharunderscore}{\kern0pt}nanson{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ classic{\isacharunderscore}{\kern0pt}nanson{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}classic{\isacharunderscore}{\kern0pt}nanson{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Classic_Nanson_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%55=21%:%
-%:%65=23%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%73=29%:%
-%:%76=30%:%
-%:%81=31%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Condorcet_Module.tex b/virage/src/test/resources/old_theories/output/document/Condorcet_Module.tex
deleted file mode 100644
index 8b6830e..0000000
--- a/virage/src/test/resources/old_theories/output/document/Condorcet_Module.tex
+++ /dev/null
@@ -1,275 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Condorcet{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Condorcet Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Condorcet{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Condorcet module used by the Condorcet (voting) rule. The Condorcet
-rule is a voting rule that implements the Condorcet criterion, i.e., it elects
-the Condorcet winner if it exists, otherwise a tie remains between all
-alternatives. The module implemented herein only rejects the alternatives not
-elected by the voting rule, and defers the alternatives that would be elected
-by the full voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}score\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ x{\isacharparenright}{\kern0pt}\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}code\ A\ p\ x{\isacharparenright}{\kern0pt}\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ condorcet\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ condorcet{\isacharunderscore}{\kern0pt}score{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymnot}\ condorcet{\isacharunderscore}{\kern0pt}rating\ f\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ rs\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ rs\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}aa{\isachardot}{\kern0pt}\ {\isasymnot}\ f\ {\isacharparenleft}{\kern0pt}aa{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ A\ rs\ {\isacharless}{\kern0pt}\ f\ a\ A\ rs\ {\isasymand}\ a\ {\isasymnoteq}\ aa\ {\isasymand}\ aa\ {\isasymin}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}rating\ f\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs\ a{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ rs\ a\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ f\ aa\ A\ rs\ {\isacharless}{\kern0pt}\ f\ a\ A\ rs\ {\isasymor}\ a\ {\isacharequal}{\kern0pt}\ aa\ {\isasymor}\ aa\ {\isasymnotin}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique\ condorcet{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\ zero{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ condorcet{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cscore{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}rating{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}cond{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}condorcet\ A\ p\ {\isasymequiv}\ max{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cscore{\isacharunderscore}{\kern0pt}dcc\ cond{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}cond\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Condorcet_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%41=14%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%55=21%:%
-%:%65=23%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%73=29%:%
-%:%74=30%:%
-%:%75=31%:%
-%:%76=31%:%
-%:%77=32%:%
-%:%78=33%:%
-%:%79=34%:%
-%:%80=34%:%
-%:%81=35%:%
-%:%89=38%:%
-%:%99=41%:%
-%:%100=41%:%
-%:%107=42%:%
-%:%108=42%:%
-%:%109=43%:%
-%:%110=43%:%
-%:%111=44%:%
-%:%118=51%:%
-%:%119=52%:%
-%:%120=52%:%
-%:%121=53%:%
-%:%122=53%:%
-%:%123=54%:%
-%:%124=54%:%
-%:%125=55%:%
-%:%126=55%:%
-%:%127=56%:%
-%:%128=56%:%
-%:%129=57%:%
-%:%135=57%:%
-%:%138=58%:%
-%:%139=59%:%
-%:%140=59%:%
-%:%147=60%:%
-%:%148=60%:%
-%:%149=61%:%
-%:%150=61%:%
-%:%151=62%:%
-%:%152=63%:%
-%:%153=63%:%
-%:%154=64%:%
-%:%155=64%:%
-%:%156=65%:%
-%:%157=65%:%
-%:%158=66%:%
-%:%159=67%:%
-%:%160=67%:%
-%:%161=68%:%
-%:%162=68%:%
-%:%163=68%:%
-%:%164=69%:%
-%:%165=69%:%
-%:%166=70%:%
-%:%167=70%:%
-%:%168=71%:%
-%:%174=71%:%
-%:%179=72%:%
-%:%184=73%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Copeland_Module.tex b/virage/src/test/resources/old_theories/output/document/Copeland_Module.tex
deleted file mode 100644
index 82760ff..0000000
--- a/virage/src/test/resources/old_theories/output/document/Copeland_Module.tex
+++ /dev/null
@@ -1,917 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Copeland{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Copeland Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Copeland{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Copeland module used by the Copeland voting rule. The Copeland
-rule elects the alternatives with the highest difference between the amount
-of simple-majority wins and the amount of simple-majority losses. The module
-implemented herein only rejects the alternatives not elected by the voting
-rule, and defers the alternatives that would be elected by the full voting
-rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}score\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ x\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ y\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}code\ x\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}code\ y\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ copeland\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland\ A\ p\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}eliminator\ copeland{\isacharunderscore}{\kern0pt}score\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ max{\isacharunderscore}{\kern0pt}eliminator\ copeland{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}M\ {\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ M\ {\isachardot}{\kern0pt}\ True{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ M{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}singleton\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ amount{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}A{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ {\isasymnot}\ wins\ x\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}irreflex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}M\ {\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ M\ {\isachardot}{\kern0pt}\ False{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ amount{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disjunct{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isasyminter}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ union{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finiteness{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finiteness{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ finiteness{\isadigit{1}}\ finiteness{\isadigit{2}}\ disjunct\ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}disjoint\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ union\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ amount{\isadigit{1}}\ amount{\isadigit{2}}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}loss{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}eq\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}iff\ wins{\isacharunderscore}{\kern0pt}antisym\ winner\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}copeland{\isacharunderscore}{\kern0pt}score{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}score\ w\ A\ p\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}loss{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \ \ \ \ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\ winner\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ a\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}loss{\isacharunderscore}{\kern0pt}count\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ {\isadigit{0}}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ f{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ loser{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}{\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner\ loser\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}wins\ w\ p\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ l\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ l\ p\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}irreflex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ M\ f\ {\isachardot}{\kern0pt}\ finite\ M\ {\isasymlongrightarrow}\ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ M\ {\isachardot}{\kern0pt}\ f\ x{\isacharbraceright}{\kern0pt}\ {\isasymle}\ card\ M{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ finite{\isacharunderscore}{\kern0pt}Diff\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{3}}\ {\isadigit{4}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{5}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isasymle}\isanewline
-\ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card{\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card{\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ loser{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{5}}\ {\isadigit{2}}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}cr{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ copeland{\isacharunderscore}{\kern0pt}score{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ copeland{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ l{\isacharbraceright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharless}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ winner\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}copeland{\isacharunderscore}{\kern0pt}score\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ winner\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ l{\isacharbraceright}{\kern0pt}\ {\isasymle}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isacharminus}{\kern0pt}{\isadigit{2}}\ {\isacharless}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}eq\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}singleton\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ diff{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ empty{\isacharunderscore}{\kern0pt}iff\ finite{\isacharunderscore}{\kern0pt}Diff\ insertE\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\ neq{\isadigit{0}}{\isacharunderscore}{\kern0pt}conv\ one{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}numeral{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ semiring{\isacharunderscore}{\kern0pt}norm{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ winner\ zero{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}diff\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ l{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ le{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ copeland{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cplscore{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ copeland{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ copeland{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}cr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ copel{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}copel{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}copeland\ A\ p\ {\isasymequiv}\ max{\isacharunderscore}{\kern0pt}eliminator\ copeland{\isacharunderscore}{\kern0pt}score\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cplscore{\isacharunderscore}{\kern0pt}dcc\ copel{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}copel\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Copeland_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%41=14%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%55=21%:%
-%:%65=23%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%73=29%:%
-%:%74=30%:%
-%:%75=31%:%
-%:%76=31%:%
-%:%77=32%:%
-%:%78=33%:%
-%:%79=34%:%
-%:%80=34%:%
-%:%81=35%:%
-%:%89=38%:%
-%:%99=41%:%
-%:%100=41%:%
-%:%101=42%:%
-%:%102=43%:%
-%:%109=44%:%
-%:%110=44%:%
-%:%111=45%:%
-%:%112=45%:%
-%:%113=46%:%
-%:%114=46%:%
-%:%115=47%:%
-%:%116=47%:%
-%:%117=48%:%
-%:%118=48%:%
-%:%119=49%:%
-%:%120=49%:%
-%:%121=50%:%
-%:%122=50%:%
-%:%123=51%:%
-%:%124=51%:%
-%:%125=52%:%
-%:%126=52%:%
-%:%127=53%:%
-%:%128=53%:%
-%:%129=54%:%
-%:%130=54%:%
-%:%131=55%:%
-%:%132=55%:%
-%:%133=56%:%
-%:%134=56%:%
-%:%135=57%:%
-%:%136=57%:%
-%:%137=58%:%
-%:%138=58%:%
-%:%139=59%:%
-%:%140=59%:%
-%:%141=60%:%
-%:%142=60%:%
-%:%143=61%:%
-%:%144=61%:%
-%:%145=62%:%
-%:%146=63%:%
-%:%147=63%:%
-%:%148=64%:%
-%:%149=64%:%
-%:%150=65%:%
-%:%151=65%:%
-%:%152=66%:%
-%:%153=66%:%
-%:%154=67%:%
-%:%155=67%:%
-%:%156=68%:%
-%:%157=68%:%
-%:%158=69%:%
-%:%159=69%:%
-%:%160=70%:%
-%:%161=70%:%
-%:%162=71%:%
-%:%163=71%:%
-%:%164=72%:%
-%:%165=72%:%
-%:%166=73%:%
-%:%167=73%:%
-%:%168=74%:%
-%:%169=74%:%
-%:%170=75%:%
-%:%171=76%:%
-%:%172=76%:%
-%:%173=77%:%
-%:%174=77%:%
-%:%175=78%:%
-%:%176=79%:%
-%:%177=80%:%
-%:%178=80%:%
-%:%179=81%:%
-%:%180=81%:%
-%:%181=82%:%
-%:%182=82%:%
-%:%183=83%:%
-%:%184=83%:%
-%:%185=84%:%
-%:%186=84%:%
-%:%187=85%:%
-%:%188=85%:%
-%:%189=86%:%
-%:%190=86%:%
-%:%191=87%:%
-%:%192=87%:%
-%:%193=88%:%
-%:%194=88%:%
-%:%195=89%:%
-%:%196=90%:%
-%:%197=91%:%
-%:%198=91%:%
-%:%199=92%:%
-%:%200=92%:%
-%:%201=93%:%
-%:%202=93%:%
-%:%203=94%:%
-%:%204=95%:%
-%:%205=95%:%
-%:%206=96%:%
-%:%207=96%:%
-%:%208=97%:%
-%:%209=97%:%
-%:%210=98%:%
-%:%211=98%:%
-%:%212=99%:%
-%:%218=99%:%
-%:%221=100%:%
-%:%222=101%:%
-%:%223=102%:%
-%:%224=102%:%
-%:%225=103%:%
-%:%226=104%:%
-%:%229=105%:%
-%:%233=105%:%
-%:%234=105%:%
-%:%235=106%:%
-%:%236=107%:%
-%:%237=107%:%
-%:%242=107%:%
-%:%245=108%:%
-%:%246=109%:%
-%:%247=110%:%
-%:%248=110%:%
-%:%249=111%:%
-%:%250=112%:%
-%:%253=113%:%
-%:%257=113%:%
-%:%258=113%:%
-%:%259=114%:%
-%:%260=114%:%
-%:%261=115%:%
-%:%262=115%:%
-%:%263=116%:%
-%:%264=117%:%
-%:%265=118%:%
-%:%266=118%:%
-%:%267=119%:%
-%:%268=120%:%
-%:%269=120%:%
-%:%270=121%:%
-%:%271=121%:%
-%:%272=122%:%
-%:%273=122%:%
-%:%274=123%:%
-%:%275=123%:%
-%:%276=124%:%
-%:%277=124%:%
-%:%278=125%:%
-%:%279=125%:%
-%:%280=126%:%
-%:%281=126%:%
-%:%282=127%:%
-%:%283=127%:%
-%:%284=128%:%
-%:%285=128%:%
-%:%286=129%:%
-%:%287=129%:%
-%:%288=130%:%
-%:%289=130%:%
-%:%290=131%:%
-%:%291=131%:%
-%:%292=132%:%
-%:%293=132%:%
-%:%294=133%:%
-%:%300=133%:%
-%:%303=134%:%
-%:%304=138%:%
-%:%305=139%:%
-%:%306=139%:%
-%:%307=140%:%
-%:%308=141%:%
-%:%309=142%:%
-%:%310=143%:%
-%:%311=144%:%
-%:%318=145%:%
-%:%319=145%:%
-%:%320=146%:%
-%:%321=146%:%
-%:%322=147%:%
-%:%323=147%:%
-%:%324=148%:%
-%:%325=148%:%
-%:%326=149%:%
-%:%327=149%:%
-%:%328=150%:%
-%:%329=150%:%
-%:%330=151%:%
-%:%331=151%:%
-%:%332=152%:%
-%:%333=152%:%
-%:%334=153%:%
-%:%335=153%:%
-%:%336=153%:%
-%:%337=154%:%
-%:%338=155%:%
-%:%339=156%:%
-%:%340=156%:%
-%:%341=157%:%
-%:%342=157%:%
-%:%343=158%:%
-%:%344=158%:%
-%:%345=159%:%
-%:%346=159%:%
-%:%347=160%:%
-%:%348=160%:%
-%:%349=161%:%
-%:%350=161%:%
-%:%351=162%:%
-%:%352=162%:%
-%:%353=162%:%
-%:%354=163%:%
-%:%355=164%:%
-%:%356=165%:%
-%:%357=165%:%
-%:%358=166%:%
-%:%359=166%:%
-%:%360=167%:%
-%:%361=167%:%
-%:%362=168%:%
-%:%363=168%:%
-%:%364=169%:%
-%:%365=169%:%
-%:%366=170%:%
-%:%367=170%:%
-%:%368=171%:%
-%:%369=171%:%
-%:%370=172%:%
-%:%371=172%:%
-%:%372=173%:%
-%:%373=173%:%
-%:%374=174%:%
-%:%375=174%:%
-%:%376=175%:%
-%:%377=175%:%
-%:%378=176%:%
-%:%379=176%:%
-%:%380=177%:%
-%:%395=179%:%
-%:%405=182%:%
-%:%406=182%:%
-%:%409=183%:%
-%:%413=183%:%
-%:%414=183%:%
-%:%415=184%:%
-%:%416=184%:%
-%:%417=185%:%
-%:%418=185%:%
-%:%419=186%:%
-%:%420=187%:%
-%:%421=188%:%
-%:%422=189%:%
-%:%423=190%:%
-%:%424=190%:%
-%:%425=191%:%
-%:%426=192%:%
-%:%427=193%:%
-%:%428=194%:%
-%:%429=194%:%
-%:%430=195%:%
-%:%431=196%:%
-%:%432=197%:%
-%:%433=197%:%
-%:%434=198%:%
-%:%435=198%:%
-%:%436=198%:%
-%:%437=199%:%
-%:%438=200%:%
-%:%439=201%:%
-%:%440=201%:%
-%:%441=202%:%
-%:%442=202%:%
-%:%443=203%:%
-%:%444=203%:%
-%:%445=203%:%
-%:%446=204%:%
-%:%447=205%:%
-%:%448=206%:%
-%:%449=206%:%
-%:%450=207%:%
-%:%451=207%:%
-%:%452=208%:%
-%:%453=208%:%
-%:%454=209%:%
-%:%455=209%:%
-%:%456=210%:%
-%:%457=211%:%
-%:%458=212%:%
-%:%459=213%:%
-%:%460=214%:%
-%:%461=214%:%
-%:%462=215%:%
-%:%463=215%:%
-%:%464=216%:%
-%:%465=217%:%
-%:%466=218%:%
-%:%467=218%:%
-%:%468=219%:%
-%:%469=219%:%
-%:%470=220%:%
-%:%471=220%:%
-%:%472=221%:%
-%:%473=221%:%
-%:%474=222%:%
-%:%475=222%:%
-%:%476=223%:%
-%:%477=223%:%
-%:%478=224%:%
-%:%484=224%:%
-%:%487=225%:%
-%:%488=226%:%
-%:%489=226%:%
-%:%496=227%:%
-%:%497=227%:%
-%:%498=228%:%
-%:%499=228%:%
-%:%500=229%:%
-%:%501=230%:%
-%:%502=230%:%
-%:%503=231%:%
-%:%504=231%:%
-%:%505=232%:%
-%:%506=232%:%
-%:%507=233%:%
-%:%508=234%:%
-%:%509=234%:%
-%:%510=235%:%
-%:%511=235%:%
-%:%512=236%:%
-%:%513=236%:%
-%:%514=237%:%
-%:%515=237%:%
-%:%516=238%:%
-%:%517=238%:%
-%:%518=239%:%
-%:%524=239%:%
-%:%529=240%:%
-%:%534=241%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Copeland_Rule.tex b/virage/src/test/resources/old_theories/output/document/Copeland_Rule.tex
deleted file mode 100644
index 6d860cb..0000000
--- a/virage/src/test/resources/old_theories/output/document/Copeland_Rule.tex
+++ /dev/null
@@ -1,179 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Copeland{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Copeland Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Copeland{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Copeland{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Copeland voting rule. The idea is to elect the alternatives with
-the highest difference between the amount of simple-majority wins and the
-amount of simple-majority losses.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ copeland\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ copeland{\isacharunderscore}{\kern0pt}code\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Condorcet Consistency Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}condorcet{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ copeland{\isacharunderscore}{\kern0pt}rule{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ copeland{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ copeland{\isacharunderscore}{\kern0pt}rule{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Copeland_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%53=19%:%
-%:%63=21%:%
-%:%64=21%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=24%:%
-%:%69=25%:%
-%:%76=27%:%
-%:%86=29%:%
-%:%87=29%:%
-%:%94=30%:%
-%:%95=30%:%
-%:%96=31%:%
-%:%97=31%:%
-%:%98=32%:%
-%:%99=33%:%
-%:%100=33%:%
-%:%101=34%:%
-%:%102=34%:%
-%:%103=35%:%
-%:%104=35%:%
-%:%105=36%:%
-%:%106=36%:%
-%:%107=37%:%
-%:%108=38%:%
-%:%109=38%:%
-%:%110=39%:%
-%:%116=39%:%
-%:%121=40%:%
-%:%126=41%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Defer_Equal_Condition.tex b/virage/src/test/resources/old_theories/output/document/Defer_Equal_Condition.tex
deleted file mode 100644
index 055725a..0000000
--- a/virage/src/test/resources/old_theories/output/document/Defer_Equal_Condition.tex
+++ /dev/null
@@ -1,101 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Defer{\isacharunderscore}{\kern0pt}Equal{\isacharunderscore}{\kern0pt}Condition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Defer Equal Condition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Defer{\isacharunderscore}{\kern0pt}Equal{\isacharunderscore}{\kern0pt}Condition\isanewline
-\ \ \isakeyword{imports}\ Termination{\isacharunderscore}{\kern0pt}Condition\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is a family of termination conditions. For a natural number n,
-the according defer-equal condition is true if and only if the given
-result's defer-set contains exactly n elements.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ n\ result\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ result\ in\ card\ d\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Component_Types/Defer_Equal_Condition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%53=19%:%
-%:%63=21%:%
-%:%64=21%:%
-%:%65=22%:%
-%:%68=23%:%
-%:%73=24%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Defer_Module.tex b/virage/src/test/resources/old_theories/output/document/Defer_Module.tex
deleted file mode 100644
index 085f602..0000000
--- a/virage/src/test/resources/old_theories/output/document/Defer_Module.tex
+++ /dev/null
@@ -1,222 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Defer{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupchapter{Basic Modules%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Defer Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Defer{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The defer module is not concerned about the voter's ballots, and
-simply defers all alternatives.
-It is primarily used for defining an empty loop.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}module\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ defer{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ defer{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ defer{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Defer_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%18=10%:%
-%:%34=12%:%
-%:%35=12%:%
-%:%36=13%:%
-%:%37=14%:%
-%:%46=17%:%
-%:%47=18%:%
-%:%48=19%:%
-%:%57=21%:%
-%:%67=23%:%
-%:%68=23%:%
-%:%69=24%:%
-%:%76=26%:%
-%:%86=28%:%
-%:%87=28%:%
-%:%90=29%:%
-%:%94=29%:%
-%:%95=29%:%
-%:%96=30%:%
-%:%97=30%:%
-%:%111=32%:%
-%:%121=34%:%
-%:%122=34%:%
-%:%125=35%:%
-%:%129=35%:%
-%:%130=35%:%
-%:%131=36%:%
-%:%132=36%:%
-%:%137=36%:%
-%:%140=37%:%
-%:%141=38%:%
-%:%142=38%:%
-%:%145=39%:%
-%:%149=39%:%
-%:%150=39%:%
-%:%151=40%:%
-%:%152=40%:%
-%:%157=40%:%
-%:%162=41%:%
-%:%167=42%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Defer_One_Loop_Composition.tex b/virage/src/test/resources/old_theories/output/document/Defer_One_Loop_Composition.tex
deleted file mode 100644
index a48fe01..0000000
--- a/virage/src/test/resources/old_theories/output/document/Defer_One_Loop_Composition.tex
+++ /dev/null
@@ -1,130 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Defer One Loop Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}Equal{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Loop{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \ \ \ \ \ \ \ \ Elect{\isacharunderscore}{\kern0pt}Composition\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is a family of loop compositions. It uses the same module in sequence
-until either no new decisions are made or only one alternative is remaining
-in the defer-set. The second family herein uses the above family and
-subsequently elects the remaining alternative.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ iter\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}iter\ m\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}let\ t\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isadigit{1}}\ in\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}loop\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}{\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d\ {\isasymequiv}\ iter\ m{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ iterelect\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}iterelect\ m\ {\isacharequal}{\kern0pt}\ elector\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Defer_One_Loop_Composition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%35=14%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%56=22%:%
-%:%66=24%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%70=27%:%
-%:%71=28%:%
-%:%72=29%:%
-%:%73=29%:%
-%:%74=30%:%
-%:%75=31%:%
-%:%76=32%:%
-%:%77=33%:%
-%:%78=34%:%
-%:%79=34%:%
-%:%80=35%:%
-%:%83=36%:%
-%:%88=37%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Drop_And_Pass_Compatibility.tex b/virage/src/test/resources/old_theories/output/document/Drop_And_Pass_Compatibility.tex
deleted file mode 100644
index a60ea2f..0000000
--- a/virage/src/test/resources/old_theories/output/document/Drop_And_Pass_Compatibility.tex
+++ /dev/null
@@ -1,459 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Drop{\isacharunderscore}{\kern0pt}And{\isacharunderscore}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Compatibility}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupchapter{Compositional Structures%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Drop And Pass Compatibility%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Drop{\isacharunderscore}{\kern0pt}And{\isacharunderscore}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Compatibility\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Drop{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is a collection of properties about the interplay and compatibility
-of both the drop module and the pass module.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}two{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}two{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\ \isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ AA\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ rrs\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x{\isadigit{0}}\ x{\isadigit{1}}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymexists}v{\isadigit{2}}\ v{\isadigit{3}}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isadigit{1}}\ {\isasymle}\ card\ v{\isadigit{2}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ v{\isadigit{2}}\ v{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ x{\isadigit{0}}\ v{\isadigit{2}}\ v{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isadigit{1}}\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}AA\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ x{\isadigit{0}}\ {\isacharparenleft}{\kern0pt}AA\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ moura\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}n\ f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnot}\ rejects\ n\ f\ {\isasymor}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnot}\ n\ {\isasymle}\ card\ A\ {\isasymor}\ infinite\ A\ {\isasymor}\ {\isasymnot}\ profile\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}rejects\ n\ f\ {\isasymor}\ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ {\isacharparenleft}{\kern0pt}n\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rejects{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}n\ f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnot}\ rejects\ n\ f\ {\isasymor}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isasymnot}\ n\ {\isasymle}\ card\ A\ {\isasymor}\ infinite\ A\ {\isasymor}\ {\isasymnot}\ profile\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}rejects\ n\ f\ {\isasymor}\ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ n\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\ profile\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ {\isadigit{2}}\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ profile\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}rrs\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isacartoucheopen}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isacartoucheclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defers{\isacharunderscore}{\kern0pt}def\ order\ pass{\isacharunderscore}{\kern0pt}two{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}two\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}pass{\isacharunderscore}{\kern0pt}disj{\isacharunderscore}{\kern0pt}compat{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ S{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{where}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ empty{\isacharunderscore}{\kern0pt}iff\ empty{\isacharunderscore}{\kern0pt}set\ fin\ profile{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p\ q{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ q{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ {\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ {\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Drop_And_Pass_Compatibility.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%18=10%:%
-%:%34=12%:%
-%:%35=12%:%
-%:%36=13%:%
-%:%37=14%:%
-%:%38=15%:%
-%:%47=18%:%
-%:%48=19%:%
-%:%57=21%:%
-%:%67=27%:%
-%:%68=27%:%
-%:%69=28%:%
-%:%70=29%:%
-%:%77=30%:%
-%:%78=30%:%
-%:%79=31%:%
-%:%80=31%:%
-%:%81=32%:%
-%:%82=32%:%
-%:%83=33%:%
-%:%84=33%:%
-%:%85=34%:%
-%:%86=34%:%
-%:%87=35%:%
-%:%88=35%:%
-%:%89=36%:%
-%:%90=37%:%
-%:%91=38%:%
-%:%94=41%:%
-%:%95=42%:%
-%:%96=42%:%
-%:%97=43%:%
-%:%98=43%:%
-%:%99=44%:%
-%:%104=49%:%
-%:%105=50%:%
-%:%106=50%:%
-%:%107=51%:%
-%:%108=51%:%
-%:%109=52%:%
-%:%110=52%:%
-%:%111=53%:%
-%:%116=58%:%
-%:%117=59%:%
-%:%118=59%:%
-%:%119=60%:%
-%:%120=60%:%
-%:%121=61%:%
-%:%125=65%:%
-%:%126=66%:%
-%:%127=66%:%
-%:%128=67%:%
-%:%129=68%:%
-%:%130=68%:%
-%:%131=69%:%
-%:%132=69%:%
-%:%133=70%:%
-%:%134=70%:%
-%:%135=71%:%
-%:%136=71%:%
-%:%137=72%:%
-%:%138=72%:%
-%:%139=73%:%
-%:%145=73%:%
-%:%148=74%:%
-%:%149=75%:%
-%:%150=76%:%
-%:%151=76%:%
-%:%152=77%:%
-%:%153=78%:%
-%:%156=79%:%
-%:%160=79%:%
-%:%161=79%:%
-%:%162=80%:%
-%:%163=80%:%
-%:%164=81%:%
-%:%165=81%:%
-%:%166=82%:%
-%:%167=82%:%
-%:%168=83%:%
-%:%169=83%:%
-%:%170=84%:%
-%:%171=84%:%
-%:%172=85%:%
-%:%173=85%:%
-%:%174=86%:%
-%:%175=86%:%
-%:%176=87%:%
-%:%177=87%:%
-%:%178=88%:%
-%:%179=88%:%
-%:%180=89%:%
-%:%181=89%:%
-%:%182=90%:%
-%:%183=91%:%
-%:%184=91%:%
-%:%185=92%:%
-%:%186=93%:%
-%:%187=93%:%
-%:%188=94%:%
-%:%189=95%:%
-%:%190=96%:%
-%:%191=96%:%
-%:%192=97%:%
-%:%193=97%:%
-%:%194=98%:%
-%:%195=98%:%
-%:%196=99%:%
-%:%202=105%:%
-%:%203=106%:%
-%:%204=106%:%
-%:%205=107%:%
-%:%206=107%:%
-%:%207=108%:%
-%:%209=110%:%
-%:%210=111%:%
-%:%211=111%:%
-%:%212=112%:%
-%:%213=112%:%
-%:%214=113%:%
-%:%215=113%:%
-%:%216=114%:%
-%:%217=114%:%
-%:%218=115%:%
-%:%219=115%:%
-%:%220=115%:%
-%:%221=116%:%
-%:%222=117%:%
-%:%223=117%:%
-%:%224=118%:%
-%:%225=118%:%
-%:%226=119%:%
-%:%227=119%:%
-%:%228=119%:%
-%:%229=120%:%
-%:%230=121%:%
-%:%231=122%:%
-%:%232=122%:%
-%:%233=123%:%
-%:%234=123%:%
-%:%235=123%:%
-%:%236=124%:%
-%:%237=125%:%
-%:%238=125%:%
-%:%239=126%:%
-%:%240=126%:%
-%:%241=127%:%
-%:%242=127%:%
-%:%243=127%:%
-%:%244=128%:%
-%:%245=129%:%
-%:%246=130%:%
-%:%247=130%:%
-%:%248=131%:%
-%:%249=131%:%
-%:%250=131%:%
-%:%251=132%:%
-%:%257=138%:%
-%:%258=139%:%
-%:%259=139%:%
-%:%260=140%:%
-%:%261=140%:%
-%:%262=141%:%
-%:%268=141%:%
-%:%273=142%:%
-%:%278=143%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Drop_Module.tex b/virage/src/test/resources/old_theories/output/document/Drop_Module.tex
deleted file mode 100644
index 134f029..0000000
--- a/virage/src/test/resources/old_theories/output/document/Drop_Module.tex
+++ /dev/null
@@ -1,358 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Drop{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Drop Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Drop{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is a family of electoral modules. For a natural number n and a
-lexicon (linear order) r of all alternatives, the according drop module
-rejects the lexicographically first n alternatives (from A) and
-defers the rest.
-It is primarily used as counterpart to the pass module in a
-parallel composition, in order to segment the alternatives into
-two groups.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}mod\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}modI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Non-Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ order{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ order\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Drop_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%48=21%:%
-%:%57=23%:%
-%:%67=25%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%79=31%:%
-%:%89=33%:%
-%:%90=33%:%
-%:%91=34%:%
-%:%92=35%:%
-%:%99=36%:%
-%:%100=36%:%
-%:%101=37%:%
-%:%102=37%:%
-%:%103=38%:%
-%:%104=38%:%
-%:%105=39%:%
-%:%107=41%:%
-%:%108=42%:%
-%:%109=42%:%
-%:%110=43%:%
-%:%111=43%:%
-%:%112=44%:%
-%:%114=46%:%
-%:%115=47%:%
-%:%116=47%:%
-%:%117=48%:%
-%:%118=48%:%
-%:%119=49%:%
-%:%120=50%:%
-%:%121=51%:%
-%:%122=51%:%
-%:%123=52%:%
-%:%124=52%:%
-%:%125=53%:%
-%:%127=55%:%
-%:%128=56%:%
-%:%129=56%:%
-%:%130=57%:%
-%:%131=57%:%
-%:%132=58%:%
-%:%134=60%:%
-%:%135=61%:%
-%:%136=61%:%
-%:%137=62%:%
-%:%138=62%:%
-%:%139=63%:%
-%:%140=63%:%
-%:%141=64%:%
-%:%142=64%:%
-%:%143=64%:%
-%:%144=65%:%
-%:%145=66%:%
-%:%146=67%:%
-%:%147=67%:%
-%:%148=68%:%
-%:%149=68%:%
-%:%150=69%:%
-%:%151=70%:%
-%:%152=71%:%
-%:%153=71%:%
-%:%154=72%:%
-%:%155=72%:%
-%:%156=73%:%
-%:%157=73%:%
-%:%158=74%:%
-%:%159=74%:%
-%:%160=75%:%
-%:%175=77%:%
-%:%185=80%:%
-%:%186=80%:%
-%:%187=81%:%
-%:%188=82%:%
-%:%191=83%:%
-%:%195=83%:%
-%:%196=83%:%
-%:%210=85%:%
-%:%220=88%:%
-%:%221=88%:%
-%:%222=89%:%
-%:%223=90%:%
-%:%226=91%:%
-%:%230=91%:%
-%:%231=91%:%
-%:%236=91%:%
-%:%241=92%:%
-%:%246=93%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Elect_Composition.tex b/virage/src/test/resources/old_theories/output/document/Elect_Composition.tex
deleted file mode 100644
index 997426a..0000000
--- a/virage/src/test/resources/old_theories/output/document/Elect_Composition.tex
+++ /dev/null
@@ -1,1049 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Elect{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Elect Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Elect{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Sequential{\isacharunderscore}{\kern0pt}Composition\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The elect composition sequences an electoral module and the elect
-module. It finalizes the module's decision as it simply elects all their
-non-rejected alternatives. Thereby, any such elect-composed module induces
-a proper voting rule in the social choice sense, as all alternatives are either
-rejected or elected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ elector\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elector\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ elect{\isacharunderscore}{\kern0pt}module{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ elector{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ elector{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}block{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electing\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ AA\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rrs\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electing\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ elect\ f\ {\isacharparenleft}{\kern0pt}AA\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ profile\ {\isacharparenleft}{\kern0pt}AA\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}AA\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ AA\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ elect\ f\ A\ rs\ {\isasymor}\ {\isasymnot}\ profile\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ infinite\ A\ {\isasymor}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electing\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}block{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}module{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}\ Profile\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}\ Result{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electing{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ AAa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ rrsa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electing\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ elect\ f\ {\isacharparenleft}{\kern0pt}AAa\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ profile\ {\isacharparenleft}{\kern0pt}AAa\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}AAa\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ AAa\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ elect\ f\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ profile\ A\ rs\ {\isasymor}\ infinite\ A\ {\isasymor}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electing\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ AAb\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ AAc\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ AAd\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AAb\ p{\isacharcomma}{\kern0pt}\ AAc\ p{\isacharcomma}{\kern0pt}\ AAd\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isadigit{3}}{\isachardot}{\kern0pt}cases\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elector{\isacharunderscore}{\kern0pt}sound\ module{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ p{\isacharcomma}{\kern0pt}\ AAc\ p{\isacharcomma}{\kern0pt}\ AAd\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ profile\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AAd\ {\isacharparenleft}{\kern0pt}elector\ m\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ AAc\ {\isacharparenleft}{\kern0pt}elector\ m\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electing\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ f{\isadigit{1}}\ Diff{\isacharunderscore}{\kern0pt}empty\ elector{\isachardot}{\kern0pt}simps\ non{\isacharunderscore}{\kern0pt}block{\isacharunderscore}{\kern0pt}m\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}block\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{{\isacharbraceleft}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ AAd\ {\isacharparenleft}{\kern0pt}elector\ m\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ profile\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{4}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{{\isacharbraceright}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{4}}\ f{\isadigit{3}}\ f{\isadigit{1}}\ fst{\isacharunderscore}{\kern0pt}conv\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Composition Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def{\isacharcomma}{\kern0pt}\ auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ elect{\isacharunderscore}{\kern0pt}module{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p\ w\ x{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ finite\ A\ {\isasymLongrightarrow}\ profile\ A\ p\ {\isasymLongrightarrow}\ w\ {\isasymin}\ A\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ elect\ m\ A\ p\ {\isasymLongrightarrow}\ x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ x\ {\isasymin}\ elect\ m\ A\ p\ {\isasymLongrightarrow}\ x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subset{\isacharunderscore}{\kern0pt}eq\ finite\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ fst{\isacharunderscore}{\kern0pt}conv\ insert{\isacharunderscore}{\kern0pt}Diff\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ order{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ subset{\isacharunderscore}{\kern0pt}Compl{\isacharunderscore}{\kern0pt}singleton\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ xa{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}xa{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ empty{\isacharunderscore}{\kern0pt}iff\ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\ DiffI\isanewline
-\ \ \ \ \ \ \ \ \ \ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ insert{\isacharunderscore}{\kern0pt}iff\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ insert{\isacharunderscore}{\kern0pt}iff\ eq{\isacharunderscore}{\kern0pt}snd{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ finite\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subsetD\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ IntI\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ IntI\isanewline
-\ \ \ \ \ \ \ \ \ \ Diff{\isacharunderscore}{\kern0pt}empty\ Diff{\isacharunderscore}{\kern0pt}iff\ finite\ prof{\isacharunderscore}{\kern0pt}A\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Elect_Composition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%56=22%:%
-%:%66=24%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%75=27%:%
-%:%85=29%:%
-%:%86=29%:%
-%:%87=30%:%
-%:%88=31%:%
-%:%91=32%:%
-%:%95=32%:%
-%:%96=32%:%
-%:%110=34%:%
-%:%120=36%:%
-%:%121=36%:%
-%:%122=37%:%
-%:%123=38%:%
-%:%124=39%:%
-%:%125=40%:%
-%:%132=41%:%
-%:%133=41%:%
-%:%134=42%:%
-%:%135=42%:%
-%:%136=43%:%
-%:%137=44%:%
-%:%138=45%:%
-%:%139=46%:%
-%:%147=54%:%
-%:%148=55%:%
-%:%149=55%:%
-%:%150=56%:%
-%:%151=56%:%
-%:%152=57%:%
-%:%153=57%:%
-%:%154=58%:%
-%:%155=59%:%
-%:%156=60%:%
-%:%157=60%:%
-%:%158=61%:%
-%:%159=61%:%
-%:%160=62%:%
-%:%160=66%:%
-%:%161=67%:%
-%:%162=67%:%
-%:%163=68%:%
-%:%164=68%:%
-%:%165=69%:%
-%:%166=70%:%
-%:%167=71%:%
-%:%168=72%:%
-%:%174=78%:%
-%:%175=79%:%
-%:%176=79%:%
-%:%177=80%:%
-%:%178=80%:%
-%:%179=81%:%
-%:%180=81%:%
-%:%181=82%:%
-%:%182=83%:%
-%:%183=84%:%
-%:%184=85%:%
-%:%185=86%:%
-%:%186=87%:%
-%:%187=87%:%
-%:%188=88%:%
-%:%189=88%:%
-%:%190=89%:%
-%:%191=89%:%
-%:%192=90%:%
-%:%193=91%:%
-%:%194=91%:%
-%:%195=92%:%
-%:%196=92%:%
-%:%197=93%:%
-%:%198=93%:%
-%:%199=94%:%
-%:%200=95%:%
-%:%201=95%:%
-%:%202=96%:%
-%:%203=96%:%
-%:%204=97%:%
-%:%205=97%:%
-%:%206=98%:%
-%:%212=104%:%
-%:%213=105%:%
-%:%214=105%:%
-%:%215=106%:%
-%:%216=107%:%
-%:%217=108%:%
-%:%218=108%:%
-%:%219=109%:%
-%:%220=109%:%
-%:%221=110%:%
-%:%222=110%:%
-%:%223=111%:%
-%:%224=111%:%
-%:%225=112%:%
-%:%226=113%:%
-%:%227=113%:%
-%:%228=114%:%
-%:%229=115%:%
-%:%230=116%:%
-%:%231=116%:%
-%:%232=117%:%
-%:%233=117%:%
-%:%234=118%:%
-%:%235=118%:%
-%:%236=119%:%
-%:%237=119%:%
-%:%238=119%:%
-%:%239=120%:%
-%:%240=120%:%
-%:%241=121%:%
-%:%242=121%:%
-%:%243=122%:%
-%:%244=122%:%
-%:%245=123%:%
-%:%260=125%:%
-%:%270=128%:%
-%:%271=128%:%
-%:%272=129%:%
-%:%273=130%:%
-%:%280=131%:%
-%:%281=131%:%
-%:%282=132%:%
-%:%283=133%:%
-%:%284=133%:%
-%:%285=134%:%
-%:%286=134%:%
-%:%287=135%:%
-%:%288=136%:%
-%:%289=136%:%
-%:%290=137%:%
-%:%291=137%:%
-%:%292=138%:%
-%:%293=138%:%
-%:%294=139%:%
-%:%298=143%:%
-%:%299=144%:%
-%:%300=144%:%
-%:%301=145%:%
-%:%302=145%:%
-%:%303=146%:%
-%:%304=147%:%
-%:%305=148%:%
-%:%306=149%:%
-%:%307=150%:%
-%:%308=150%:%
-%:%309=151%:%
-%:%310=152%:%
-%:%311=153%:%
-%:%312=153%:%
-%:%313=154%:%
-%:%316=157%:%
-%:%317=158%:%
-%:%318=158%:%
-%:%319=159%:%
-%:%320=160%:%
-%:%321=160%:%
-%:%322=161%:%
-%:%323=161%:%
-%:%324=162%:%
-%:%325=162%:%
-%:%326=163%:%
-%:%327=163%:%
-%:%328=164%:%
-%:%329=165%:%
-%:%330=166%:%
-%:%331=167%:%
-%:%332=168%:%
-%:%333=169%:%
-%:%334=169%:%
-%:%335=170%:%
-%:%336=171%:%
-%:%337=172%:%
-%:%338=173%:%
-%:%339=174%:%
-%:%341=176%:%
-%:%342=177%:%
-%:%343=177%:%
-%:%344=178%:%
-%:%345=178%:%
-%:%346=179%:%
-%:%347=179%:%
-%:%348=180%:%
-%:%349=180%:%
-%:%350=181%:%
-%:%351=181%:%
-%:%352=182%:%
-%:%353=183%:%
-%:%354=183%:%
-%:%355=184%:%
-%:%356=184%:%
-%:%357=185%:%
-%:%358=185%:%
-%:%359=186%:%
-%:%360=187%:%
-%:%361=188%:%
-%:%362=189%:%
-%:%363=190%:%
-%:%364=190%:%
-%:%365=191%:%
-%:%366=192%:%
-%:%367=193%:%
-%:%368=194%:%
-%:%370=196%:%
-%:%371=197%:%
-%:%372=198%:%
-%:%373=198%:%
-%:%374=199%:%
-%:%375=199%:%
-%:%376=200%:%
-%:%377=200%:%
-%:%378=201%:%
-%:%379=201%:%
-%:%380=202%:%
-%:%381=202%:%
-%:%382=203%:%
-%:%383=204%:%
-%:%384=205%:%
-%:%385=205%:%
-%:%386=206%:%
-%:%387=206%:%
-%:%388=207%:%
-%:%389=207%:%
-%:%390=208%:%
-%:%391=209%:%
-%:%392=210%:%
-%:%393=211%:%
-%:%394=212%:%
-%:%395=213%:%
-%:%396=213%:%
-%:%397=214%:%
-%:%398=215%:%
-%:%399=216%:%
-%:%400=217%:%
-%:%401=218%:%
-%:%402=219%:%
-%:%404=221%:%
-%:%405=222%:%
-%:%406=223%:%
-%:%407=224%:%
-%:%408=224%:%
-%:%409=225%:%
-%:%410=225%:%
-%:%411=226%:%
-%:%412=226%:%
-%:%413=227%:%
-%:%414=227%:%
-%:%415=228%:%
-%:%416=228%:%
-%:%417=229%:%
-%:%418=230%:%
-%:%419=231%:%
-%:%420=231%:%
-%:%421=232%:%
-%:%422=232%:%
-%:%423=233%:%
-%:%424=233%:%
-%:%425=234%:%
-%:%426=235%:%
-%:%427=236%:%
-%:%428=237%:%
-%:%429=238%:%
-%:%430=238%:%
-%:%431=239%:%
-%:%432=240%:%
-%:%433=241%:%
-%:%434=242%:%
-%:%435=243%:%
-%:%436=244%:%
-%:%438=246%:%
-%:%439=247%:%
-%:%441=249%:%
-%:%442=250%:%
-%:%443=250%:%
-%:%444=251%:%
-%:%445=251%:%
-%:%446=252%:%
-%:%447=252%:%
-%:%448=253%:%
-%:%449=253%:%
-%:%450=253%:%
-%:%451=254%:%
-%:%452=254%:%
-%:%453=255%:%
-%:%454=255%:%
-%:%455=256%:%
-%:%456=256%:%
-%:%457=256%:%
-%:%458=257%:%
-%:%459=257%:%
-%:%460=258%:%
-%:%461=259%:%
-%:%462=260%:%
-%:%463=260%:%
-%:%464=261%:%
-%:%465=261%:%
-%:%466=262%:%
-%:%467=262%:%
-%:%468=263%:%
-%:%469=264%:%
-%:%470=265%:%
-%:%471=266%:%
-%:%472=267%:%
-%:%473=267%:%
-%:%474=268%:%
-%:%475=269%:%
-%:%476=270%:%
-%:%477=271%:%
-%:%478=272%:%
-%:%480=274%:%
-%:%481=275%:%
-%:%482=275%:%
-%:%483=276%:%
-%:%484=276%:%
-%:%485=277%:%
-%:%486=277%:%
-%:%487=278%:%
-%:%488=278%:%
-%:%489=279%:%
-%:%490=279%:%
-%:%491=280%:%
-%:%492=281%:%
-%:%493=281%:%
-%:%494=282%:%
-%:%495=282%:%
-%:%496=283%:%
-%:%497=283%:%
-%:%498=284%:%
-%:%499=285%:%
-%:%500=286%:%
-%:%501=287%:%
-%:%502=288%:%
-%:%503=288%:%
-%:%504=289%:%
-%:%505=290%:%
-%:%506=291%:%
-%:%507=292%:%
-%:%508=293%:%
-%:%509=294%:%
-%:%511=296%:%
-%:%512=297%:%
-%:%513=297%:%
-%:%514=298%:%
-%:%515=298%:%
-%:%516=299%:%
-%:%517=299%:%
-%:%518=300%:%
-%:%519=300%:%
-%:%520=301%:%
-%:%521=301%:%
-%:%522=302%:%
-%:%523=303%:%
-%:%524=303%:%
-%:%525=304%:%
-%:%526=304%:%
-%:%527=305%:%
-%:%528=305%:%
-%:%529=306%:%
-%:%530=307%:%
-%:%531=308%:%
-%:%532=309%:%
-%:%533=310%:%
-%:%534=310%:%
-%:%535=311%:%
-%:%536=312%:%
-%:%537=313%:%
-%:%538=314%:%
-%:%539=315%:%
-%:%540=316%:%
-%:%542=318%:%
-%:%543=319%:%
-%:%544=319%:%
-%:%545=320%:%
-%:%546=320%:%
-%:%547=321%:%
-%:%548=321%:%
-%:%549=322%:%
-%:%550=322%:%
-%:%551=323%:%
-%:%552=323%:%
-%:%553=324%:%
-%:%554=325%:%
-%:%555=325%:%
-%:%556=326%:%
-%:%557=326%:%
-%:%558=327%:%
-%:%559=327%:%
-%:%560=328%:%
-%:%561=329%:%
-%:%562=330%:%
-%:%563=331%:%
-%:%564=332%:%
-%:%565=332%:%
-%:%566=333%:%
-%:%567=334%:%
-%:%568=335%:%
-%:%569=336%:%
-%:%570=337%:%
-%:%571=338%:%
-%:%572=339%:%
-%:%574=341%:%
-%:%575=342%:%
-%:%576=342%:%
-%:%577=343%:%
-%:%578=343%:%
-%:%579=344%:%
-%:%580=344%:%
-%:%581=345%:%
-%:%582=345%:%
-%:%583=346%:%
-%:%584=346%:%
-%:%585=347%:%
-%:%586=348%:%
-%:%587=348%:%
-%:%588=349%:%
-%:%594=349%:%
-%:%599=350%:%
-%:%604=351%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Elect_Module.tex b/virage/src/test/resources/old_theories/output/document/Elect_Module.tex
deleted file mode 100644
index 34d1c97..0000000
--- a/virage/src/test/resources/old_theories/output/document/Elect_Module.tex
+++ /dev/null
@@ -1,190 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Elect{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Elect Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Elect{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The elect module is not concerned about the voter's ballots, and
-just elects all alternatives. It is primarily used in sequence after
-an electoral module that only defers alternatives to finalize the decision,
-thereby inducing a proper voting rule in the social choice sense.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elect{\isacharunderscore}{\kern0pt}module\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Elect_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%54=20%:%
-%:%64=22%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%73=25%:%
-%:%83=27%:%
-%:%84=27%:%
-%:%87=28%:%
-%:%91=28%:%
-%:%92=28%:%
-%:%93=29%:%
-%:%94=29%:%
-%:%108=31%:%
-%:%118=33%:%
-%:%119=33%:%
-%:%122=34%:%
-%:%126=34%:%
-%:%127=34%:%
-%:%128=35%:%
-%:%129=35%:%
-%:%134=35%:%
-%:%139=36%:%
-%:%144=37%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Electoral_Module.tex b/virage/src/test/resources/old_theories/output/document/Electoral_Module.tex
deleted file mode 100644
index b44bb38..0000000
--- a/virage/src/test/resources/old_theories/output/document/Electoral_Module.tex
+++ /dev/null
@@ -1,3358 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Electoral{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-\isanewline
-%
-\isamarkupchapter{Component Types%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Electoral Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Electoral{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Result{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-Electoral modules are the principal component type of the composable modules
-voting framework, as they are a generalization of voting rules in the sense of
-social choice functions.
-These are only the types used for electoral modules. Further restrictions are
-encompassed by the electoral-module predicate.
-
-An electoral module does not need to make final decisions for all alternatives,
-but can instead defer the decision for some or all of them to other modules.
-Hence, electoral modules partition the received (possibly empty) set of
-alternatives into elected, rejected and deferred alternatives. In particular,
-any of those sets, e.g., the set of winning (elected) alternatives, may also
-be left empty, as long as they collectively still hold all the received
-alternatives. Just like a voting rule, an electoral module also receives a
-profile which holds the voters’ preferences, which, unlike a voting rule,
-consider only the (sub-)set of alternatives that the module receives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Definitions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymequiv}\ {\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}modI{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isasymlbrakk}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ elect\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymequiv}\ elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ reject\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isasymequiv}\ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer{\isachardoublequoteclose}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymequiv}\ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defers\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defers\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ rejects\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}rejects\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ eliminates\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}eliminates\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isachargreater}{\kern0pt}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ elects\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elects\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ A\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ q{\isachardot}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ q\ a\ {\isasymlongrightarrow}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Equivalence Definitions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ reject\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ defer\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}leq{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prof{\isacharunderscore}{\kern0pt}leq{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ reject\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymnotin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prof{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymnotin}\ reject\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ n\ A\ p\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ reject\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ ele{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rej{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ def\ ele\ rej\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ UnI{\isadigit{1}}\ asm\ fstI\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ UnI{\isadigit{1}}\ asm\ fstI\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \ \ \ \ \ \ sndI\ subsetD\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ set{\isacharunderscore}{\kern0pt}partit\ partit\ sndI\ subsetD\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ asm{\isadigit{3}}\ fst{\isacharunderscore}{\kern0pt}conv\ partit\isanewline
-\ \ \ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}partit\ snd{\isacharunderscore}{\kern0pt}conv\ Un{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ profile\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ prof{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}exhaust{\isacharunderscore}{\kern0pt}sel\ DiffE\ UnCI\ asm{\isadigit{1}}\ asm{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ module\ profile\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}p\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isasymnot}\ disjoint{\isadigit{3}}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D{\isachardot}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharcomma}{\kern0pt}\ C{\isacharcomma}{\kern0pt}\ D{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ B\ {\isasyminter}\ C\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ B\ {\isasyminter}\ D\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ C\ {\isasyminter}\ D\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ prof{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ disj{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disj{\isadigit{3}}\ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ AA\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ AAa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ AAb\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{where}\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ AAa\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ AAb\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ AA\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ AAa\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ AA\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ AAb\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ AAa\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ AAb\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disj{\isadigit{3}}\ eq{\isacharunderscore}{\kern0pt}snd{\isacharunderscore}{\kern0pt}iff\ fstI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ module\ profile\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m\ prof{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}partit\ partit\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ profile\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ prof{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}exhaust{\isacharunderscore}{\kern0pt}sel\ DiffE\ UnCI\ asm{\isadigit{1}}\ asm{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ module\ profile\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}p\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}supI{\isadigit{1}}\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{1}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}supI{\isadigit{1}}\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{2}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ new{\isacharunderscore}{\kern0pt}A\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ new{\isacharunderscore}{\kern0pt}A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ infinite{\isacharunderscore}{\kern0pt}super\isanewline
-\ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound\ module\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ upper{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}bounds{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}result{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ A\ {\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ elec{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymunion}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ alternative{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}elected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ DiffI\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ alternative\isanewline
-\ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}elected\ not{\isacharunderscore}{\kern0pt}rejected\isanewline
-\ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ n\ A\ p\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ m\ A\ p\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ IntI\ assms\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ alternative{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p\ {\isasymor}\ x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ alternative\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \ \ e{\isacharunderscore}{\kern0pt}mod\ not{\isacharunderscore}{\kern0pt}rejected\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ eliminating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ leftover{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymsubset}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}iff\ Diff{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ eliminates{\isacharunderscore}{\kern0pt}def\ eliminating\ eq{\isacharunderscore}{\kern0pt}iff\ leftover{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}zero\ f{\isacharunderscore}{\kern0pt}prof\ psubsetI\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}alts{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}profs{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}results{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ input{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ elect\ m\ A\ p{\isachardot}{\kern0pt}\ a\ {\isasymin}\ elect\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ elect\ m\ A\ q{\isachardot}{\kern0pt}\ a\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ reject\ m\ A\ p{\isachardot}{\kern0pt}\ a\ {\isasymin}\ reject\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ reject\ m\ A\ q{\isachardot}{\kern0pt}\ a\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ input\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ defer\ m\ A\ p{\isachardot}{\kern0pt}\ a\ {\isasymin}\ defer\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ defer\ m\ A\ q{\isachardot}{\kern0pt}\ a\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ input\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}collapse\ subsetI\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ elect\ n\ A\ q{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ n\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ n\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}n\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}m\ prod{\isachardot}{\kern0pt}collapse\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}n\ prod{\isachardot}{\kern0pt}collapse\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ n\ A\ q\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}p\ reject{\isacharunderscore}{\kern0pt}q\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ prod{\isacharunderscore}{\kern0pt}eqI{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Non-Blocking%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ reject\ m\ A\ p\ {\isasymnoteq}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ electing\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}electing\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}alt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ one{\isacharunderscore}{\kern0pt}alt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}right\ Int{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}right\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ electing\ electing{\isacharunderscore}{\kern0pt}def\ inf{\isachardot}{\kern0pt}orderE\isanewline
-\ \ \ \ \ \ \ \ one{\isacharunderscore}{\kern0pt}alt\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}disjoint\ Diff{\isacharunderscore}{\kern0pt}empty\ Int{\isacharunderscore}{\kern0pt}absorb{\isadigit{2}}\ electing\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ rejecting{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ finite{\isacharunderscore}{\kern0pt}subset\ non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}empty\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ rejecting\isanewline
-\ \ \ \ \ \ \ \ rejects{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ eliminating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ finite{\isacharunderscore}{\kern0pt}subset\ non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}empty\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ eliminating\ eliminates{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}deciding\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}deciding\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ m\ {\isasymand}\ defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ decrementing\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}decrementing\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymforall}\ A\ p\ {\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ A\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymand}\ finite\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}compatibility\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymand}\ finite\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}w\ {\isasymnotin}\ reject\ m\ A\ p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}l{\isachardot}{\kern0pt}\ {\isasymnot}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ l\ {\isasymlongrightarrow}\ l\ {\isasymnotin}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}w\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}l{\isachardot}{\kern0pt}\ {\isasymnot}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ l\ {\isasymlongrightarrow}\ l\ {\isasymin}\ reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite\ A\ {\isasymand}\ w\ {\isasymin}\ defer\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ w\ {\isasymin}\ defer\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}S{\isachardot}{\kern0pt}\ finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ m\ A\ p\ {\isasymor}\ elect\ m\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p\ {\isasymor}\ defer\ m\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Inference Rules%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ ccomp{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}dd{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}winner{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ ccomp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ dd{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}deciding\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ dd\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ c{\isacharunderscore}{\kern0pt}win{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ equals{\isadigit{0}}D\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ insert{\isacharunderscore}{\kern0pt}subset\ c{\isacharunderscore}{\kern0pt}win\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}w\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}l\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ l\ {\isasymnoteq}\ w\ {\isasymand}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnotin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ c{\isacharunderscore}{\kern0pt}win\ equals{\isadigit{0}}D\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}elect\ \isacommand{have}\isamarkupfalse%
-\ one{\isacharunderscore}{\kern0pt}side{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}w\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ccomp\ condorcet{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ c{\isacharunderscore}{\kern0pt}win\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ ccomp\ \isacommand{have}\isamarkupfalse%
-\ other{\isacharunderscore}{\kern0pt}side{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ c{\isacharunderscore}{\kern0pt}win\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ one{\isacharunderscore}{\kern0pt}side{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ ccomp{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}dd{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ ccomp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ dd{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}deciding\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def{\isacharcomma}{\kern0pt}\ auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finiteness{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ assm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm\ finiteness\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ dd\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ dd\ ccomp\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ccomp{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}dd{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}winner\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ wins\ d\ p\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finiteness\ prof{\isacharunderscore}{\kern0pt}A\ winner\ Collect{\isacharunderscore}{\kern0pt}cong\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ d\ {\isacharless}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ d\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}d\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}compat{\isacharunderscore}{\kern0pt}comm{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ n\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}S{\isachardot}{\kern0pt}\ finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ A\ \isakeyword{where}\ old{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharparenleft}{\kern0pt}S{\isacharminus}{\kern0pt}A{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ double{\isacharunderscore}{\kern0pt}diff\ order{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ dl{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ defer{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Social Choice Properties%
-}
-\isamarkuptrue%
-%
-\isamarkupsubsubsection{Condorcet Consistency%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}e\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ e{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymLongrightarrow}\ electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ cc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ assm{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymLongrightarrow}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ cc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ x\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ wins{\isachardot}{\kern0pt}elims\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ assm{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ {\isasymforall}A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p\ w{\isachardot}{\kern0pt}\ finite\ A\ {\isasymand}\ profile\ A\ p\ {\isasymand}\ w\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm{\isadigit{0}}\ e{\isacharunderscore}{\kern0pt}mod\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsubsection{(Weak) Monotonicity%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite\ A\ {\isasymand}\ w\ {\isasymin}\ elect\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ w\ {\isasymin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsubsection{Homogeneity%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ times\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}times\ n\ l\ {\isacharequal}{\kern0pt}\ concat\ {\isacharparenleft}{\kern0pt}replicate\ n\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ homogeneity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-{\isachardoublequoteopen}homogeneity\ m\ {\isasymequiv}\isanewline
-\ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ n\ {\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ n\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ {\isacharparenleft}{\kern0pt}times\ n\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Component_Types/Electoral_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%13=6%:%
-%:%15=9%:%
-%:%19=11%:%
-%:%35=13%:%
-%:%36=13%:%
-%:%37=14%:%
-%:%38=15%:%
-%:%39=16%:%
-%:%40=17%:%
-%:%49=20%:%
-%:%50=21%:%
-%:%51=22%:%
-%:%52=23%:%
-%:%53=24%:%
-%:%54=25%:%
-%:%55=26%:%
-%:%56=27%:%
-%:%57=28%:%
-%:%58=29%:%
-%:%59=30%:%
-%:%60=31%:%
-%:%61=32%:%
-%:%62=33%:%
-%:%63=34%:%
-%:%72=36%:%
-%:%82=39%:%
-%:%83=39%:%
-%:%90=41%:%
-%:%100=51%:%
-%:%101=51%:%
-%:%102=52%:%
-%:%103=53%:%
-%:%104=54%:%
-%:%105=54%:%
-%:%106=55%:%
-%:%107=56%:%
-%:%110=57%:%
-%:%114=57%:%
-%:%115=57%:%
-%:%116=58%:%
-%:%117=58%:%
-%:%122=58%:%
-%:%125=59%:%
-%:%126=63%:%
-%:%127=64%:%
-%:%128=64%:%
-%:%129=65%:%
-%:%130=66%:%
-%:%131=67%:%
-%:%132=68%:%
-%:%133=68%:%
-%:%134=69%:%
-%:%135=70%:%
-%:%136=71%:%
-%:%137=72%:%
-%:%138=72%:%
-%:%139=73%:%
-%:%140=74%:%
-%:%141=75%:%
-%:%142=79%:%
-%:%143=80%:%
-%:%144=80%:%
-%:%145=81%:%
-%:%148=84%:%
-%:%149=85%:%
-%:%150=89%:%
-%:%151=90%:%
-%:%152=90%:%
-%:%153=91%:%
-%:%155=93%:%
-%:%156=94%:%
-%:%157=98%:%
-%:%158=99%:%
-%:%159=99%:%
-%:%160=100%:%
-%:%162=102%:%
-%:%163=103%:%
-%:%164=107%:%
-%:%165=108%:%
-%:%166=108%:%
-%:%167=109%:%
-%:%169=111%:%
-%:%170=112%:%
-%:%171=130%:%
-%:%172=131%:%
-%:%173=135%:%
-%:%174=136%:%
-%:%175=136%:%
-%:%176=137%:%
-%:%184=140%:%
-%:%194=142%:%
-%:%195=142%:%
-%:%196=143%:%
-%:%197=144%:%
-%:%201=148%:%
-%:%202=149%:%
-%:%203=150%:%
-%:%204=150%:%
-%:%205=151%:%
-%:%206=152%:%
-%:%209=155%:%
-%:%210=156%:%
-%:%211=157%:%
-%:%212=157%:%
-%:%213=158%:%
-%:%214=159%:%
-%:%217=162%:%
-%:%218=163%:%
-%:%219=164%:%
-%:%220=164%:%
-%:%221=165%:%
-%:%222=166%:%
-%:%233=172%:%
-%:%243=174%:%
-%:%244=174%:%
-%:%245=175%:%
-%:%246=176%:%
-%:%247=177%:%
-%:%248=178%:%
-%:%249=179%:%
-%:%252=180%:%
-%:%256=180%:%
-%:%257=180%:%
-%:%258=181%:%
-%:%259=181%:%
-%:%264=181%:%
-%:%267=182%:%
-%:%268=183%:%
-%:%269=183%:%
-%:%270=184%:%
-%:%271=185%:%
-%:%272=186%:%
-%:%273=187%:%
-%:%276=188%:%
-%:%280=188%:%
-%:%281=188%:%
-%:%282=189%:%
-%:%283=189%:%
-%:%288=189%:%
-%:%291=190%:%
-%:%292=191%:%
-%:%293=191%:%
-%:%294=192%:%
-%:%295=193%:%
-%:%296=194%:%
-%:%297=195%:%
-%:%304=196%:%
-%:%305=196%:%
-%:%306=197%:%
-%:%307=197%:%
-%:%308=198%:%
-%:%309=199%:%
-%:%310=199%:%
-%:%311=200%:%
-%:%312=201%:%
-%:%313=201%:%
-%:%314=202%:%
-%:%316=204%:%
-%:%317=205%:%
-%:%318=205%:%
-%:%319=206%:%
-%:%320=206%:%
-%:%321=206%:%
-%:%322=207%:%
-%:%323=208%:%
-%:%324=208%:%
-%:%325=209%:%
-%:%326=209%:%
-%:%327=210%:%
-%:%328=210%:%
-%:%329=211%:%
-%:%330=211%:%
-%:%331=212%:%
-%:%332=212%:%
-%:%333=213%:%
-%:%334=213%:%
-%:%335=214%:%
-%:%336=214%:%
-%:%337=215%:%
-%:%338=216%:%
-%:%339=216%:%
-%:%340=217%:%
-%:%341=218%:%
-%:%342=218%:%
-%:%343=219%:%
-%:%345=221%:%
-%:%346=222%:%
-%:%347=222%:%
-%:%348=223%:%
-%:%349=223%:%
-%:%350=223%:%
-%:%351=224%:%
-%:%352=225%:%
-%:%353=225%:%
-%:%354=226%:%
-%:%355=226%:%
-%:%356=227%:%
-%:%357=227%:%
-%:%358=228%:%
-%:%359=228%:%
-%:%360=229%:%
-%:%361=230%:%
-%:%362=230%:%
-%:%363=231%:%
-%:%364=231%:%
-%:%365=232%:%
-%:%366=232%:%
-%:%367=233%:%
-%:%368=234%:%
-%:%369=234%:%
-%:%370=235%:%
-%:%371=236%:%
-%:%372=236%:%
-%:%373=237%:%
-%:%375=239%:%
-%:%376=240%:%
-%:%377=240%:%
-%:%378=241%:%
-%:%379=241%:%
-%:%380=241%:%
-%:%381=242%:%
-%:%382=243%:%
-%:%383=243%:%
-%:%384=244%:%
-%:%385=244%:%
-%:%386=245%:%
-%:%387=245%:%
-%:%388=246%:%
-%:%389=246%:%
-%:%390=247%:%
-%:%391=247%:%
-%:%392=248%:%
-%:%393=248%:%
-%:%394=249%:%
-%:%395=249%:%
-%:%396=250%:%
-%:%397=251%:%
-%:%398=251%:%
-%:%399=252%:%
-%:%400=253%:%
-%:%401=254%:%
-%:%402=255%:%
-%:%403=255%:%
-%:%404=256%:%
-%:%406=258%:%
-%:%407=259%:%
-%:%408=259%:%
-%:%409=260%:%
-%:%410=260%:%
-%:%411=260%:%
-%:%412=261%:%
-%:%413=262%:%
-%:%414=262%:%
-%:%415=263%:%
-%:%416=263%:%
-%:%417=264%:%
-%:%418=264%:%
-%:%419=265%:%
-%:%420=265%:%
-%:%421=266%:%
-%:%422=267%:%
-%:%423=267%:%
-%:%424=268%:%
-%:%430=268%:%
-%:%433=269%:%
-%:%434=270%:%
-%:%435=270%:%
-%:%436=271%:%
-%:%437=272%:%
-%:%438=273%:%
-%:%439=274%:%
-%:%440=275%:%
-%:%442=277%:%
-%:%449=278%:%
-%:%450=278%:%
-%:%451=279%:%
-%:%452=279%:%
-%:%453=280%:%
-%:%454=281%:%
-%:%455=281%:%
-%:%456=282%:%
-%:%457=283%:%
-%:%458=284%:%
-%:%459=284%:%
-%:%460=285%:%
-%:%462=287%:%
-%:%463=288%:%
-%:%464=288%:%
-%:%465=289%:%
-%:%466=289%:%
-%:%467=289%:%
-%:%468=290%:%
-%:%469=291%:%
-%:%470=291%:%
-%:%471=292%:%
-%:%472=292%:%
-%:%473=293%:%
-%:%474=293%:%
-%:%475=293%:%
-%:%476=294%:%
-%:%477=295%:%
-%:%478=295%:%
-%:%479=296%:%
-%:%480=296%:%
-%:%481=296%:%
-%:%482=297%:%
-%:%483=298%:%
-%:%484=298%:%
-%:%485=299%:%
-%:%486=299%:%
-%:%487=300%:%
-%:%488=300%:%
-%:%489=301%:%
-%:%490=301%:%
-%:%491=302%:%
-%:%492=303%:%
-%:%493=304%:%
-%:%494=304%:%
-%:%495=305%:%
-%:%496=305%:%
-%:%497=306%:%
-%:%498=306%:%
-%:%499=307%:%
-%:%500=308%:%
-%:%501=308%:%
-%:%502=309%:%
-%:%503=310%:%
-%:%504=311%:%
-%:%505=311%:%
-%:%506=312%:%
-%:%508=314%:%
-%:%509=315%:%
-%:%510=315%:%
-%:%511=316%:%
-%:%512=316%:%
-%:%513=317%:%
-%:%515=319%:%
-%:%516=320%:%
-%:%517=320%:%
-%:%518=321%:%
-%:%519=321%:%
-%:%520=321%:%
-%:%521=322%:%
-%:%522=323%:%
-%:%523=323%:%
-%:%524=324%:%
-%:%525=324%:%
-%:%526=324%:%
-%:%527=325%:%
-%:%528=326%:%
-%:%529=326%:%
-%:%530=327%:%
-%:%531=327%:%
-%:%532=328%:%
-%:%533=328%:%
-%:%534=329%:%
-%:%535=330%:%
-%:%536=330%:%
-%:%537=331%:%
-%:%538=331%:%
-%:%539=332%:%
-%:%540=333%:%
-%:%541=333%:%
-%:%542=334%:%
-%:%543=334%:%
-%:%544=335%:%
-%:%545=336%:%
-%:%546=336%:%
-%:%547=337%:%
-%:%548=337%:%
-%:%549=338%:%
-%:%550=338%:%
-%:%551=338%:%
-%:%552=339%:%
-%:%553=340%:%
-%:%554=341%:%
-%:%555=342%:%
-%:%556=343%:%
-%:%560=347%:%
-%:%561=348%:%
-%:%562=348%:%
-%:%563=349%:%
-%:%564=349%:%
-%:%565=350%:%
-%:%566=350%:%
-%:%568=352%:%
-%:%569=353%:%
-%:%570=353%:%
-%:%571=354%:%
-%:%572=354%:%
-%:%573=355%:%
-%:%574=355%:%
-%:%575=356%:%
-%:%576=356%:%
-%:%577=357%:%
-%:%578=358%:%
-%:%579=358%:%
-%:%580=359%:%
-%:%581=359%:%
-%:%582=360%:%
-%:%583=360%:%
-%:%584=361%:%
-%:%585=362%:%
-%:%586=362%:%
-%:%587=363%:%
-%:%588=364%:%
-%:%589=365%:%
-%:%590=365%:%
-%:%591=366%:%
-%:%593=368%:%
-%:%594=369%:%
-%:%595=369%:%
-%:%596=370%:%
-%:%597=370%:%
-%:%598=370%:%
-%:%599=371%:%
-%:%600=372%:%
-%:%601=372%:%
-%:%602=373%:%
-%:%603=373%:%
-%:%604=374%:%
-%:%605=374%:%
-%:%606=374%:%
-%:%607=375%:%
-%:%608=376%:%
-%:%609=376%:%
-%:%610=377%:%
-%:%611=377%:%
-%:%612=377%:%
-%:%613=378%:%
-%:%614=379%:%
-%:%615=379%:%
-%:%616=380%:%
-%:%617=380%:%
-%:%618=381%:%
-%:%619=381%:%
-%:%620=382%:%
-%:%621=382%:%
-%:%622=383%:%
-%:%623=384%:%
-%:%624=385%:%
-%:%625=385%:%
-%:%626=386%:%
-%:%632=386%:%
-%:%635=387%:%
-%:%636=388%:%
-%:%637=388%:%
-%:%638=389%:%
-%:%639=390%:%
-%:%640=391%:%
-%:%641=392%:%
-%:%644=393%:%
-%:%648=393%:%
-%:%649=393%:%
-%:%650=394%:%
-%:%651=394%:%
-%:%656=394%:%
-%:%659=395%:%
-%:%660=396%:%
-%:%661=396%:%
-%:%662=397%:%
-%:%663=398%:%
-%:%664=399%:%
-%:%665=400%:%
-%:%668=401%:%
-%:%672=401%:%
-%:%673=401%:%
-%:%674=402%:%
-%:%675=402%:%
-%:%680=402%:%
-%:%683=403%:%
-%:%684=404%:%
-%:%685=404%:%
-%:%686=405%:%
-%:%687=406%:%
-%:%688=407%:%
-%:%689=408%:%
-%:%692=409%:%
-%:%696=409%:%
-%:%697=409%:%
-%:%698=410%:%
-%:%699=410%:%
-%:%704=410%:%
-%:%707=411%:%
-%:%708=412%:%
-%:%709=412%:%
-%:%710=413%:%
-%:%711=414%:%
-%:%712=415%:%
-%:%713=416%:%
-%:%714=417%:%
-%:%717=418%:%
-%:%721=418%:%
-%:%722=418%:%
-%:%723=419%:%
-%:%724=420%:%
-%:%725=420%:%
-%:%730=420%:%
-%:%733=421%:%
-%:%734=425%:%
-%:%735=426%:%
-%:%736=426%:%
-%:%737=427%:%
-%:%738=428%:%
-%:%739=429%:%
-%:%740=430%:%
-%:%741=431%:%
-%:%743=433%:%
-%:%746=434%:%
-%:%750=434%:%
-%:%751=434%:%
-%:%752=435%:%
-%:%757=435%:%
-%:%760=436%:%
-%:%761=437%:%
-%:%762=437%:%
-%:%763=438%:%
-%:%764=439%:%
-%:%765=440%:%
-%:%766=441%:%
-%:%773=442%:%
-%:%774=442%:%
-%:%775=443%:%
-%:%776=443%:%
-%:%777=443%:%
-%:%778=444%:%
-%:%779=444%:%
-%:%780=445%:%
-%:%781=445%:%
-%:%782=446%:%
-%:%783=446%:%
-%:%784=447%:%
-%:%785=447%:%
-%:%786=448%:%
-%:%787=448%:%
-%:%788=449%:%
-%:%789=449%:%
-%:%790=449%:%
-%:%791=449%:%
-%:%792=450%:%
-%:%793=451%:%
-%:%794=452%:%
-%:%795=452%:%
-%:%796=453%:%
-%:%797=453%:%
-%:%798=454%:%
-%:%799=454%:%
-%:%800=454%:%
-%:%801=455%:%
-%:%802=455%:%
-%:%803=456%:%
-%:%809=456%:%
-%:%812=457%:%
-%:%813=458%:%
-%:%814=458%:%
-%:%815=459%:%
-%:%816=460%:%
-%:%817=461%:%
-%:%818=462%:%
-%:%825=463%:%
-%:%826=463%:%
-%:%827=464%:%
-%:%828=464%:%
-%:%829=464%:%
-%:%830=465%:%
-%:%831=465%:%
-%:%832=466%:%
-%:%833=466%:%
-%:%834=467%:%
-%:%835=468%:%
-%:%836=468%:%
-%:%837=469%:%
-%:%838=469%:%
-%:%839=470%:%
-%:%840=470%:%
-%:%841=471%:%
-%:%842=471%:%
-%:%843=472%:%
-%:%844=472%:%
-%:%845=473%:%
-%:%846=473%:%
-%:%847=473%:%
-%:%848=473%:%
-%:%849=474%:%
-%:%850=475%:%
-%:%851=476%:%
-%:%852=476%:%
-%:%853=477%:%
-%:%854=477%:%
-%:%855=478%:%
-%:%856=478%:%
-%:%857=478%:%
-%:%858=479%:%
-%:%859=479%:%
-%:%860=480%:%
-%:%866=480%:%
-%:%869=481%:%
-%:%870=482%:%
-%:%871=482%:%
-%:%872=483%:%
-%:%873=484%:%
-%:%874=485%:%
-%:%875=486%:%
-%:%882=487%:%
-%:%883=487%:%
-%:%884=488%:%
-%:%885=488%:%
-%:%886=488%:%
-%:%887=489%:%
-%:%888=489%:%
-%:%889=490%:%
-%:%890=490%:%
-%:%891=491%:%
-%:%892=491%:%
-%:%893=492%:%
-%:%894=492%:%
-%:%895=493%:%
-%:%896=493%:%
-%:%897=493%:%
-%:%898=493%:%
-%:%899=494%:%
-%:%900=495%:%
-%:%901=496%:%
-%:%902=496%:%
-%:%903=497%:%
-%:%904=497%:%
-%:%905=498%:%
-%:%906=498%:%
-%:%907=498%:%
-%:%908=499%:%
-%:%909=499%:%
-%:%910=500%:%
-%:%916=500%:%
-%:%919=501%:%
-%:%920=502%:%
-%:%921=502%:%
-%:%922=503%:%
-%:%923=504%:%
-%:%924=505%:%
-%:%925=506%:%
-%:%926=507%:%
-%:%927=508%:%
-%:%928=509%:%
-%:%931=510%:%
-%:%935=510%:%
-%:%936=510%:%
-%:%937=511%:%
-%:%938=512%:%
-%:%939=513%:%
-%:%940=513%:%
-%:%945=513%:%
-%:%948=514%:%
-%:%949=515%:%
-%:%950=515%:%
-%:%951=516%:%
-%:%952=517%:%
-%:%955=518%:%
-%:%959=518%:%
-%:%960=518%:%
-%:%961=519%:%
-%:%962=520%:%
-%:%963=520%:%
-%:%968=520%:%
-%:%971=521%:%
-%:%972=522%:%
-%:%973=522%:%
-%:%974=523%:%
-%:%975=524%:%
-%:%976=525%:%
-%:%977=526%:%
-%:%978=527%:%
-%:%979=528%:%
-%:%982=529%:%
-%:%986=529%:%
-%:%987=529%:%
-%:%988=530%:%
-%:%989=531%:%
-%:%990=531%:%
-%:%995=531%:%
-%:%998=532%:%
-%:%999=533%:%
-%:%1000=533%:%
-%:%1001=534%:%
-%:%1002=535%:%
-%:%1003=536%:%
-%:%1004=537%:%
-%:%1005=538%:%
-%:%1008=539%:%
-%:%1012=539%:%
-%:%1013=539%:%
-%:%1014=540%:%
-%:%1015=541%:%
-%:%1016=542%:%
-%:%1017=542%:%
-%:%1022=542%:%
-%:%1025=543%:%
-%:%1026=544%:%
-%:%1027=544%:%
-%:%1028=545%:%
-%:%1029=546%:%
-%:%1030=547%:%
-%:%1031=548%:%
-%:%1032=549%:%
-%:%1039=550%:%
-%:%1040=550%:%
-%:%1041=551%:%
-%:%1042=551%:%
-%:%1043=552%:%
-%:%1044=552%:%
-%:%1045=553%:%
-%:%1046=553%:%
-%:%1047=554%:%
-%:%1048=554%:%
-%:%1049=554%:%
-%:%1050=555%:%
-%:%1051=555%:%
-%:%1052=556%:%
-%:%1053=557%:%
-%:%1054=558%:%
-%:%1055=558%:%
-%:%1056=559%:%
-%:%1057=559%:%
-%:%1058=559%:%
-%:%1059=560%:%
-%:%1060=560%:%
-%:%1061=561%:%
-%:%1062=561%:%
-%:%1063=562%:%
-%:%1064=562%:%
-%:%1065=562%:%
-%:%1066=563%:%
-%:%1067=563%:%
-%:%1068=564%:%
-%:%1069=565%:%
-%:%1070=566%:%
-%:%1071=566%:%
-%:%1072=567%:%
-%:%1073=567%:%
-%:%1074=567%:%
-%:%1075=568%:%
-%:%1076=568%:%
-%:%1077=569%:%
-%:%1078=569%:%
-%:%1079=570%:%
-%:%1080=570%:%
-%:%1081=570%:%
-%:%1082=571%:%
-%:%1083=571%:%
-%:%1084=572%:%
-%:%1085=573%:%
-%:%1086=574%:%
-%:%1087=574%:%
-%:%1088=575%:%
-%:%1089=575%:%
-%:%1090=575%:%
-%:%1091=576%:%
-%:%1092=576%:%
-%:%1093=577%:%
-%:%1094=577%:%
-%:%1095=578%:%
-%:%1101=578%:%
-%:%1104=579%:%
-%:%1105=580%:%
-%:%1106=580%:%
-%:%1107=581%:%
-%:%1108=582%:%
-%:%1109=583%:%
-%:%1110=584%:%
-%:%1111=585%:%
-%:%1112=586%:%
-%:%1113=587%:%
-%:%1114=588%:%
-%:%1121=589%:%
-%:%1122=589%:%
-%:%1123=590%:%
-%:%1124=590%:%
-%:%1125=591%:%
-%:%1126=592%:%
-%:%1127=592%:%
-%:%1128=593%:%
-%:%1129=593%:%
-%:%1130=594%:%
-%:%1131=594%:%
-%:%1132=595%:%
-%:%1133=596%:%
-%:%1134=596%:%
-%:%1135=597%:%
-%:%1136=597%:%
-%:%1137=598%:%
-%:%1138=598%:%
-%:%1139=599%:%
-%:%1140=600%:%
-%:%1141=600%:%
-%:%1142=601%:%
-%:%1143=601%:%
-%:%1144=602%:%
-%:%1145=602%:%
-%:%1146=602%:%
-%:%1147=603%:%
-%:%1148=604%:%
-%:%1149=604%:%
-%:%1150=605%:%
-%:%1151=605%:%
-%:%1152=606%:%
-%:%1153=606%:%
-%:%1154=607%:%
-%:%1155=608%:%
-%:%1156=608%:%
-%:%1157=609%:%
-%:%1158=609%:%
-%:%1159=610%:%
-%:%1160=610%:%
-%:%1161=610%:%
-%:%1162=611%:%
-%:%1163=612%:%
-%:%1164=612%:%
-%:%1165=613%:%
-%:%1166=613%:%
-%:%1167=614%:%
-%:%1168=614%:%
-%:%1169=615%:%
-%:%1170=616%:%
-%:%1171=616%:%
-%:%1172=617%:%
-%:%1173=618%:%
-%:%1174=618%:%
-%:%1175=619%:%
-%:%1176=619%:%
-%:%1177=620%:%
-%:%1178=621%:%
-%:%1179=621%:%
-%:%1180=622%:%
-%:%1181=623%:%
-%:%1182=623%:%
-%:%1183=624%:%
-%:%1184=624%:%
-%:%1185=624%:%
-%:%1186=625%:%
-%:%1187=625%:%
-%:%1188=626%:%
-%:%1203=628%:%
-%:%1213=634%:%
-%:%1214=634%:%
-%:%1215=635%:%
-%:%1225=640%:%
-%:%1235=646%:%
-%:%1236=646%:%
-%:%1237=647%:%
-%:%1239=649%:%
-%:%1240=650%:%
-%:%1241=651%:%
-%:%1242=651%:%
-%:%1243=652%:%
-%:%1244=653%:%
-%:%1245=654%:%
-%:%1246=655%:%
-%:%1247=656%:%
-%:%1250=657%:%
-%:%1254=657%:%
-%:%1255=657%:%
-%:%1256=658%:%
-%:%1257=659%:%
-%:%1258=660%:%
-%:%1259=660%:%
-%:%1264=660%:%
-%:%1267=661%:%
-%:%1268=662%:%
-%:%1269=662%:%
-%:%1270=663%:%
-%:%1271=664%:%
-%:%1274=665%:%
-%:%1278=665%:%
-%:%1279=665%:%
-%:%1280=666%:%
-%:%1281=667%:%
-%:%1282=668%:%
-%:%1283=668%:%
-%:%1297=670%:%
-%:%1307=676%:%
-%:%1308=676%:%
-%:%1309=677%:%
-%:%1310=678%:%
-%:%1311=679%:%
-%:%1312=680%:%
-%:%1313=680%:%
-%:%1314=681%:%
-%:%1315=682%:%
-%:%1316=683%:%
-%:%1317=684%:%
-%:%1318=685%:%
-%:%1319=686%:%
-%:%1322=687%:%
-%:%1326=687%:%
-%:%1327=687%:%
-%:%1328=688%:%
-%:%1329=689%:%
-%:%1330=690%:%
-%:%1331=691%:%
-%:%1332=691%:%
-%:%1337=691%:%
-%:%1340=692%:%
-%:%1341=693%:%
-%:%1342=693%:%
-%:%1343=694%:%
-%:%1344=695%:%
-%:%1345=696%:%
-%:%1346=697%:%
-%:%1347=698%:%
-%:%1348=699%:%
-%:%1351=700%:%
-%:%1355=700%:%
-%:%1356=700%:%
-%:%1357=701%:%
-%:%1358=702%:%
-%:%1359=703%:%
-%:%1360=704%:%
-%:%1361=704%:%
-%:%1366=704%:%
-%:%1369=705%:%
-%:%1370=712%:%
-%:%1371=713%:%
-%:%1372=713%:%
-%:%1373=714%:%
-%:%1374=715%:%
-%:%1375=716%:%
-%:%1376=720%:%
-%:%1377=721%:%
-%:%1378=721%:%
-%:%1379=722%:%
-%:%1382=725%:%
-%:%1383=726%:%
-%:%1384=727%:%
-%:%1385=727%:%
-%:%1386=728%:%
-%:%1392=734%:%
-%:%1393=735%:%
-%:%1394=736%:%
-%:%1395=736%:%
-%:%1396=737%:%
-%:%1402=743%:%
-%:%1403=744%:%
-%:%1404=748%:%
-%:%1405=749%:%
-%:%1406=749%:%
-%:%1407=750%:%
-%:%1410=753%:%
-%:%1411=754%:%
-%:%1412=758%:%
-%:%1413=759%:%
-%:%1414=759%:%
-%:%1415=760%:%
-%:%1418=763%:%
-%:%1419=764%:%
-%:%1420=769%:%
-%:%1421=770%:%
-%:%1422=770%:%
-%:%1423=771%:%
-%:%1424=772%:%
-%:%1431=779%:%
-%:%1432=780%:%
-%:%1433=785%:%
-%:%1434=786%:%
-%:%1435=786%:%
-%:%1436=787%:%
-%:%1439=790%:%
-%:%1440=791%:%
-%:%1441=796%:%
-%:%1442=797%:%
-%:%1443=797%:%
-%:%1444=798%:%
-%:%1454=803%:%
-%:%1464=805%:%
-%:%1465=805%:%
-%:%1466=806%:%
-%:%1467=807%:%
-%:%1468=808%:%
-%:%1469=809%:%
-%:%1476=810%:%
-%:%1477=810%:%
-%:%1478=811%:%
-%:%1479=811%:%
-%:%1480=812%:%
-%:%1481=812%:%
-%:%1482=812%:%
-%:%1483=813%:%
-%:%1484=814%:%
-%:%1485=814%:%
-%:%1486=815%:%
-%:%1487=815%:%
-%:%1488=816%:%
-%:%1489=816%:%
-%:%1490=817%:%
-%:%1491=818%:%
-%:%1492=818%:%
-%:%1493=819%:%
-%:%1494=819%:%
-%:%1495=820%:%
-%:%1496=820%:%
-%:%1497=821%:%
-%:%1498=821%:%
-%:%1499=822%:%
-%:%1500=823%:%
-%:%1501=823%:%
-%:%1502=824%:%
-%:%1503=824%:%
-%:%1504=825%:%
-%:%1505=825%:%
-%:%1506=826%:%
-%:%1507=827%:%
-%:%1508=827%:%
-%:%1509=828%:%
-%:%1510=828%:%
-%:%1511=828%:%
-%:%1512=829%:%
-%:%1513=829%:%
-%:%1514=830%:%
-%:%1515=830%:%
-%:%1516=831%:%
-%:%1517=831%:%
-%:%1518=832%:%
-%:%1519=832%:%
-%:%1520=833%:%
-%:%1521=833%:%
-%:%1522=834%:%
-%:%1523=834%:%
-%:%1524=835%:%
-%:%1525=835%:%
-%:%1526=836%:%
-%:%1527=836%:%
-%:%1528=837%:%
-%:%1529=837%:%
-%:%1530=838%:%
-%:%1531=838%:%
-%:%1532=838%:%
-%:%1533=839%:%
-%:%1534=840%:%
-%:%1535=840%:%
-%:%1536=841%:%
-%:%1537=842%:%
-%:%1538=842%:%
-%:%1539=843%:%
-%:%1540=843%:%
-%:%1541=843%:%
-%:%1542=844%:%
-%:%1543=844%:%
-%:%1544=845%:%
-%:%1545=845%:%
-%:%1546=846%:%
-%:%1547=846%:%
-%:%1548=847%:%
-%:%1549=847%:%
-%:%1550=848%:%
-%:%1556=848%:%
-%:%1559=849%:%
-%:%1560=850%:%
-%:%1561=850%:%
-%:%1562=851%:%
-%:%1563=852%:%
-%:%1564=853%:%
-%:%1571=854%:%
-%:%1572=854%:%
-%:%1573=855%:%
-%:%1574=855%:%
-%:%1575=856%:%
-%:%1576=856%:%
-%:%1577=857%:%
-%:%1578=857%:%
-%:%1579=858%:%
-%:%1580=858%:%
-%:%1581=859%:%
-%:%1582=859%:%
-%:%1583=860%:%
-%:%1584=861%:%
-%:%1585=862%:%
-%:%1586=863%:%
-%:%1587=863%:%
-%:%1588=864%:%
-%:%1589=865%:%
-%:%1590=866%:%
-%:%1591=867%:%
-%:%1593=869%:%
-%:%1594=870%:%
-%:%1595=870%:%
-%:%1596=871%:%
-%:%1597=871%:%
-%:%1598=872%:%
-%:%1599=872%:%
-%:%1600=873%:%
-%:%1601=873%:%
-%:%1602=874%:%
-%:%1605=877%:%
-%:%1606=878%:%
-%:%1607=878%:%
-%:%1608=879%:%
-%:%1609=880%:%
-%:%1610=880%:%
-%:%1611=880%:%
-%:%1612=881%:%
-%:%1613=882%:%
-%:%1614=882%:%
-%:%1615=883%:%
-%:%1616=884%:%
-%:%1617=884%:%
-%:%1618=885%:%
-%:%1619=886%:%
-%:%1620=886%:%
-%:%1621=886%:%
-%:%1622=887%:%
-%:%1623=887%:%
-%:%1624=888%:%
-%:%1625=888%:%
-%:%1626=889%:%
-%:%1627=890%:%
-%:%1628=890%:%
-%:%1629=890%:%
-%:%1630=891%:%
-%:%1631=891%:%
-%:%1632=892%:%
-%:%1633=893%:%
-%:%1634=893%:%
-%:%1635=894%:%
-%:%1636=894%:%
-%:%1637=894%:%
-%:%1638=895%:%
-%:%1639=895%:%
-%:%1640=896%:%
-%:%1641=896%:%
-%:%1642=897%:%
-%:%1643=897%:%
-%:%1644=898%:%
-%:%1645=898%:%
-%:%1646=899%:%
-%:%1647=899%:%
-%:%1648=900%:%
-%:%1649=900%:%
-%:%1650=901%:%
-%:%1651=901%:%
-%:%1652=902%:%
-%:%1653=902%:%
-%:%1654=903%:%
-%:%1655=903%:%
-%:%1656=904%:%
-%:%1657=904%:%
-%:%1658=905%:%
-%:%1661=908%:%
-%:%1662=909%:%
-%:%1663=909%:%
-%:%1664=910%:%
-%:%1665=910%:%
-%:%1666=911%:%
-%:%1667=911%:%
-%:%1668=912%:%
-%:%1672=916%:%
-%:%1673=917%:%
-%:%1674=917%:%
-%:%1675=918%:%
-%:%1676=918%:%
-%:%1677=919%:%
-%:%1682=924%:%
-%:%1683=925%:%
-%:%1684=925%:%
-%:%1685=926%:%
-%:%1686=926%:%
-%:%1687=927%:%
-%:%1692=932%:%
-%:%1693=933%:%
-%:%1694=933%:%
-%:%1695=934%:%
-%:%1701=934%:%
-%:%1704=935%:%
-%:%1705=936%:%
-%:%1706=937%:%
-%:%1707=937%:%
-%:%1708=938%:%
-%:%1709=939%:%
-%:%1716=940%:%
-%:%1717=940%:%
-%:%1718=941%:%
-%:%1719=941%:%
-%:%1720=942%:%
-%:%1725=947%:%
-%:%1726=948%:%
-%:%1727=948%:%
-%:%1728=949%:%
-%:%1729=949%:%
-%:%1730=950%:%
-%:%1731=951%:%
-%:%1732=951%:%
-%:%1733=952%:%
-%:%1738=957%:%
-%:%1739=958%:%
-%:%1740=958%:%
-%:%1741=959%:%
-%:%1742=959%:%
-%:%1743=960%:%
-%:%1744=960%:%
-%:%1745=961%:%
-%:%1750=966%:%
-%:%1751=967%:%
-%:%1752=967%:%
-%:%1753=968%:%
-%:%1754=968%:%
-%:%1755=969%:%
-%:%1760=974%:%
-%:%1761=975%:%
-%:%1762=975%:%
-%:%1763=976%:%
-%:%1764=976%:%
-%:%1765=977%:%
-%:%1766=977%:%
-%:%1767=978%:%
-%:%1772=983%:%
-%:%1773=984%:%
-%:%1774=984%:%
-%:%1775=985%:%
-%:%1776=985%:%
-%:%1777=986%:%
-%:%1778=986%:%
-%:%1779=986%:%
-%:%1780=987%:%
-%:%1781=987%:%
-%:%1782=988%:%
-%:%1783=988%:%
-%:%1784=989%:%
-%:%1785=989%:%
-%:%1786=989%:%
-%:%1787=990%:%
-%:%1788=990%:%
-%:%1789=991%:%
-%:%1795=991%:%
-%:%1798=992%:%
-%:%1799=996%:%
-%:%1800=997%:%
-%:%1801=997%:%
-%:%1802=998%:%
-%:%1803=999%:%
-%:%1806=1000%:%
-%:%1810=1000%:%
-%:%1811=1000%:%
-%:%1812=1001%:%
-%:%1813=1001%:%
-%:%1827=1003%:%
-%:%1831=1005%:%
-%:%1841=1007%:%
-%:%1842=1007%:%
-%:%1843=1008%:%
-%:%1849=1014%:%
-%:%1850=1015%:%
-%:%1851=1016%:%
-%:%1852=1016%:%
-%:%1853=1017%:%
-%:%1857=1021%:%
-%:%1864=1022%:%
-%:%1865=1022%:%
-%:%1866=1023%:%
-%:%1867=1023%:%
-%:%1868=1024%:%
-%:%1869=1024%:%
-%:%1870=1025%:%
-%:%1871=1025%:%
-%:%1872=1026%:%
-%:%1873=1026%:%
-%:%1874=1027%:%
-%:%1875=1027%:%
-%:%1876=1028%:%
-%:%1877=1029%:%
-%:%1878=1030%:%
-%:%1879=1031%:%
-%:%1880=1031%:%
-%:%1881=1032%:%
-%:%1882=1033%:%
-%:%1883=1033%:%
-%:%1884=1034%:%
-%:%1885=1035%:%
-%:%1886=1035%:%
-%:%1887=1036%:%
-%:%1888=1036%:%
-%:%1889=1037%:%
-%:%1890=1037%:%
-%:%1891=1038%:%
-%:%1892=1039%:%
-%:%1893=1040%:%
-%:%1894=1041%:%
-%:%1895=1041%:%
-%:%1896=1041%:%
-%:%1897=1042%:%
-%:%1899=1044%:%
-%:%1900=1045%:%
-%:%1901=1045%:%
-%:%1902=1046%:%
-%:%1903=1046%:%
-%:%1904=1047%:%
-%:%1905=1047%:%
-%:%1906=1047%:%
-%:%1907=1048%:%
-%:%1910=1051%:%
-%:%1911=1052%:%
-%:%1912=1052%:%
-%:%1913=1053%:%
-%:%1914=1053%:%
-%:%1915=1054%:%
-%:%1916=1054%:%
-%:%1917=1055%:%
-%:%1918=1055%:%
-%:%1919=1056%:%
-%:%1922=1059%:%
-%:%1923=1060%:%
-%:%1924=1060%:%
-%:%1925=1061%:%
-%:%1926=1061%:%
-%:%1927=1062%:%
-%:%1928=1062%:%
-%:%1929=1063%:%
-%:%1930=1064%:%
-%:%1931=1064%:%
-%:%1932=1065%:%
-%:%1937=1070%:%
-%:%1938=1071%:%
-%:%1939=1071%:%
-%:%1940=1072%:%
-%:%1941=1072%:%
-%:%1942=1073%:%
-%:%1957=1075%:%
-%:%1967=1081%:%
-%:%1968=1081%:%
-%:%1969=1082%:%
-%:%1979=1087%:%
-%:%1989=1089%:%
-%:%1990=1089%:%
-%:%1991=1090%:%
-%:%1992=1091%:%
-%:%1993=1092%:%
-%:%1994=1092%:%
-%:%1995=1093%:%
-%:%1999=1097%:%
-%:%2002=1098%:%
-%:%2007=1099%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Elimination_Module.tex b/virage/src/test/resources/old_theories/output/document/Elimination_Module.tex
deleted file mode 100644
index 2a12805..0000000
--- a/virage/src/test/resources/old_theories/output/document/Elimination_Module.tex
+++ /dev/null
@@ -1,1358 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Elimination{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Elimination Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Elimination{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ Evaluation{\isacharunderscore}{\kern0pt}Function\isanewline
-\ \ \ \ \ \ \ \ \ \ Electoral{\isacharunderscore}{\kern0pt}Module\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the elimination module. It rejects a set of alternatives only if these
-are not all alternatives. The alternatives potentially to be rejected are put
-in a so-called elimination set. These are all alternatives that score below
-a preset threshold value that depends on the specific voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isacharequal}{\kern0pt}\ nat\isanewline
-\isanewline
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Set\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ elimination{\isacharunderscore}{\kern0pt}set\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}nat\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}e\ a\ A\ p{\isacharparenright}{\kern0pt}\ t\ {\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ elimination{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ A\isanewline
-\ \ \ \ \ \ \ \ then\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ else\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Common Eliminators%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}less{\isacharunderscore}{\kern0pt}eliminator\ e\ t\ A\ p\ {\isacharequal}{\kern0pt}\ elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ less{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}leq{\isacharunderscore}{\kern0pt}eliminator\ e\ t\ A\ p\ {\isacharequal}{\kern0pt}\ elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}min{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ leq{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}Min\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ average\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Threshold{\isacharunderscore}{\kern0pt}Value{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}average\ e\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ {\isacharparenleft}{\kern0pt}card\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}average\ e\ A\ p{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\ leq{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}average\ e\ A\ p{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ elim{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ t{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ t{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ t{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ t{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ Min\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ Min\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Non-Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ elim{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ {\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ elim{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\ profile\ less{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Inference Rules%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}ccomp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p\ w\ x{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ finite\ A\ {\isasymLongrightarrow}\ w\ {\isasymin}\ A\ {\isasymLongrightarrow}\ e\ w\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isasymLongrightarrow}\ e\ x\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rating\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p\ w\ x{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ profile\ A\ p\ {\isasymLongrightarrow}\ w\ {\isasymin}\ A\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite\ A\ {\isasymLongrightarrow}\ e\ w\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isasymLongrightarrow}\ e\ x\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}helper{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharparenleft}{\kern0pt}Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}e\ w\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val\isanewline
-\ \ \ \ \ \ \ \ \ \ rating\ winner\ f{\isacharunderscore}{\kern0pt}prof\ max\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ e\ x\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ y\ A\ p\ {\isacharbar}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ rating\ winner\ f{\isacharunderscore}{\kern0pt}prof\ not{\isacharunderscore}{\kern0pt}max\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}trsh\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}Max\ {\isacharbraceleft}{\kern0pt}e\ y\ A\ p\ {\isacharbar}{\kern0pt}\ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}trsh{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ rating\ winner\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharquery}{\kern0pt}trsh\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}helper{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharquery}{\kern0pt}trsh\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharquery}{\kern0pt}trsh\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ calculation\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ winner\ Collect{\isacharunderscore}{\kern0pt}cong\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p\ {\isasymand}\ w\ {\isasymin}\ A\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isasymor}\ wins\ w\ p\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}trsh\ {\isacharequal}{\kern0pt}\ e\ w\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rating\ winner\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ False\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Component_Types/Elimination_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%54=20%:%
-%:%64=22%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=24%:%
-%:%69=25%:%
-%:%70=26%:%
-%:%71=26%:%
-%:%73=28%:%
-%:%74=29%:%
-%:%75=30%:%
-%:%76=31%:%
-%:%77=31%:%
-%:%78=32%:%
-%:%79=33%:%
-%:%89=38%:%
-%:%99=40%:%
-%:%100=40%:%
-%:%101=41%:%
-%:%102=42%:%
-%:%103=43%:%
-%:%104=44%:%
-%:%105=44%:%
-%:%106=45%:%
-%:%107=46%:%
-%:%108=47%:%
-%:%109=48%:%
-%:%110=48%:%
-%:%111=49%:%
-%:%112=50%:%
-%:%113=51%:%
-%:%114=52%:%
-%:%115=52%:%
-%:%116=53%:%
-%:%117=54%:%
-%:%118=55%:%
-%:%119=56%:%
-%:%120=56%:%
-%:%121=57%:%
-%:%122=58%:%
-%:%123=59%:%
-%:%124=60%:%
-%:%125=60%:%
-%:%126=61%:%
-%:%127=62%:%
-%:%128=63%:%
-%:%129=64%:%
-%:%130=64%:%
-%:%131=65%:%
-%:%132=66%:%
-%:%139=68%:%
-%:%149=70%:%
-%:%150=70%:%
-%:%157=71%:%
-%:%158=71%:%
-%:%159=72%:%
-%:%160=72%:%
-%:%161=73%:%
-%:%162=74%:%
-%:%163=75%:%
-%:%164=75%:%
-%:%165=76%:%
-%:%166=76%:%
-%:%167=77%:%
-%:%168=77%:%
-%:%169=78%:%
-%:%170=78%:%
-%:%171=79%:%
-%:%177=79%:%
-%:%180=80%:%
-%:%181=81%:%
-%:%182=81%:%
-%:%185=82%:%
-%:%189=82%:%
-%:%190=82%:%
-%:%191=83%:%
-%:%192=83%:%
-%:%193=84%:%
-%:%194=84%:%
-%:%195=85%:%
-%:%196=86%:%
-%:%197=87%:%
-%:%198=87%:%
-%:%199=88%:%
-%:%200=89%:%
-%:%201=90%:%
-%:%202=90%:%
-%:%203=91%:%
-%:%209=91%:%
-%:%212=92%:%
-%:%213=93%:%
-%:%214=93%:%
-%:%217=94%:%
-%:%221=94%:%
-%:%222=94%:%
-%:%223=95%:%
-%:%224=95%:%
-%:%225=96%:%
-%:%226=96%:%
-%:%227=97%:%
-%:%228=98%:%
-%:%229=99%:%
-%:%230=99%:%
-%:%231=100%:%
-%:%232=101%:%
-%:%233=102%:%
-%:%234=102%:%
-%:%235=103%:%
-%:%241=103%:%
-%:%244=104%:%
-%:%245=105%:%
-%:%246=105%:%
-%:%249=106%:%
-%:%253=106%:%
-%:%254=106%:%
-%:%255=107%:%
-%:%256=107%:%
-%:%257=108%:%
-%:%258=108%:%
-%:%259=109%:%
-%:%260=110%:%
-%:%261=111%:%
-%:%262=111%:%
-%:%263=112%:%
-%:%264=113%:%
-%:%265=114%:%
-%:%266=114%:%
-%:%267=115%:%
-%:%273=115%:%
-%:%276=116%:%
-%:%277=117%:%
-%:%278=117%:%
-%:%281=118%:%
-%:%285=118%:%
-%:%286=118%:%
-%:%287=119%:%
-%:%288=119%:%
-%:%289=120%:%
-%:%290=120%:%
-%:%291=121%:%
-%:%292=122%:%
-%:%293=123%:%
-%:%294=123%:%
-%:%295=124%:%
-%:%296=125%:%
-%:%297=126%:%
-%:%298=126%:%
-%:%299=127%:%
-%:%305=127%:%
-%:%308=128%:%
-%:%309=129%:%
-%:%310=129%:%
-%:%313=130%:%
-%:%317=130%:%
-%:%318=130%:%
-%:%319=131%:%
-%:%320=131%:%
-%:%321=132%:%
-%:%322=132%:%
-%:%323=133%:%
-%:%324=134%:%
-%:%325=135%:%
-%:%326=135%:%
-%:%327=136%:%
-%:%328=137%:%
-%:%329=138%:%
-%:%330=138%:%
-%:%331=139%:%
-%:%337=139%:%
-%:%340=140%:%
-%:%341=141%:%
-%:%342=141%:%
-%:%345=142%:%
-%:%349=142%:%
-%:%350=142%:%
-%:%351=143%:%
-%:%352=143%:%
-%:%353=144%:%
-%:%354=144%:%
-%:%355=145%:%
-%:%356=146%:%
-%:%357=147%:%
-%:%358=147%:%
-%:%359=148%:%
-%:%360=149%:%
-%:%361=150%:%
-%:%362=150%:%
-%:%363=151%:%
-%:%378=153%:%
-%:%388=155%:%
-%:%389=155%:%
-%:%390=156%:%
-%:%391=157%:%
-%:%394=158%:%
-%:%398=158%:%
-%:%399=158%:%
-%:%404=158%:%
-%:%407=159%:%
-%:%408=160%:%
-%:%409=160%:%
-%:%410=161%:%
-%:%411=162%:%
-%:%414=163%:%
-%:%418=163%:%
-%:%419=163%:%
-%:%420=164%:%
-%:%421=164%:%
-%:%426=164%:%
-%:%429=165%:%
-%:%430=166%:%
-%:%431=166%:%
-%:%432=167%:%
-%:%433=168%:%
-%:%440=169%:%
-%:%441=169%:%
-%:%442=170%:%
-%:%443=170%:%
-%:%444=171%:%
-%:%445=171%:%
-%:%446=172%:%
-%:%447=172%:%
-%:%448=173%:%
-%:%449=173%:%
-%:%450=174%:%
-%:%456=174%:%
-%:%459=175%:%
-%:%460=176%:%
-%:%461=176%:%
-%:%462=177%:%
-%:%463=178%:%
-%:%470=179%:%
-%:%471=179%:%
-%:%472=180%:%
-%:%473=180%:%
-%:%474=181%:%
-%:%475=181%:%
-%:%476=182%:%
-%:%477=182%:%
-%:%478=183%:%
-%:%479=183%:%
-%:%480=184%:%
-%:%486=184%:%
-%:%489=185%:%
-%:%490=186%:%
-%:%491=186%:%
-%:%492=187%:%
-%:%493=188%:%
-%:%500=189%:%
-%:%501=189%:%
-%:%502=190%:%
-%:%503=190%:%
-%:%504=191%:%
-%:%505=191%:%
-%:%506=192%:%
-%:%507=192%:%
-%:%508=193%:%
-%:%509=193%:%
-%:%510=194%:%
-%:%516=194%:%
-%:%519=195%:%
-%:%520=196%:%
-%:%521=196%:%
-%:%522=197%:%
-%:%523=198%:%
-%:%530=199%:%
-%:%531=199%:%
-%:%532=200%:%
-%:%533=200%:%
-%:%534=201%:%
-%:%535=201%:%
-%:%536=202%:%
-%:%537=202%:%
-%:%538=203%:%
-%:%539=203%:%
-%:%540=204%:%
-%:%546=204%:%
-%:%549=205%:%
-%:%550=206%:%
-%:%551=206%:%
-%:%552=207%:%
-%:%553=208%:%
-%:%560=209%:%
-%:%561=209%:%
-%:%562=210%:%
-%:%563=210%:%
-%:%564=211%:%
-%:%565=211%:%
-%:%566=212%:%
-%:%567=212%:%
-%:%568=213%:%
-%:%569=213%:%
-%:%570=214%:%
-%:%585=216%:%
-%:%595=220%:%
-%:%596=220%:%
-%:%597=221%:%
-%:%598=222%:%
-%:%599=223%:%
-%:%600=224%:%
-%:%601=225%:%
-%:%604=226%:%
-%:%608=226%:%
-%:%609=226%:%
-%:%610=227%:%
-%:%611=227%:%
-%:%612=228%:%
-%:%613=228%:%
-%:%614=229%:%
-%:%616=231%:%
-%:%617=232%:%
-%:%618=232%:%
-%:%619=233%:%
-%:%620=233%:%
-%:%621=234%:%
-%:%622=234%:%
-%:%623=235%:%
-%:%629=241%:%
-%:%630=242%:%
-%:%631=242%:%
-%:%632=243%:%
-%:%638=243%:%
-%:%641=244%:%
-%:%642=245%:%
-%:%643=245%:%
-%:%644=246%:%
-%:%645=247%:%
-%:%646=248%:%
-%:%647=249%:%
-%:%648=250%:%
-%:%655=251%:%
-%:%656=251%:%
-%:%657=252%:%
-%:%658=252%:%
-%:%659=253%:%
-%:%660=254%:%
-%:%661=255%:%
-%:%662=255%:%
-%:%663=256%:%
-%:%664=256%:%
-%:%665=257%:%
-%:%666=258%:%
-%:%667=258%:%
-%:%668=259%:%
-%:%669=259%:%
-%:%670=260%:%
-%:%671=260%:%
-%:%672=261%:%
-%:%673=262%:%
-%:%674=262%:%
-%:%675=263%:%
-%:%676=264%:%
-%:%677=265%:%
-%:%678=265%:%
-%:%679=266%:%
-%:%680=266%:%
-%:%681=267%:%
-%:%682=268%:%
-%:%683=268%:%
-%:%684=269%:%
-%:%690=269%:%
-%:%693=270%:%
-%:%694=274%:%
-%:%695=275%:%
-%:%696=275%:%
-%:%697=276%:%
-%:%698=277%:%
-%:%701=278%:%
-%:%705=278%:%
-%:%706=278%:%
-%:%707=279%:%
-%:%708=279%:%
-%:%709=280%:%
-%:%710=280%:%
-%:%711=281%:%
-%:%712=282%:%
-%:%713=283%:%
-%:%714=284%:%
-%:%715=284%:%
-%:%716=285%:%
-%:%717=286%:%
-%:%718=287%:%
-%:%719=287%:%
-%:%720=288%:%
-%:%721=288%:%
-%:%722=289%:%
-%:%725=292%:%
-%:%726=293%:%
-%:%727=293%:%
-%:%728=294%:%
-%:%729=294%:%
-%:%730=295%:%
-%:%731=295%:%
-%:%732=296%:%
-%:%733=296%:%
-%:%734=297%:%
-%:%735=297%:%
-%:%736=298%:%
-%:%737=298%:%
-%:%738=298%:%
-%:%739=299%:%
-%:%740=300%:%
-%:%741=300%:%
-%:%742=301%:%
-%:%743=301%:%
-%:%744=302%:%
-%:%745=302%:%
-%:%746=303%:%
-%:%749=306%:%
-%:%750=307%:%
-%:%751=307%:%
-%:%752=308%:%
-%:%753=308%:%
-%:%754=309%:%
-%:%755=309%:%
-%:%756=309%:%
-%:%757=310%:%
-%:%758=310%:%
-%:%759=311%:%
-%:%760=311%:%
-%:%761=312%:%
-%:%762=312%:%
-%:%763=312%:%
-%:%764=313%:%
-%:%765=313%:%
-%:%766=314%:%
-%:%767=314%:%
-%:%768=315%:%
-%:%769=315%:%
-%:%770=315%:%
-%:%771=316%:%
-%:%772=316%:%
-%:%773=317%:%
-%:%774=317%:%
-%:%775=318%:%
-%:%776=318%:%
-%:%777=318%:%
-%:%778=319%:%
-%:%781=322%:%
-%:%782=323%:%
-%:%783=323%:%
-%:%784=324%:%
-%:%785=324%:%
-%:%786=325%:%
-%:%787=325%:%
-%:%788=325%:%
-%:%789=326%:%
-%:%790=326%:%
-%:%791=327%:%
-%:%792=327%:%
-%:%793=328%:%
-%:%794=328%:%
-%:%795=329%:%
-%:%796=329%:%
-%:%797=330%:%
-%:%798=330%:%
-%:%799=331%:%
-%:%800=331%:%
-%:%801=332%:%
-%:%802=332%:%
-%:%803=333%:%
-%:%804=334%:%
-%:%805=334%:%
-%:%806=335%:%
-%:%807=335%:%
-%:%808=336%:%
-%:%809=336%:%
-%:%810=337%:%
-%:%811=338%:%
-%:%812=338%:%
-%:%813=339%:%
-%:%814=339%:%
-%:%815=340%:%
-%:%816=340%:%
-%:%817=341%:%
-%:%818=341%:%
-%:%819=342%:%
-%:%820=342%:%
-%:%821=343%:%
-%:%822=343%:%
-%:%823=344%:%
-%:%824=344%:%
-%:%825=345%:%
-%:%826=345%:%
-%:%827=346%:%
-%:%828=346%:%
-%:%829=347%:%
-%:%835=347%:%
-%:%840=348%:%
-%:%845=349%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Evaluation_Function.tex b/virage/src/test/resources/old_theories/output/document/Evaluation_Function.tex
deleted file mode 100644
index fc6ad16..0000000
--- a/virage/src/test/resources/old_theories/output/document/Evaluation_Function.tex
+++ /dev/null
@@ -1,344 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Evaluation{\isacharunderscore}{\kern0pt}Function}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Evaluation Function%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Evaluation{\isacharunderscore}{\kern0pt}Function\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Profile{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the evaluation function. From a set of currently eligible alternatives,
-the evaluation function computes a numerical value that is then to be used for
-further (s)election, e.g., by the elimination module.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ \ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ f\ {\isasymequiv}\isanewline
-\ \ \ \ {\isasymforall}A\ p\ w\ {\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}l\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ l\ {\isasymnoteq}\ w\ {\isasymlongrightarrow}\ f\ l\ A\ p\ {\isacharless}{\kern0pt}\ f\ w\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Theorems%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}e\ w\ A\ p\ {\isacharequal}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}set\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isacharquery}{\kern0pt}eMax\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isacharquery}{\kern0pt}eW\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}e\ w\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}set\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}eW\ {\isasymin}\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectI\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ e\ {\isasymin}\ {\isacharquery}{\kern0pt}set\ {\isachardot}{\kern0pt}\ e\ {\isasymle}\ {\isacharquery}{\kern0pt}eW{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\ eq{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}order\ rating\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}eW\ {\isasymin}\ {\isacharquery}{\kern0pt}set\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}set{\isachardot}{\kern0pt}\ a\ {\isasymle}\ {\isacharquery}{\kern0pt}eW{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{4}}\ Max{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ linA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ loser{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnoteq}\ l{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}e\ l\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}e\ l\ A\ p\ {\isacharless}{\kern0pt}\ e\ w\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\ linA\ loser\ rating\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}e\ w\ A\ p\ {\isacharequal}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val\ f{\isacharunderscore}{\kern0pt}prof\ rating\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Component_Types/Evaluation_Function.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%41=14%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%52=18%:%
-%:%62=20%:%
-%:%63=20%:%
-%:%70=22%:%
-%:%80=28%:%
-%:%81=28%:%
-%:%82=29%:%
-%:%91=33%:%
-%:%101=39%:%
-%:%102=39%:%
-%:%103=40%:%
-%:%104=41%:%
-%:%105=42%:%
-%:%106=43%:%
-%:%107=44%:%
-%:%114=45%:%
-%:%115=45%:%
-%:%116=46%:%
-%:%116=49%:%
-%:%117=50%:%
-%:%118=50%:%
-%:%119=51%:%
-%:%120=52%:%
-%:%121=53%:%
-%:%122=54%:%
-%:%123=54%:%
-%:%124=54%:%
-%:%125=55%:%
-%:%126=55%:%
-%:%127=56%:%
-%:%128=57%:%
-%:%129=57%:%
-%:%130=58%:%
-%:%131=58%:%
-%:%132=59%:%
-%:%133=59%:%
-%:%134=60%:%
-%:%135=61%:%
-%:%136=61%:%
-%:%137=62%:%
-%:%138=62%:%
-%:%139=63%:%
-%:%140=63%:%
-%:%141=64%:%
-%:%142=65%:%
-%:%143=65%:%
-%:%144=66%:%
-%:%145=66%:%
-%:%146=67%:%
-%:%147=68%:%
-%:%148=68%:%
-%:%149=69%:%
-%:%150=70%:%
-%:%151=70%:%
-%:%152=70%:%
-%:%153=71%:%
-%:%154=72%:%
-%:%155=72%:%
-%:%156=73%:%
-%:%157=73%:%
-%:%158=73%:%
-%:%159=74%:%
-%:%160=74%:%
-%:%161=75%:%
-%:%167=75%:%
-%:%170=76%:%
-%:%171=82%:%
-%:%172=83%:%
-%:%173=83%:%
-%:%174=84%:%
-%:%175=85%:%
-%:%176=86%:%
-%:%177=87%:%
-%:%178=88%:%
-%:%179=89%:%
-%:%180=90%:%
-%:%187=91%:%
-%:%188=91%:%
-%:%189=92%:%
-%:%190=92%:%
-%:%191=93%:%
-%:%192=93%:%
-%:%193=94%:%
-%:%194=94%:%
-%:%195=95%:%
-%:%196=95%:%
-%:%197=95%:%
-%:%198=96%:%
-%:%199=96%:%
-%:%200=97%:%
-%:%201=97%:%
-%:%202=98%:%
-%:%203=98%:%
-%:%204=98%:%
-%:%205=99%:%
-%:%206=99%:%
-%:%207=100%:%
-%:%213=100%:%
-%:%218=101%:%
-%:%223=102%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Loop_Composition.tex b/virage/src/test/resources/old_theories/output/document/Loop_Composition.tex
deleted file mode 100644
index 75f4efe..0000000
--- a/virage/src/test/resources/old_theories/output/document/Loop_Composition.tex
+++ /dev/null
@@ -1,2502 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Loop{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Loop Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Loop{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Sequential{\isacharunderscore}{\kern0pt}Composition\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The loop composition uses the same module in sequence,
-combined with a termination condition, until either
-  (1) the termination condition is met or
-  (2) no new decisions are made (i.e., a fixed point is reached).%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}termination{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}term{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}inf{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymin}\isanewline
-\ \ \ \ \ \ \ \ measure\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubset}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ acc\ A\ p{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubset}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ P\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ bool\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymlbrakk}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymlbrakk}{\isasymnot}\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}f\ A\ p\ rs\ fa{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fa{\isacharcomma}{\kern0pt}\ f{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isacharunderscore}{\kern0pt}cases{\isadigit{5}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ P\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{2}}\ a{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m\ ta\ acca\ Aa\ pa\ ma{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ acc\ A\ p\ {\isacharequal}{\kern0pt}\ acca\ Aa\ pa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m\ ta\ acca\ Aa\ pa\ ma{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ acc\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ t\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ acc\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ m\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ ta\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ acca\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ Aa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ pa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ ma\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{2}}\ a{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}acc\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m\ ta\ acca\ Aa\ pa\ ma{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}r{\isachardot}{\kern0pt}\ wf\ r\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ f\ A\ rs\ fa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ p\ {\isacharparenleft}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharparenright}{\kern0pt}\ A\ rs\ {\isasymsubset}\ defer\ f\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}termination{\isacharunderscore}{\kern0pt}helper\ wf{\isacharunderscore}{\kern0pt}measure\ {\isachardoublequoteopen}termination{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}r\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ Ex\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymlambda}ra{\isachardot}{\kern0pt}\ {\isasymforall}f\ A\ rs\ pa\ fa{\isachardot}{\kern0pt}\ {\isasymexists}ra\ pb\ rb\ pc\ pd\ fb\ Aa\ rsa\ fc\ pe{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ wf\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}p{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ f\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ pa\ {\isacharparenleft}{\kern0pt}f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ wf\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}ra{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ set{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\ {\isacharparenleft}{\kern0pt}pb{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ wf\ rb\ {\isasymand}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharparenright}{\kern0pt}\ A\ rs\ {\isasymsubset}\ defer\ f\ A\ rs\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}pc{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ pa{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ f{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ pa{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymin}\ rb\ {\isasymand}\ wf\ rb\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}pd{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}defer\ fb\ {\isacharparenleft}{\kern0pt}Aa{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ rsa{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}fb\ {\isasymtriangleright}\ fc{\isacharparenright}{\kern0pt}\ Aa\ rsa\ {\isasymsubset}\ defer\ fb\ Aa\ rsa\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ pe\ {\isacharparenleft}{\kern0pt}fb\ Aa\ rsa{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}fb\ {\isasymtriangleright}\ fc{\isacharcomma}{\kern0pt}\ fc{\isacharcomma}{\kern0pt}\ pe{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ rsa{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ fb{\isacharcomma}{\kern0pt}\ fc{\isacharcomma}{\kern0pt}\ pe{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ rsa{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isacharparenright}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ set\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ rr\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}wf\ rr\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ f\ A\ rs\ fa{\isachardot}{\kern0pt}\ p\ {\isacharparenleft}{\kern0pt}f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharparenright}{\kern0pt}\ A\ rs\ {\isasymsubset}\ defer\ f\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ f{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymin}\ rr{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}termination{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}helper{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubset}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ then\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ else\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ loop{\isacharunderscore}{\kern0pt}composition\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}module\ A\ p{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ loop{\isacharunderscore}{\kern0pt}composition\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ m\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ \ {\isachardoublequoteopen}termination{\isachardoublequoteclose}\ wf{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ loop\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymcirclearrowleft}\isactrlsub t\ {\isasymequiv}\ loop{\isacharunderscore}{\kern0pt}composition\ m\ t{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}loop{\isacharunderscore}{\kern0pt}composition\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ then\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}module\ A\ p{\isacharparenright}{\kern0pt}\ else\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ m\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ {\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m\ profile\isanewline
-\ \ \ \ \ \ \ \ \ \ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ m{\isacharunderscore}{\kern0pt}module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}partit\ m{\isacharunderscore}{\kern0pt}module\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}no{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}incr{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}trans\ eq{\isacharunderscore}{\kern0pt}iff\ less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymand}\ n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ n\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}help\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}unchanged{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}eq\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ less{\isacharunderscore}{\kern0pt}irrefl\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}helper\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}unchanged\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}order\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}bounded\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ order{\isachardot}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}strict\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ t\ {\isacharparenleft}{\kern0pt}acc\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isasymnoteq}\ f\ A\ rsa\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isacharequal}{\kern0pt}\ f\ A\ rsa\ {\isasymor}\ {\isasymnot}\ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymnotin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ {\isasymnot}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed\ monotone{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q\ {\isasymsubset}\ defer\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ monotone{\isacharunderscore}{\kern0pt}m\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ psubsetI\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ subsetCE\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ q\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ monotone{\isacharunderscore}{\kern0pt}m\ subsetCE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ card{\isacharunderscore}{\kern0pt}changed{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ gr{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}not{\isadigit{0}}\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ monotone{\isacharunderscore}{\kern0pt}m\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ psubsetI\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p\ less{\isachardot}{\kern0pt}hyps\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}no{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}incr\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ f{\isacharunderscore}{\kern0pt}prof\ rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ subsetCE\ subset{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ {\isasymnot}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}lifted\ A\ p\ q\ a\ {\isasymand}\ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}helper\isanewline
-\ \ \ \ \ \ \ \ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}acc\ profile\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ lifted\ A\ p\ q\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ monotone{\isacharunderscore}{\kern0pt}acc\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isasymnoteq}\ f\ A\ rsa\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isacharequal}{\kern0pt}\ f\ A\ rsa\ {\isasymor}\ {\isasymnot}\ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ a\ {\isasymnotin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ {\isasymnot}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}partit\isanewline
-\ \ \ \ \ \ \ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ elect\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ n\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}less\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ n\ {\isasymge}\ x\ {\isasymand}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\isanewline
-\ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ n\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}less\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ subset{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ step{\isacharunderscore}{\kern0pt}reduces{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ term{\isacharunderscore}{\kern0pt}satisfied{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ term{\isacharunderscore}{\kern0pt}satisfied\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ term{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ rec{\isacharunderscore}{\kern0pt}step{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ step{\isacharunderscore}{\kern0pt}reduces{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}set\ term{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}too{\isacharunderscore}{\kern0pt}small{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ old{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}at{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ i\ \isakeyword{where}\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}i\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}x\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}too{\isacharunderscore}{\kern0pt}big{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nat{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}iff\ old{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}at{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}leftover{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ f{\isacharunderscore}{\kern0pt}prof{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ step{\isacharunderscore}{\kern0pt}profile{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isadigit{2}}\ enough{\isacharunderscore}{\kern0pt}leftover\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ snd{\isacharunderscore}{\kern0pt}conv\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isasymge}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}too{\isacharunderscore}{\kern0pt}big\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ x\ {\isasymle}\ i{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}leftover\ f{\isacharunderscore}{\kern0pt}prof\ subset\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ i\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ step{\isacharunderscore}{\kern0pt}profile\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}i\ {\isacharequal}{\kern0pt}\ x\ {\isasymor}\ i\ {\isachargreater}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nat{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}le\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isachargreater}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ {\isadigit{1}}\ {\isacharless}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ x\ {\isasymle}\ i{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isacharless}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ subset\ step{\isacharunderscore}{\kern0pt}profile\ enough{\isacharunderscore}{\kern0pt}leftover\ f{\isacharunderscore}{\kern0pt}prof\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}new{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ acc\ {\isasymlongrightarrow}\ non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isachardot}{\kern0pt}hyps\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ rec{\isacharunderscore}{\kern0pt}step\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isachargreater}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\ i\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\ t\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}x\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}order\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof\ rec{\isacharunderscore}{\kern0pt}step\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ acc{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}enough{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ acc{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ acc{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}enough\ gr{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}not{\isadigit{0}}\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}less\isanewline
-\ \ \ \ \ \ \ \ less{\isacharunderscore}{\kern0pt}one\ linorder{\isacharunderscore}{\kern0pt}neqE{\isacharunderscore}{\kern0pt}nat\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}acc\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ single{\isacharunderscore}{\kern0pt}elimination\ nat{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\ less{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ enough{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isasymge}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ card\ A\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ A\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}alternatives\ not{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}card\ A\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isachargreater}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isadigit{2}}\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ m\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}x\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Composition Rules%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}module{\isachardot}{\kern0pt}simps\ monotone{\isacharunderscore}{\kern0pt}m\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ monotone{\isacharunderscore}{\kern0pt}m\ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ x{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}helper\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ empty{\isacharunderscore}{\kern0pt}iff\ fin{\isacharunderscore}{\kern0pt}A\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}elect\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ card\ A\ {\isasymge}\ n\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ eliminates{\isacharunderscore}{\kern0pt}def\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ calculation\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Loop_Composition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%35=14%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%56=22%:%
-%:%66=24%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=28%:%
-%:%72=29%:%
-%:%73=30%:%
-%:%74=31%:%
-%:%77=32%:%
-%:%81=32%:%
-%:%82=32%:%
-%:%83=33%:%
-%:%84=33%:%
-%:%89=33%:%
-%:%92=34%:%
-%:%93=38%:%
-%:%94=39%:%
-%:%95=39%:%
-%:%96=40%:%
-%:%97=41%:%
-%:%98=42%:%
-%:%100=44%:%
-%:%101=45%:%
-%:%103=47%:%
-%:%110=48%:%
-%:%111=48%:%
-%:%112=49%:%
-%:%113=49%:%
-%:%114=50%:%
-%:%115=51%:%
-%:%116=52%:%
-%:%117=53%:%
-%:%118=53%:%
-%:%119=54%:%
-%:%122=57%:%
-%:%123=58%:%
-%:%126=61%:%
-%:%127=62%:%
-%:%128=62%:%
-%:%129=63%:%
-%:%130=63%:%
-%:%131=64%:%
-%:%132=64%:%
-%:%133=65%:%
-%:%134=65%:%
-%:%135=65%:%
-%:%136=66%:%
-%:%137=66%:%
-%:%138=67%:%
-%:%139=67%:%
-%:%140=68%:%
-%:%141=68%:%
-%:%142=69%:%
-%:%143=69%:%
-%:%144=70%:%
-%:%150=76%:%
-%:%151=77%:%
-%:%152=77%:%
-%:%153=78%:%
-%:%154=78%:%
-%:%155=79%:%
-%:%156=79%:%
-%:%157=80%:%
-%:%163=86%:%
-%:%164=87%:%
-%:%165=87%:%
-%:%166=88%:%
-%:%167=88%:%
-%:%168=89%:%
-%:%169=90%:%
-%:%170=91%:%
-%:%171=92%:%
-%:%172=93%:%
-%:%173=94%:%
-%:%174=95%:%
-%:%175=96%:%
-%:%176=97%:%
-%:%177=98%:%
-%:%178=99%:%
-%:%179=99%:%
-%:%180=100%:%
-%:%181=101%:%
-%:%182=102%:%
-%:%183=103%:%
-%:%184=104%:%
-%:%185=105%:%
-%:%186=105%:%
-%:%187=106%:%
-%:%188=106%:%
-%:%189=107%:%
-%:%190=107%:%
-%:%191=108%:%
-%:%192=108%:%
-%:%193=109%:%
-%:%194=109%:%
-%:%195=110%:%
-%:%196=110%:%
-%:%197=111%:%
-%:%198=111%:%
-%:%199=112%:%
-%:%200=112%:%
-%:%201=113%:%
-%:%208=120%:%
-%:%209=121%:%
-%:%210=121%:%
-%:%211=122%:%
-%:%217=122%:%
-%:%220=123%:%
-%:%221=123%:%
-%:%228=124%:%
-%:%229=124%:%
-%:%230=125%:%
-%:%231=125%:%
-%:%232=126%:%
-%:%237=131%:%
-%:%238=132%:%
-%:%239=132%:%
-%:%240=133%:%
-%:%241=133%:%
-%:%242=134%:%
-%:%243=134%:%
-%:%244=135%:%
-%:%274=165%:%
-%:%275=166%:%
-%:%276=166%:%
-%:%277=167%:%
-%:%278=167%:%
-%:%279=168%:%
-%:%282=171%:%
-%:%283=172%:%
-%:%287=176%:%
-%:%288=177%:%
-%:%289=177%:%
-%:%290=178%:%
-%:%291=178%:%
-%:%292=179%:%
-%:%293=179%:%
-%:%294=180%:%
-%:%295=180%:%
-%:%296=181%:%
-%:%297=181%:%
-%:%298=182%:%
-%:%304=182%:%
-%:%307=183%:%
-%:%308=184%:%
-%:%309=184%:%
-%:%310=185%:%
-%:%313=188%:%
-%:%316=189%:%
-%:%320=189%:%
-%:%321=189%:%
-%:%326=189%:%
-%:%329=190%:%
-%:%330=191%:%
-%:%331=191%:%
-%:%332=192%:%
-%:%333=193%:%
-%:%334=194%:%
-%:%335=195%:%
-%:%336=196%:%
-%:%337=197%:%
-%:%340=198%:%
-%:%344=198%:%
-%:%345=198%:%
-%:%350=198%:%
-%:%353=199%:%
-%:%354=199%:%
-%:%357=200%:%
-%:%361=200%:%
-%:%362=200%:%
-%:%363=201%:%
-%:%364=201%:%
-%:%369=201%:%
-%:%372=202%:%
-%:%373=203%:%
-%:%374=203%:%
-%:%375=204%:%
-%:%376=205%:%
-%:%377=206%:%
-%:%378=207%:%
-%:%379=208%:%
-%:%380=208%:%
-%:%381=209%:%
-%:%383=211%:%
-%:%386=212%:%
-%:%390=212%:%
-%:%391=212%:%
-%:%396=212%:%
-%:%399=213%:%
-%:%400=214%:%
-%:%401=214%:%
-%:%402=215%:%
-%:%403=216%:%
-%:%404=217%:%
-%:%405=218%:%
-%:%406=219%:%
-%:%407=220%:%
-%:%414=221%:%
-%:%415=221%:%
-%:%416=222%:%
-%:%417=222%:%
-%:%418=223%:%
-%:%419=223%:%
-%:%420=224%:%
-%:%421=224%:%
-%:%422=225%:%
-%:%423=226%:%
-%:%424=227%:%
-%:%425=227%:%
-%:%426=228%:%
-%:%441=230%:%
-%:%451=232%:%
-%:%452=232%:%
-%:%453=233%:%
-%:%454=234%:%
-%:%457=235%:%
-%:%461=235%:%
-%:%462=235%:%
-%:%463=236%:%
-%:%464=237%:%
-%:%465=237%:%
-%:%470=237%:%
-%:%473=238%:%
-%:%474=239%:%
-%:%475=239%:%
-%:%476=240%:%
-%:%477=241%:%
-%:%478=242%:%
-%:%479=243%:%
-%:%480=244%:%
-%:%481=245%:%
-%:%488=246%:%
-%:%489=246%:%
-%:%490=247%:%
-%:%491=247%:%
-%:%492=248%:%
-%:%493=248%:%
-%:%494=249%:%
-%:%495=249%:%
-%:%496=250%:%
-%:%497=251%:%
-%:%498=252%:%
-%:%499=252%:%
-%:%500=253%:%
-%:%515=255%:%
-%:%525=257%:%
-%:%526=257%:%
-%:%527=258%:%
-%:%528=259%:%
-%:%529=260%:%
-%:%530=261%:%
-%:%531=262%:%
-%:%536=267%:%
-%:%543=268%:%
-%:%544=268%:%
-%:%545=269%:%
-%:%546=269%:%
-%:%547=270%:%
-%:%548=270%:%
-%:%549=271%:%
-%:%551=273%:%
-%:%552=274%:%
-%:%553=274%:%
-%:%554=275%:%
-%:%555=275%:%
-%:%556=276%:%
-%:%557=276%:%
-%:%558=277%:%
-%:%560=279%:%
-%:%561=280%:%
-%:%562=280%:%
-%:%563=281%:%
-%:%564=281%:%
-%:%565=282%:%
-%:%567=284%:%
-%:%568=285%:%
-%:%569=285%:%
-%:%570=286%:%
-%:%571=286%:%
-%:%572=287%:%
-%:%573=287%:%
-%:%574=288%:%
-%:%575=288%:%
-%:%576=289%:%
-%:%577=289%:%
-%:%578=290%:%
-%:%579=290%:%
-%:%580=291%:%
-%:%581=291%:%
-%:%582=292%:%
-%:%584=294%:%
-%:%585=295%:%
-%:%586=295%:%
-%:%587=296%:%
-%:%588=297%:%
-%:%589=298%:%
-%:%590=299%:%
-%:%591=300%:%
-%:%592=300%:%
-%:%593=301%:%
-%:%594=301%:%
-%:%595=301%:%
-%:%596=301%:%
-%:%597=302%:%
-%:%598=303%:%
-%:%599=303%:%
-%:%600=304%:%
-%:%601=305%:%
-%:%602=305%:%
-%:%603=306%:%
-%:%604=306%:%
-%:%605=306%:%
-%:%606=307%:%
-%:%610=311%:%
-%:%611=312%:%
-%:%612=312%:%
-%:%613=313%:%
-%:%614=313%:%
-%:%615=314%:%
-%:%616=314%:%
-%:%617=315%:%
-%:%618=315%:%
-%:%619=316%:%
-%:%620=316%:%
-%:%621=317%:%
-%:%622=317%:%
-%:%623=318%:%
-%:%624=318%:%
-%:%625=319%:%
-%:%626=320%:%
-%:%627=320%:%
-%:%628=320%:%
-%:%629=321%:%
-%:%630=322%:%
-%:%631=323%:%
-%:%632=323%:%
-%:%633=324%:%
-%:%634=325%:%
-%:%635=325%:%
-%:%636=326%:%
-%:%637=326%:%
-%:%638=326%:%
-%:%639=327%:%
-%:%641=329%:%
-%:%642=330%:%
-%:%643=330%:%
-%:%644=331%:%
-%:%645=331%:%
-%:%646=332%:%
-%:%647=332%:%
-%:%648=333%:%
-%:%649=333%:%
-%:%650=334%:%
-%:%651=334%:%
-%:%652=335%:%
-%:%653=335%:%
-%:%654=336%:%
-%:%656=338%:%
-%:%657=339%:%
-%:%658=339%:%
-%:%659=340%:%
-%:%660=340%:%
-%:%661=341%:%
-%:%662=341%:%
-%:%663=342%:%
-%:%664=342%:%
-%:%665=343%:%
-%:%667=345%:%
-%:%668=346%:%
-%:%669=346%:%
-%:%670=347%:%
-%:%671=347%:%
-%:%672=348%:%
-%:%677=353%:%
-%:%678=354%:%
-%:%679=354%:%
-%:%680=355%:%
-%:%681=355%:%
-%:%682=356%:%
-%:%683=356%:%
-%:%684=357%:%
-%:%685=357%:%
-%:%686=358%:%
-%:%687=358%:%
-%:%688=359%:%
-%:%689=359%:%
-%:%690=360%:%
-%:%691=360%:%
-%:%692=361%:%
-%:%694=363%:%
-%:%695=364%:%
-%:%696=364%:%
-%:%697=365%:%
-%:%698=366%:%
-%:%699=367%:%
-%:%700=368%:%
-%:%701=368%:%
-%:%702=369%:%
-%:%703=369%:%
-%:%704=369%:%
-%:%705=370%:%
-%:%708=373%:%
-%:%709=374%:%
-%:%710=374%:%
-%:%711=375%:%
-%:%712=376%:%
-%:%713=377%:%
-%:%714=378%:%
-%:%715=378%:%
-%:%716=379%:%
-%:%717=379%:%
-%:%718=380%:%
-%:%719=381%:%
-%:%720=382%:%
-%:%721=382%:%
-%:%722=383%:%
-%:%723=384%:%
-%:%724=385%:%
-%:%725=386%:%
-%:%726=386%:%
-%:%727=387%:%
-%:%728=387%:%
-%:%729=388%:%
-%:%730=388%:%
-%:%731=389%:%
-%:%732=390%:%
-%:%733=391%:%
-%:%734=392%:%
-%:%735=392%:%
-%:%736=393%:%
-%:%737=393%:%
-%:%738=394%:%
-%:%739=394%:%
-%:%740=395%:%
-%:%741=395%:%
-%:%742=396%:%
-%:%743=396%:%
-%:%744=397%:%
-%:%745=397%:%
-%:%746=398%:%
-%:%747=398%:%
-%:%748=399%:%
-%:%749=399%:%
-%:%750=400%:%
-%:%756=400%:%
-%:%759=401%:%
-%:%760=402%:%
-%:%761=402%:%
-%:%762=403%:%
-%:%763=404%:%
-%:%764=405%:%
-%:%765=406%:%
-%:%766=407%:%
-%:%767=408%:%
-%:%768=409%:%
-%:%771=410%:%
-%:%775=410%:%
-%:%776=410%:%
-%:%777=411%:%
-%:%778=412%:%
-%:%779=412%:%
-%:%784=412%:%
-%:%787=413%:%
-%:%788=414%:%
-%:%789=414%:%
-%:%790=415%:%
-%:%791=416%:%
-%:%792=417%:%
-%:%793=418%:%
-%:%794=419%:%
-%:%797=422%:%
-%:%800=423%:%
-%:%804=423%:%
-%:%805=423%:%
-%:%806=424%:%
-%:%807=424%:%
-%:%812=424%:%
-%:%815=425%:%
-%:%816=426%:%
-%:%817=426%:%
-%:%818=427%:%
-%:%819=428%:%
-%:%822=429%:%
-%:%826=429%:%
-%:%827=429%:%
-%:%828=430%:%
-%:%829=430%:%
-%:%834=430%:%
-%:%837=431%:%
-%:%838=432%:%
-%:%839=432%:%
-%:%840=433%:%
-%:%841=434%:%
-%:%842=435%:%
-%:%843=436%:%
-%:%850=437%:%
-%:%851=437%:%
-%:%852=438%:%
-%:%853=438%:%
-%:%854=439%:%
-%:%861=446%:%
-%:%862=447%:%
-%:%863=447%:%
-%:%864=448%:%
-%:%865=448%:%
-%:%866=449%:%
-%:%867=449%:%
-%:%868=450%:%
-%:%869=450%:%
-%:%870=451%:%
-%:%871=452%:%
-%:%872=453%:%
-%:%873=453%:%
-%:%874=454%:%
-%:%880=454%:%
-%:%883=455%:%
-%:%884=456%:%
-%:%885=456%:%
-%:%886=457%:%
-%:%887=458%:%
-%:%888=459%:%
-%:%889=460%:%
-%:%890=461%:%
-%:%891=462%:%
-%:%898=463%:%
-%:%899=463%:%
-%:%900=464%:%
-%:%901=464%:%
-%:%902=465%:%
-%:%903=465%:%
-%:%904=466%:%
-%:%905=466%:%
-%:%906=467%:%
-%:%907=468%:%
-%:%908=469%:%
-%:%909=469%:%
-%:%910=470%:%
-%:%916=470%:%
-%:%919=471%:%
-%:%920=472%:%
-%:%921=472%:%
-%:%922=473%:%
-%:%923=474%:%
-%:%924=475%:%
-%:%925=476%:%
-%:%926=477%:%
-%:%927=478%:%
-%:%928=479%:%
-%:%929=480%:%
-%:%931=482%:%
-%:%938=483%:%
-%:%939=483%:%
-%:%940=484%:%
-%:%941=484%:%
-%:%942=485%:%
-%:%943=485%:%
-%:%944=486%:%
-%:%945=487%:%
-%:%946=488%:%
-%:%947=488%:%
-%:%948=489%:%
-%:%949=489%:%
-%:%950=490%:%
-%:%951=490%:%
-%:%952=491%:%
-%:%953=492%:%
-%:%954=493%:%
-%:%955=493%:%
-%:%956=494%:%
-%:%957=494%:%
-%:%958=495%:%
-%:%959=495%:%
-%:%960=496%:%
-%:%961=496%:%
-%:%962=497%:%
-%:%963=497%:%
-%:%964=498%:%
-%:%965=498%:%
-%:%966=499%:%
-%:%967=499%:%
-%:%968=500%:%
-%:%969=500%:%
-%:%970=501%:%
-%:%971=501%:%
-%:%972=502%:%
-%:%973=502%:%
-%:%974=503%:%
-%:%975=503%:%
-%:%976=504%:%
-%:%977=504%:%
-%:%978=505%:%
-%:%979=505%:%
-%:%980=506%:%
-%:%981=506%:%
-%:%982=507%:%
-%:%983=507%:%
-%:%984=508%:%
-%:%986=510%:%
-%:%987=511%:%
-%:%988=511%:%
-%:%989=512%:%
-%:%990=513%:%
-%:%991=513%:%
-%:%992=514%:%
-%:%993=514%:%
-%:%994=515%:%
-%:%995=515%:%
-%:%996=516%:%
-%:%997=516%:%
-%:%998=517%:%
-%:%999=517%:%
-%:%1000=518%:%
-%:%1001=518%:%
-%:%1002=519%:%
-%:%1003=519%:%
-%:%1004=520%:%
-%:%1005=520%:%
-%:%1006=521%:%
-%:%1007=521%:%
-%:%1008=522%:%
-%:%1009=522%:%
-%:%1010=523%:%
-%:%1011=523%:%
-%:%1012=524%:%
-%:%1013=524%:%
-%:%1014=524%:%
-%:%1015=525%:%
-%:%1016=526%:%
-%:%1017=526%:%
-%:%1018=527%:%
-%:%1019=527%:%
-%:%1020=528%:%
-%:%1021=528%:%
-%:%1022=529%:%
-%:%1023=529%:%
-%:%1024=530%:%
-%:%1025=530%:%
-%:%1026=531%:%
-%:%1027=531%:%
-%:%1028=532%:%
-%:%1029=532%:%
-%:%1030=533%:%
-%:%1031=533%:%
-%:%1032=534%:%
-%:%1034=536%:%
-%:%1035=537%:%
-%:%1036=537%:%
-%:%1037=538%:%
-%:%1038=538%:%
-%:%1039=539%:%
-%:%1040=539%:%
-%:%1041=540%:%
-%:%1044=543%:%
-%:%1045=544%:%
-%:%1046=544%:%
-%:%1047=545%:%
-%:%1048=546%:%
-%:%1049=546%:%
-%:%1050=547%:%
-%:%1051=547%:%
-%:%1052=548%:%
-%:%1053=548%:%
-%:%1054=549%:%
-%:%1055=549%:%
-%:%1056=550%:%
-%:%1057=550%:%
-%:%1058=551%:%
-%:%1059=551%:%
-%:%1060=552%:%
-%:%1061=552%:%
-%:%1062=553%:%
-%:%1063=553%:%
-%:%1064=554%:%
-%:%1065=554%:%
-%:%1066=555%:%
-%:%1067=555%:%
-%:%1068=556%:%
-%:%1069=557%:%
-%:%1070=558%:%
-%:%1071=558%:%
-%:%1072=559%:%
-%:%1073=559%:%
-%:%1074=560%:%
-%:%1075=560%:%
-%:%1076=561%:%
-%:%1077=562%:%
-%:%1078=563%:%
-%:%1079=563%:%
-%:%1080=564%:%
-%:%1081=564%:%
-%:%1082=565%:%
-%:%1083=565%:%
-%:%1084=566%:%
-%:%1085=567%:%
-%:%1086=567%:%
-%:%1087=568%:%
-%:%1088=568%:%
-%:%1089=569%:%
-%:%1090=569%:%
-%:%1091=570%:%
-%:%1092=570%:%
-%:%1093=571%:%
-%:%1094=571%:%
-%:%1095=572%:%
-%:%1096=572%:%
-%:%1097=573%:%
-%:%1098=573%:%
-%:%1099=574%:%
-%:%1100=574%:%
-%:%1101=575%:%
-%:%1102=575%:%
-%:%1103=575%:%
-%:%1104=576%:%
-%:%1105=577%:%
-%:%1106=577%:%
-%:%1107=578%:%
-%:%1108=578%:%
-%:%1109=579%:%
-%:%1110=579%:%
-%:%1111=579%:%
-%:%1112=580%:%
-%:%1113=581%:%
-%:%1114=582%:%
-%:%1115=582%:%
-%:%1116=583%:%
-%:%1117=584%:%
-%:%1118=584%:%
-%:%1119=585%:%
-%:%1120=585%:%
-%:%1121=585%:%
-%:%1122=586%:%
-%:%1123=587%:%
-%:%1124=588%:%
-%:%1125=588%:%
-%:%1126=589%:%
-%:%1127=589%:%
-%:%1128=590%:%
-%:%1129=590%:%
-%:%1130=590%:%
-%:%1131=591%:%
-%:%1132=592%:%
-%:%1133=592%:%
-%:%1134=593%:%
-%:%1135=593%:%
-%:%1136=593%:%
-%:%1137=594%:%
-%:%1139=596%:%
-%:%1140=597%:%
-%:%1141=597%:%
-%:%1142=598%:%
-%:%1143=598%:%
-%:%1144=599%:%
-%:%1145=599%:%
-%:%1146=600%:%
-%:%1147=600%:%
-%:%1148=601%:%
-%:%1149=601%:%
-%:%1150=602%:%
-%:%1151=602%:%
-%:%1152=603%:%
-%:%1153=603%:%
-%:%1154=604%:%
-%:%1155=604%:%
-%:%1156=605%:%
-%:%1157=605%:%
-%:%1158=606%:%
-%:%1159=606%:%
-%:%1160=607%:%
-%:%1161=607%:%
-%:%1162=608%:%
-%:%1163=608%:%
-%:%1164=609%:%
-%:%1165=609%:%
-%:%1166=610%:%
-%:%1167=610%:%
-%:%1168=611%:%
-%:%1169=612%:%
-%:%1170=613%:%
-%:%1171=613%:%
-%:%1172=614%:%
-%:%1173=614%:%
-%:%1174=615%:%
-%:%1175=615%:%
-%:%1176=616%:%
-%:%1177=616%:%
-%:%1178=617%:%
-%:%1179=618%:%
-%:%1180=619%:%
-%:%1181=619%:%
-%:%1182=620%:%
-%:%1183=620%:%
-%:%1184=621%:%
-%:%1185=621%:%
-%:%1186=622%:%
-%:%1187=622%:%
-%:%1188=623%:%
-%:%1194=623%:%
-%:%1197=624%:%
-%:%1198=625%:%
-%:%1199=625%:%
-%:%1200=626%:%
-%:%1201=627%:%
-%:%1202=628%:%
-%:%1203=629%:%
-%:%1204=630%:%
-%:%1205=631%:%
-%:%1206=632%:%
-%:%1207=633%:%
-%:%1208=634%:%
-%:%1211=635%:%
-%:%1215=635%:%
-%:%1216=635%:%
-%:%1217=636%:%
-%:%1218=637%:%
-%:%1219=638%:%
-%:%1220=639%:%
-%:%1221=640%:%
-%:%1222=640%:%
-%:%1227=640%:%
-%:%1230=641%:%
-%:%1231=642%:%
-%:%1232=642%:%
-%:%1233=643%:%
-%:%1234=644%:%
-%:%1235=645%:%
-%:%1236=646%:%
-%:%1237=647%:%
-%:%1238=648%:%
-%:%1239=649%:%
-%:%1240=650%:%
-%:%1247=651%:%
-%:%1248=651%:%
-%:%1249=652%:%
-%:%1250=652%:%
-%:%1251=653%:%
-%:%1252=653%:%
-%:%1253=654%:%
-%:%1254=654%:%
-%:%1255=655%:%
-%:%1256=655%:%
-%:%1257=656%:%
-%:%1258=656%:%
-%:%1259=657%:%
-%:%1260=657%:%
-%:%1261=658%:%
-%:%1262=658%:%
-%:%1263=659%:%
-%:%1264=659%:%
-%:%1265=660%:%
-%:%1266=660%:%
-%:%1267=661%:%
-%:%1268=661%:%
-%:%1269=662%:%
-%:%1270=662%:%
-%:%1271=663%:%
-%:%1272=663%:%
-%:%1273=664%:%
-%:%1274=664%:%
-%:%1275=665%:%
-%:%1276=665%:%
-%:%1277=666%:%
-%:%1278=666%:%
-%:%1279=667%:%
-%:%1280=667%:%
-%:%1281=668%:%
-%:%1282=668%:%
-%:%1283=669%:%
-%:%1284=669%:%
-%:%1285=670%:%
-%:%1286=671%:%
-%:%1287=671%:%
-%:%1288=672%:%
-%:%1289=672%:%
-%:%1290=673%:%
-%:%1291=673%:%
-%:%1292=674%:%
-%:%1293=674%:%
-%:%1294=675%:%
-%:%1295=675%:%
-%:%1296=676%:%
-%:%1297=676%:%
-%:%1298=677%:%
-%:%1299=677%:%
-%:%1300=678%:%
-%:%1301=678%:%
-%:%1302=679%:%
-%:%1303=680%:%
-%:%1304=680%:%
-%:%1305=681%:%
-%:%1306=681%:%
-%:%1307=682%:%
-%:%1322=684%:%
-%:%1332=687%:%
-%:%1333=687%:%
-%:%1334=688%:%
-%:%1335=689%:%
-%:%1342=690%:%
-%:%1343=690%:%
-%:%1344=691%:%
-%:%1345=691%:%
-%:%1346=692%:%
-%:%1347=693%:%
-%:%1348=693%:%
-%:%1349=694%:%
-%:%1350=695%:%
-%:%1351=696%:%
-%:%1352=696%:%
-%:%1353=697%:%
-%:%1354=698%:%
-%:%1355=699%:%
-%:%1356=699%:%
-%:%1357=700%:%
-%:%1358=700%:%
-%:%1359=701%:%
-%:%1360=701%:%
-%:%1361=702%:%
-%:%1362=703%:%
-%:%1363=704%:%
-%:%1364=705%:%
-%:%1365=705%:%
-%:%1366=706%:%
-%:%1372=706%:%
-%:%1375=707%:%
-%:%1376=708%:%
-%:%1377=709%:%
-%:%1378=709%:%
-%:%1379=710%:%
-%:%1380=711%:%
-%:%1383=712%:%
-%:%1387=712%:%
-%:%1388=712%:%
-%:%1389=713%:%
-%:%1390=713%:%
-%:%1391=714%:%
-%:%1392=714%:%
-%:%1393=715%:%
-%:%1394=715%:%
-%:%1395=716%:%
-%:%1396=716%:%
-%:%1397=717%:%
-%:%1398=717%:%
-%:%1399=718%:%
-%:%1400=718%:%
-%:%1401=719%:%
-%:%1402=720%:%
-%:%1403=721%:%
-%:%1404=722%:%
-%:%1405=722%:%
-%:%1406=723%:%
-%:%1407=724%:%
-%:%1408=725%:%
-%:%1409=726%:%
-%:%1410=726%:%
-%:%1411=727%:%
-%:%1412=727%:%
-%:%1413=728%:%
-%:%1414=729%:%
-%:%1415=730%:%
-%:%1416=730%:%
-%:%1417=731%:%
-%:%1423=731%:%
-%:%1426=732%:%
-%:%1427=733%:%
-%:%1428=733%:%
-%:%1429=734%:%
-%:%1430=735%:%
-%:%1431=736%:%
-%:%1432=737%:%
-%:%1433=738%:%
-%:%1434=739%:%
-%:%1441=740%:%
-%:%1442=740%:%
-%:%1443=741%:%
-%:%1444=741%:%
-%:%1445=742%:%
-%:%1446=743%:%
-%:%1447=744%:%
-%:%1448=744%:%
-%:%1449=745%:%
-%:%1450=746%:%
-%:%1451=746%:%
-%:%1452=747%:%
-%:%1453=747%:%
-%:%1454=747%:%
-%:%1455=748%:%
-%:%1456=748%:%
-%:%1457=749%:%
-%:%1458=749%:%
-%:%1459=750%:%
-%:%1460=750%:%
-%:%1461=751%:%
-%:%1462=751%:%
-%:%1463=752%:%
-%:%1469=752%:%
-%:%1474=753%:%
-%:%1479=754%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Maximum_Aggregator.tex b/virage/src/test/resources/old_theories/output/document/Maximum_Aggregator.tex
deleted file mode 100644
index 7d741d6..0000000
--- a/virage/src/test/resources/old_theories/output/document/Maximum_Aggregator.tex
+++ /dev/null
@@ -1,561 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Maximum{\isacharunderscore}{\kern0pt}Aggregator}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Maximum Aggregator%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Maximum{\isacharunderscore}{\kern0pt}Aggregator\isanewline
-\ \ \isakeyword{imports}\ Aggregator\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The max(imum) aggregator takes two partitions of an alternative set A as
-input. It returns a partition where every alternative receives the maximum
-result of the two input partitions.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}aggregator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemma%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}l\ {\isasymin}\ A{\isachardot}{\kern0pt}\ l\ {\isasymnotin}\ e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ set{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}aggregator\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ aggregator{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}e{\isadigit{2}}\ {\isasymunion}\ r{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ e{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ r{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ asm{\isadigit{3}}\ asm{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}e{\isadigit{2}}\ {\isasymunion}\ r{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ e{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ r{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ d{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ asm{\isadigit{3}}\ asm{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}consv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ subset{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}comm{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}commutative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}commutative{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}aggregator\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Component_Types/Maximum_Aggregator.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%53=19%:%
-%:%63=21%:%
-%:%64=21%:%
-%:%65=22%:%
-%:%75=27%:%
-%:%85=29%:%
-%:%86=29%:%
-%:%88=31%:%
-%:%95=32%:%
-%:%96=32%:%
-%:%97=33%:%
-%:%98=33%:%
-%:%99=34%:%
-%:%100=34%:%
-%:%101=35%:%
-%:%102=35%:%
-%:%103=35%:%
-%:%104=36%:%
-%:%105=37%:%
-%:%106=37%:%
-%:%107=38%:%
-%:%108=38%:%
-%:%109=38%:%
-%:%110=39%:%
-%:%111=40%:%
-%:%112=41%:%
-%:%113=41%:%
-%:%114=42%:%
-%:%115=42%:%
-%:%116=42%:%
-%:%117=43%:%
-%:%118=44%:%
-%:%119=44%:%
-%:%120=45%:%
-%:%121=45%:%
-%:%122=45%:%
-%:%123=46%:%
-%:%124=46%:%
-%:%125=47%:%
-%:%140=49%:%
-%:%150=51%:%
-%:%151=51%:%
-%:%154=52%:%
-%:%158=52%:%
-%:%159=52%:%
-%:%160=53%:%
-%:%161=53%:%
-%:%162=54%:%
-%:%163=54%:%
-%:%164=55%:%
-%:%165=56%:%
-%:%166=57%:%
-%:%167=58%:%
-%:%168=59%:%
-%:%169=60%:%
-%:%170=61%:%
-%:%171=62%:%
-%:%172=63%:%
-%:%173=63%:%
-%:%174=64%:%
-%:%175=65%:%
-%:%176=66%:%
-%:%177=67%:%
-%:%178=68%:%
-%:%179=68%:%
-%:%180=69%:%
-%:%181=69%:%
-%:%182=70%:%
-%:%183=70%:%
-%:%184=71%:%
-%:%185=71%:%
-%:%186=72%:%
-%:%187=72%:%
-%:%188=73%:%
-%:%189=74%:%
-%:%190=75%:%
-%:%191=76%:%
-%:%192=77%:%
-%:%193=78%:%
-%:%194=79%:%
-%:%195=80%:%
-%:%196=81%:%
-%:%197=81%:%
-%:%198=82%:%
-%:%199=83%:%
-%:%200=84%:%
-%:%201=85%:%
-%:%202=86%:%
-%:%203=86%:%
-%:%204=87%:%
-%:%205=87%:%
-%:%206=88%:%
-%:%207=88%:%
-%:%208=89%:%
-%:%223=91%:%
-%:%233=94%:%
-%:%234=94%:%
-%:%241=95%:%
-%:%242=95%:%
-%:%243=96%:%
-%:%244=96%:%
-%:%245=97%:%
-%:%247=99%:%
-%:%248=100%:%
-%:%249=100%:%
-%:%250=101%:%
-%:%251=101%:%
-%:%252=102%:%
-%:%253=102%:%
-%:%254=103%:%
-%:%256=105%:%
-%:%257=106%:%
-%:%258=106%:%
-%:%259=107%:%
-%:%260=107%:%
-%:%261=107%:%
-%:%262=108%:%
-%:%264=110%:%
-%:%265=111%:%
-%:%266=111%:%
-%:%267=112%:%
-%:%268=112%:%
-%:%269=112%:%
-%:%270=113%:%
-%:%273=116%:%
-%:%274=117%:%
-%:%275=117%:%
-%:%276=118%:%
-%:%277=118%:%
-%:%278=118%:%
-%:%279=119%:%
-%:%281=121%:%
-%:%282=122%:%
-%:%283=122%:%
-%:%284=123%:%
-%:%285=123%:%
-%:%286=123%:%
-%:%287=124%:%
-%:%291=128%:%
-%:%292=129%:%
-%:%293=129%:%
-%:%294=130%:%
-%:%295=130%:%
-%:%296=131%:%
-%:%297=131%:%
-%:%298=132%:%
-%:%304=132%:%
-%:%307=133%:%
-%:%308=134%:%
-%:%309=135%:%
-%:%310=135%:%
-%:%313=136%:%
-%:%317=136%:%
-%:%318=136%:%
-%:%319=137%:%
-%:%320=137%:%
-%:%321=138%:%
-%:%322=138%:%
-%:%323=139%:%
-%:%324=139%:%
-%:%325=140%:%
-%:%326=140%:%
-%:%327=141%:%
-%:%328=141%:%
-%:%329=142%:%
-%:%330=143%:%
-%:%331=144%:%
-%:%332=145%:%
-%:%333=146%:%
-%:%334=147%:%
-%:%335=148%:%
-%:%336=149%:%
-%:%337=149%:%
-%:%338=150%:%
-%:%339=151%:%
-%:%340=152%:%
-%:%341=152%:%
-%:%342=153%:%
-%:%348=153%:%
-%:%353=154%:%
-%:%358=155%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Maximum_Parallel_Composition.tex b/virage/src/test/resources/old_theories/output/document/Maximum_Parallel_Composition.tex
deleted file mode 100644
index 69a4e07..0000000
--- a/virage/src/test/resources/old_theories/output/document/Maximum_Parallel_Composition.tex
+++ /dev/null
@@ -1,1960 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Maximum{\isacharunderscore}{\kern0pt}Parallel{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Maximum Parallel Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Maximum{\isacharunderscore}{\kern0pt}Parallel{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Maximum{\isacharunderscore}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Parallel{\isacharunderscore}{\kern0pt}Composition\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is a family of parallel compositions. It composes a new electoral module
-from two electoral modules combined with the maximum aggregator. Therein, the
-two modules each make a decision and then a partition is returned where every
-alternative receives the maximum result of the two input partitions. This means
-that, if any alternative is elected by at least one of the modules, then it
-gets elected, if any non-elected alternative is deferred by at least one of the
-modules, then it gets deferred, only alternatives rejected by both modules get
-rejected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition\ m\ n\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}let\ a\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}aggregator\ in\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}parallel\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}\isakeyword{infix}\ {\isachardoublequoteopen}{\isasymparallel}\isactrlsub {\isasymup}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition\ m\ n{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}m\ mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}result{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ A\ p\ x\ {\isasymor}\isanewline
-\ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ x\ {\isasymin}\ e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ Un{\isacharunderscore}{\kern0pt}iff\ a{\isadigit{1}}\ empty{\isacharunderscore}{\kern0pt}iff\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ DiffD{\isadigit{1}}\ DiffD{\isadigit{2}}\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardot}{\kern0pt}simps\ Un{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}conv\ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ prod{\isachardot}{\kern0pt}collapse\ f{\isacharunderscore}{\kern0pt}prof\ sndI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Int{\isacharunderscore}{\kern0pt}iff\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}a{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ a{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ in{\isacharunderscore}{\kern0pt}A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ f{\isacharunderscore}{\kern0pt}prof\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ fst\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}unfold\ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ surjective{\isacharunderscore}{\kern0pt}pairing\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}iff\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}consv\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\ not{\isacharunderscore}{\kern0pt}a{\isadigit{1}}\ not{\isacharunderscore}{\kern0pt}a{\isadigit{2}}\ Int{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}both{\isacharunderscore}{\kern0pt}reject{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ fst\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ case{\isacharunderscore}{\kern0pt}prodI{\isadigit{2}}\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ split{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ prod{\isachardot}{\kern0pt}collapse\ split{\isacharunderscore}{\kern0pt}beta\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}iff\ a\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\ module{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n\ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ surjective{\isacharunderscore}{\kern0pt}pairing\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ elect\ m\ A\ p\ {\isasymand}\ x\ {\isasymnotin}\ defer\ m\ A\ p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymnotin}\ elect\ n\ A\ p\ {\isasymand}\ x\ {\isasymnotin}\ defer\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ empty{\isacharunderscore}{\kern0pt}iff\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ DiffD{\isadigit{1}}\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardot}{\kern0pt}simps\ Un{\isacharunderscore}{\kern0pt}iff\ a\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ prod{\isachardot}{\kern0pt}simps\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ old{\isachardot}{\kern0pt}prod{\isachardot}{\kern0pt}inject\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ prod{\isachardot}{\kern0pt}collapse\ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Set{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}insert\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}insert\isanewline
-\ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}result\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}both{\isacharunderscore}{\kern0pt}reject\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ rejected\isanewline
-\ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ rejected\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}result\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}both{\isacharunderscore}{\kern0pt}reject\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ rejected\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ rejected\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}intersect{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ A\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dcompat{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}by{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ DiffI\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Composition Rules%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ conserv{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Profile{\isachardot}{\kern0pt}lifted\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}profs{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ compatible\ \isacommand{obtain}\isamarkupfalse%
-\ A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\ A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ x\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ x\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ x\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ x\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}x\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ n\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}profs\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}n\ lifted{\isacharunderscore}{\kern0pt}x\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ monotone{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{2}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ lifted{\isacharunderscore}{\kern0pt}x\ a{\isadigit{0}}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ f{\isacharunderscore}{\kern0pt}profs\ IntI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ empty{\isacharunderscore}{\kern0pt}iff\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffI\ lifted{\isacharunderscore}{\kern0pt}x\ compatible\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ n\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}x\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ DiffD{\isadigit{2}}\ compatible\ dcompat{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}by{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ lifted{\isacharunderscore}{\kern0pt}x\ a{\isadigit{1}}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{2}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}profs\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}x\ defer{\isacharunderscore}{\kern0pt}n\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}profs\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}profs\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ eq{\isacharunderscore}{\kern0pt}alts{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}profs{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}results\ max{\isacharunderscore}{\kern0pt}par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ x\ y{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}sum{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ S\ {\isacharplus}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ compatible\ \isacommand{obtain}\isamarkupfalse%
-\ A\ \isakeyword{where}\ A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ x\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ x\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ y\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ compatible\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}representation{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}intersect\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ x\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ subsets{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ S\ {\isasymand}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ finite\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}subset\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ S\ {\isacharplus}{\kern0pt}\ n\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}Int\ reject{\isacharunderscore}{\kern0pt}representation\ reject{\isacharunderscore}{\kern0pt}sum\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ S{\isachardot}{\kern0pt}\ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ subsets\ subset{\isacharunderscore}{\kern0pt}eq\ sup{\isachardot}{\kern0pt}absorb{\isacharunderscore}{\kern0pt}iff{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isachardot}{\kern0pt}cobounded{\isadigit{1}}\ sup{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}commute\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ defers{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejec{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ disj{\isacharunderscore}{\kern0pt}comp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ eliminates{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rejec{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ rejects{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharless}{\kern0pt}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{2}}\ less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}nat\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ elec{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ module\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ fin{\isacharunderscore}{\kern0pt}A{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ reject\ m\ A\ p{\isacharcomma}{\kern0pt}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ module\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ elec{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ case{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isasymge}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ case{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ rejec{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ rejects{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}m\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ A\ {\isacharplus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}comp\ prof{\isacharunderscore}{\kern0pt}A\ fin{\isacharunderscore}{\kern0pt}A\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}m\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Maximum_Parallel_Composition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%48=21%:%
-%:%49=22%:%
-%:%50=23%:%
-%:%59=25%:%
-%:%69=27%:%
-%:%70=27%:%
-%:%71=28%:%
-%:%72=29%:%
-%:%73=30%:%
-%:%74=31%:%
-%:%75=32%:%
-%:%76=32%:%
-%:%77=33%:%
-%:%78=34%:%
-%:%85=36%:%
-%:%95=38%:%
-%:%96=38%:%
-%:%97=39%:%
-%:%98=40%:%
-%:%99=41%:%
-%:%100=42%:%
-%:%103=43%:%
-%:%107=43%:%
-%:%108=43%:%
-%:%109=44%:%
-%:%110=44%:%
-%:%124=46%:%
-%:%134=48%:%
-%:%135=48%:%
-%:%136=49%:%
-%:%137=50%:%
-%:%138=51%:%
-%:%139=52%:%
-%:%140=53%:%
-%:%141=54%:%
-%:%142=55%:%
-%:%143=56%:%
-%:%150=57%:%
-%:%151=57%:%
-%:%152=58%:%
-%:%153=58%:%
-%:%154=59%:%
-%:%155=59%:%
-%:%156=60%:%
-%:%158=62%:%
-%:%159=63%:%
-%:%160=63%:%
-%:%161=64%:%
-%:%162=64%:%
-%:%163=65%:%
-%:%164=65%:%
-%:%165=66%:%
-%:%166=66%:%
-%:%167=67%:%
-%:%168=67%:%
-%:%169=68%:%
-%:%170=69%:%
-%:%171=70%:%
-%:%172=70%:%
-%:%173=71%:%
-%:%174=71%:%
-%:%175=72%:%
-%:%176=72%:%
-%:%177=73%:%
-%:%178=73%:%
-%:%179=74%:%
-%:%180=74%:%
-%:%181=75%:%
-%:%182=75%:%
-%:%183=76%:%
-%:%184=76%:%
-%:%185=77%:%
-%:%186=77%:%
-%:%187=78%:%
-%:%188=79%:%
-%:%189=80%:%
-%:%190=81%:%
-%:%191=82%:%
-%:%192=83%:%
-%:%193=83%:%
-%:%194=84%:%
-%:%195=84%:%
-%:%196=85%:%
-%:%197=85%:%
-%:%198=86%:%
-%:%199=86%:%
-%:%200=86%:%
-%:%201=87%:%
-%:%202=88%:%
-%:%203=88%:%
-%:%204=89%:%
-%:%205=90%:%
-%:%206=90%:%
-%:%207=91%:%
-%:%208=91%:%
-%:%209=92%:%
-%:%211=94%:%
-%:%212=95%:%
-%:%213=95%:%
-%:%214=96%:%
-%:%215=97%:%
-%:%216=97%:%
-%:%217=98%:%
-%:%218=98%:%
-%:%219=99%:%
-%:%221=101%:%
-%:%222=102%:%
-%:%223=102%:%
-%:%224=103%:%
-%:%225=103%:%
-%:%226=104%:%
-%:%227=104%:%
-%:%228=105%:%
-%:%229=106%:%
-%:%230=107%:%
-%:%231=108%:%
-%:%232=109%:%
-%:%233=110%:%
-%:%234=111%:%
-%:%235=111%:%
-%:%236=112%:%
-%:%237=112%:%
-%:%238=113%:%
-%:%244=113%:%
-%:%247=114%:%
-%:%248=115%:%
-%:%249=115%:%
-%:%250=116%:%
-%:%251=117%:%
-%:%252=118%:%
-%:%253=119%:%
-%:%254=120%:%
-%:%255=121%:%
-%:%256=122%:%
-%:%263=123%:%
-%:%264=123%:%
-%:%265=124%:%
-%:%266=124%:%
-%:%267=125%:%
-%:%268=126%:%
-%:%269=127%:%
-%:%270=127%:%
-%:%271=128%:%
-%:%272=128%:%
-%:%273=129%:%
-%:%274=129%:%
-%:%275=130%:%
-%:%277=132%:%
-%:%278=133%:%
-%:%279=133%:%
-%:%280=134%:%
-%:%281=135%:%
-%:%282=135%:%
-%:%283=136%:%
-%:%284=136%:%
-%:%285=137%:%
-%:%287=139%:%
-%:%288=140%:%
-%:%289=140%:%
-%:%290=141%:%
-%:%291=141%:%
-%:%292=142%:%
-%:%293=142%:%
-%:%294=143%:%
-%:%295=144%:%
-%:%296=145%:%
-%:%297=145%:%
-%:%298=146%:%
-%:%299=146%:%
-%:%300=147%:%
-%:%301=147%:%
-%:%302=147%:%
-%:%303=148%:%
-%:%304=149%:%
-%:%305=150%:%
-%:%306=150%:%
-%:%307=151%:%
-%:%308=151%:%
-%:%309=152%:%
-%:%310=152%:%
-%:%311=153%:%
-%:%312=154%:%
-%:%313=155%:%
-%:%314=155%:%
-%:%315=156%:%
-%:%316=156%:%
-%:%317=157%:%
-%:%318=157%:%
-%:%319=158%:%
-%:%320=158%:%
-%:%321=159%:%
-%:%322=160%:%
-%:%323=161%:%
-%:%324=162%:%
-%:%325=163%:%
-%:%326=163%:%
-%:%327=164%:%
-%:%328=164%:%
-%:%329=165%:%
-%:%330=165%:%
-%:%331=165%:%
-%:%332=166%:%
-%:%333=166%:%
-%:%334=167%:%
-%:%340=167%:%
-%:%343=168%:%
-%:%344=169%:%
-%:%345=169%:%
-%:%346=170%:%
-%:%347=171%:%
-%:%348=172%:%
-%:%349=173%:%
-%:%350=174%:%
-%:%351=175%:%
-%:%352=176%:%
-%:%355=177%:%
-%:%359=177%:%
-%:%360=177%:%
-%:%361=178%:%
-%:%362=179%:%
-%:%363=180%:%
-%:%364=181%:%
-%:%365=182%:%
-%:%366=182%:%
-%:%371=182%:%
-%:%374=183%:%
-%:%375=184%:%
-%:%376=184%:%
-%:%377=185%:%
-%:%378=186%:%
-%:%379=187%:%
-%:%380=188%:%
-%:%381=189%:%
-%:%382=190%:%
-%:%383=191%:%
-%:%386=192%:%
-%:%390=192%:%
-%:%391=192%:%
-%:%392=193%:%
-%:%393=194%:%
-%:%394=194%:%
-%:%399=194%:%
-%:%402=195%:%
-%:%403=196%:%
-%:%404=196%:%
-%:%405=197%:%
-%:%406=198%:%
-%:%407=199%:%
-%:%408=200%:%
-%:%409=201%:%
-%:%410=202%:%
-%:%411=203%:%
-%:%414=204%:%
-%:%418=204%:%
-%:%419=204%:%
-%:%420=205%:%
-%:%421=206%:%
-%:%422=207%:%
-%:%423=208%:%
-%:%424=208%:%
-%:%429=208%:%
-%:%432=209%:%
-%:%433=210%:%
-%:%434=210%:%
-%:%435=211%:%
-%:%436=212%:%
-%:%437=213%:%
-%:%438=214%:%
-%:%439=215%:%
-%:%440=216%:%
-%:%441=217%:%
-%:%444=218%:%
-%:%448=218%:%
-%:%449=218%:%
-%:%450=219%:%
-%:%451=220%:%
-%:%452=220%:%
-%:%457=220%:%
-%:%460=221%:%
-%:%461=222%:%
-%:%462=222%:%
-%:%463=223%:%
-%:%464=224%:%
-%:%465=225%:%
-%:%466=226%:%
-%:%467=227%:%
-%:%468=228%:%
-%:%469=229%:%
-%:%476=230%:%
-%:%477=230%:%
-%:%478=231%:%
-%:%479=231%:%
-%:%480=232%:%
-%:%481=233%:%
-%:%482=234%:%
-%:%483=234%:%
-%:%484=235%:%
-%:%485=235%:%
-%:%486=236%:%
-%:%487=237%:%
-%:%488=238%:%
-%:%489=238%:%
-%:%490=239%:%
-%:%491=239%:%
-%:%492=240%:%
-%:%493=240%:%
-%:%494=241%:%
-%:%495=242%:%
-%:%496=243%:%
-%:%497=243%:%
-%:%498=244%:%
-%:%499=244%:%
-%:%500=245%:%
-%:%502=247%:%
-%:%503=248%:%
-%:%504=248%:%
-%:%505=249%:%
-%:%506=249%:%
-%:%507=250%:%
-%:%508=250%:%
-%:%509=251%:%
-%:%515=251%:%
-%:%518=252%:%
-%:%519=253%:%
-%:%520=253%:%
-%:%521=254%:%
-%:%522=255%:%
-%:%523=256%:%
-%:%524=257%:%
-%:%525=258%:%
-%:%528=261%:%
-%:%531=262%:%
-%:%535=262%:%
-%:%536=262%:%
-%:%537=263%:%
-%:%538=264%:%
-%:%539=264%:%
-%:%553=266%:%
-%:%563=272%:%
-%:%564=272%:%
-%:%565=273%:%
-%:%566=274%:%
-%:%567=275%:%
-%:%568=276%:%
-%:%571=277%:%
-%:%575=277%:%
-%:%576=277%:%
-%:%577=278%:%
-%:%578=278%:%
-%:%583=278%:%
-%:%586=279%:%
-%:%587=283%:%
-%:%588=284%:%
-%:%589=284%:%
-%:%590=285%:%
-%:%591=286%:%
-%:%592=287%:%
-%:%593=288%:%
-%:%594=289%:%
-%:%597=290%:%
-%:%601=290%:%
-%:%602=290%:%
-%:%603=291%:%
-%:%604=291%:%
-%:%605=292%:%
-%:%606=292%:%
-%:%607=293%:%
-%:%608=293%:%
-%:%609=294%:%
-%:%610=294%:%
-%:%611=295%:%
-%:%612=295%:%
-%:%613=296%:%
-%:%614=296%:%
-%:%615=297%:%
-%:%616=297%:%
-%:%617=298%:%
-%:%618=298%:%
-%:%619=299%:%
-%:%620=299%:%
-%:%621=300%:%
-%:%622=300%:%
-%:%623=301%:%
-%:%624=301%:%
-%:%625=302%:%
-%:%626=302%:%
-%:%627=303%:%
-%:%628=304%:%
-%:%629=305%:%
-%:%630=306%:%
-%:%631=307%:%
-%:%632=307%:%
-%:%633=308%:%
-%:%634=309%:%
-%:%635=310%:%
-%:%636=310%:%
-%:%637=311%:%
-%:%638=311%:%
-%:%639=312%:%
-%:%640=312%:%
-%:%641=312%:%
-%:%642=313%:%
-%:%645=316%:%
-%:%646=317%:%
-%:%647=317%:%
-%:%648=318%:%
-%:%649=318%:%
-%:%650=319%:%
-%:%651=319%:%
-%:%652=320%:%
-%:%653=321%:%
-%:%654=321%:%
-%:%655=322%:%
-%:%656=322%:%
-%:%657=323%:%
-%:%658=323%:%
-%:%659=324%:%
-%:%660=324%:%
-%:%661=325%:%
-%:%662=325%:%
-%:%663=326%:%
-%:%664=326%:%
-%:%665=326%:%
-%:%666=327%:%
-%:%667=327%:%
-%:%668=328%:%
-%:%669=329%:%
-%:%670=329%:%
-%:%671=330%:%
-%:%672=330%:%
-%:%673=331%:%
-%:%674=332%:%
-%:%675=332%:%
-%:%676=333%:%
-%:%677=334%:%
-%:%678=334%:%
-%:%679=335%:%
-%:%680=335%:%
-%:%681=335%:%
-%:%682=336%:%
-%:%683=336%:%
-%:%684=337%:%
-%:%685=338%:%
-%:%686=338%:%
-%:%687=339%:%
-%:%688=339%:%
-%:%689=339%:%
-%:%690=340%:%
-%:%691=341%:%
-%:%692=341%:%
-%:%693=342%:%
-%:%694=343%:%
-%:%695=343%:%
-%:%696=344%:%
-%:%697=344%:%
-%:%698=344%:%
-%:%699=345%:%
-%:%700=346%:%
-%:%701=346%:%
-%:%702=347%:%
-%:%703=347%:%
-%:%704=348%:%
-%:%705=349%:%
-%:%706=349%:%
-%:%707=350%:%
-%:%708=351%:%
-%:%709=351%:%
-%:%710=352%:%
-%:%711=352%:%
-%:%712=352%:%
-%:%713=353%:%
-%:%714=353%:%
-%:%715=354%:%
-%:%716=355%:%
-%:%717=356%:%
-%:%718=356%:%
-%:%719=357%:%
-%:%720=357%:%
-%:%721=357%:%
-%:%722=358%:%
-%:%723=359%:%
-%:%724=359%:%
-%:%725=360%:%
-%:%726=361%:%
-%:%727=361%:%
-%:%728=362%:%
-%:%729=362%:%
-%:%730=362%:%
-%:%731=363%:%
-%:%732=364%:%
-%:%733=364%:%
-%:%734=365%:%
-%:%735=365%:%
-%:%736=366%:%
-%:%737=366%:%
-%:%738=367%:%
-%:%739=367%:%
-%:%740=368%:%
-%:%741=368%:%
-%:%742=369%:%
-%:%743=369%:%
-%:%744=370%:%
-%:%745=370%:%
-%:%746=371%:%
-%:%747=371%:%
-%:%748=372%:%
-%:%749=373%:%
-%:%750=373%:%
-%:%751=374%:%
-%:%752=374%:%
-%:%753=375%:%
-%:%754=375%:%
-%:%755=376%:%
-%:%756=376%:%
-%:%757=377%:%
-%:%758=377%:%
-%:%759=377%:%
-%:%760=378%:%
-%:%761=378%:%
-%:%762=379%:%
-%:%763=380%:%
-%:%764=381%:%
-%:%765=382%:%
-%:%766=383%:%
-%:%767=383%:%
-%:%768=384%:%
-%:%769=384%:%
-%:%770=385%:%
-%:%771=386%:%
-%:%772=386%:%
-%:%773=387%:%
-%:%774=388%:%
-%:%775=388%:%
-%:%776=389%:%
-%:%777=389%:%
-%:%778=389%:%
-%:%779=390%:%
-%:%780=390%:%
-%:%781=391%:%
-%:%782=392%:%
-%:%783=393%:%
-%:%784=393%:%
-%:%785=394%:%
-%:%786=394%:%
-%:%787=394%:%
-%:%788=395%:%
-%:%789=396%:%
-%:%790=396%:%
-%:%791=397%:%
-%:%792=398%:%
-%:%793=398%:%
-%:%794=399%:%
-%:%795=399%:%
-%:%796=399%:%
-%:%797=400%:%
-%:%798=401%:%
-%:%799=401%:%
-%:%800=402%:%
-%:%801=402%:%
-%:%802=403%:%
-%:%803=404%:%
-%:%804=404%:%
-%:%805=405%:%
-%:%806=406%:%
-%:%807=406%:%
-%:%808=407%:%
-%:%809=407%:%
-%:%810=407%:%
-%:%811=408%:%
-%:%812=408%:%
-%:%813=409%:%
-%:%814=410%:%
-%:%815=410%:%
-%:%816=411%:%
-%:%817=411%:%
-%:%818=411%:%
-%:%819=412%:%
-%:%820=413%:%
-%:%821=413%:%
-%:%822=414%:%
-%:%823=415%:%
-%:%824=415%:%
-%:%825=416%:%
-%:%826=416%:%
-%:%827=416%:%
-%:%828=417%:%
-%:%829=418%:%
-%:%830=418%:%
-%:%831=419%:%
-%:%832=419%:%
-%:%833=420%:%
-%:%834=420%:%
-%:%835=421%:%
-%:%836=421%:%
-%:%837=422%:%
-%:%838=422%:%
-%:%839=423%:%
-%:%840=423%:%
-%:%841=424%:%
-%:%842=424%:%
-%:%843=425%:%
-%:%844=425%:%
-%:%845=426%:%
-%:%846=427%:%
-%:%847=427%:%
-%:%848=428%:%
-%:%854=428%:%
-%:%857=429%:%
-%:%858=430%:%
-%:%859=430%:%
-%:%860=431%:%
-%:%861=432%:%
-%:%862=433%:%
-%:%863=434%:%
-%:%864=435%:%
-%:%871=436%:%
-%:%872=436%:%
-%:%873=437%:%
-%:%874=437%:%
-%:%875=437%:%
-%:%876=438%:%
-%:%880=442%:%
-%:%881=443%:%
-%:%882=443%:%
-%:%883=444%:%
-%:%884=444%:%
-%:%885=445%:%
-%:%886=445%:%
-%:%887=446%:%
-%:%888=446%:%
-%:%889=447%:%
-%:%890=448%:%
-%:%891=448%:%
-%:%892=449%:%
-%:%893=449%:%
-%:%894=450%:%
-%:%895=450%:%
-%:%896=451%:%
-%:%897=451%:%
-%:%898=452%:%
-%:%899=452%:%
-%:%900=453%:%
-%:%901=453%:%
-%:%902=454%:%
-%:%903=454%:%
-%:%904=455%:%
-%:%905=455%:%
-%:%906=456%:%
-%:%907=456%:%
-%:%908=457%:%
-%:%909=457%:%
-%:%910=458%:%
-%:%911=458%:%
-%:%912=459%:%
-%:%914=461%:%
-%:%915=462%:%
-%:%916=462%:%
-%:%917=463%:%
-%:%918=463%:%
-%:%919=464%:%
-%:%920=464%:%
-%:%921=465%:%
-%:%922=465%:%
-%:%923=466%:%
-%:%924=466%:%
-%:%925=467%:%
-%:%926=467%:%
-%:%927=468%:%
-%:%928=468%:%
-%:%929=469%:%
-%:%930=470%:%
-%:%931=470%:%
-%:%932=471%:%
-%:%933=471%:%
-%:%934=472%:%
-%:%935=472%:%
-%:%936=473%:%
-%:%937=473%:%
-%:%938=474%:%
-%:%944=474%:%
-%:%947=475%:%
-%:%948=481%:%
-%:%949=482%:%
-%:%950=482%:%
-%:%951=483%:%
-%:%952=484%:%
-%:%953=485%:%
-%:%954=486%:%
-%:%955=487%:%
-%:%956=488%:%
-%:%959=489%:%
-%:%963=489%:%
-%:%964=489%:%
-%:%965=490%:%
-%:%966=490%:%
-%:%967=491%:%
-%:%968=491%:%
-%:%969=492%:%
-%:%970=492%:%
-%:%971=493%:%
-%:%972=493%:%
-%:%973=494%:%
-%:%974=494%:%
-%:%975=495%:%
-%:%976=495%:%
-%:%977=496%:%
-%:%978=496%:%
-%:%979=497%:%
-%:%980=497%:%
-%:%981=498%:%
-%:%982=498%:%
-%:%983=499%:%
-%:%984=499%:%
-%:%985=500%:%
-%:%986=500%:%
-%:%987=501%:%
-%:%988=501%:%
-%:%989=502%:%
-%:%990=503%:%
-%:%991=504%:%
-%:%992=504%:%
-%:%993=505%:%
-%:%994=506%:%
-%:%995=507%:%
-%:%996=508%:%
-%:%997=508%:%
-%:%998=509%:%
-%:%999=509%:%
-%:%1000=510%:%
-%:%1001=510%:%
-%:%1002=511%:%
-%:%1003=511%:%
-%:%1004=512%:%
-%:%1005=512%:%
-%:%1006=513%:%
-%:%1007=513%:%
-%:%1008=514%:%
-%:%1009=514%:%
-%:%1010=515%:%
-%:%1011=515%:%
-%:%1012=516%:%
-%:%1013=516%:%
-%:%1014=517%:%
-%:%1015=517%:%
-%:%1016=518%:%
-%:%1017=518%:%
-%:%1018=518%:%
-%:%1019=519%:%
-%:%1020=520%:%
-%:%1021=520%:%
-%:%1022=521%:%
-%:%1023=521%:%
-%:%1024=522%:%
-%:%1025=522%:%
-%:%1026=522%:%
-%:%1027=523%:%
-%:%1028=524%:%
-%:%1029=524%:%
-%:%1030=525%:%
-%:%1031=525%:%
-%:%1032=526%:%
-%:%1033=526%:%
-%:%1034=527%:%
-%:%1035=527%:%
-%:%1036=527%:%
-%:%1037=528%:%
-%:%1038=529%:%
-%:%1039=530%:%
-%:%1040=530%:%
-%:%1041=531%:%
-%:%1042=531%:%
-%:%1043=532%:%
-%:%1044=532%:%
-%:%1045=532%:%
-%:%1046=533%:%
-%:%1048=535%:%
-%:%1049=536%:%
-%:%1050=536%:%
-%:%1051=537%:%
-%:%1052=537%:%
-%:%1053=538%:%
-%:%1054=538%:%
-%:%1055=539%:%
-%:%1056=539%:%
-%:%1057=540%:%
-%:%1058=540%:%
-%:%1059=541%:%
-%:%1060=541%:%
-%:%1061=542%:%
-%:%1062=542%:%
-%:%1063=543%:%
-%:%1064=543%:%
-%:%1065=544%:%
-%:%1066=544%:%
-%:%1067=545%:%
-%:%1068=545%:%
-%:%1069=545%:%
-%:%1070=546%:%
-%:%1071=547%:%
-%:%1072=547%:%
-%:%1073=548%:%
-%:%1074=548%:%
-%:%1075=549%:%
-%:%1076=549%:%
-%:%1077=550%:%
-%:%1078=550%:%
-%:%1079=551%:%
-%:%1080=552%:%
-%:%1081=553%:%
-%:%1082=553%:%
-%:%1083=554%:%
-%:%1084=554%:%
-%:%1085=555%:%
-%:%1086=555%:%
-%:%1087=556%:%
-%:%1088=556%:%
-%:%1089=557%:%
-%:%1090=558%:%
-%:%1091=558%:%
-%:%1092=559%:%
-%:%1093=559%:%
-%:%1094=560%:%
-%:%1100=560%:%
-%:%1105=561%:%
-%:%1110=562%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Minimax_Module.tex b/virage/src/test/resources/old_theories/output/document/Minimax_Module.tex
deleted file mode 100644
index 328683a..0000000
--- a/virage/src/test/resources/old_theories/output/document/Minimax_Module.tex
+++ /dev/null
@@ -1,690 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Minimax{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Minimax Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Minimax{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Minimax module used by the Minimax voting rule. The Minimax rule
-elects the alternatives with the highest Minimax score. The module implemented
-herein only rejects the alternatives not elected by the voting rule, and defers
-the alternatives that would be elected by the full voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}score\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ minimax\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax\ A\ p\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}eliminator\ minimax{\isacharunderscore}{\kern0pt}score\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}eliminator\ minimax{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemma%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}minimax{\isacharunderscore}{\kern0pt}score{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}score\ l\ A\ p\ {\isasymle}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isacharquery}{\kern0pt}set\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isacharquery}{\kern0pt}lscore\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}score\ l\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prof\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}set\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}lscore\ {\isacharequal}{\kern0pt}\ Min\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}lscore\ {\isasymin}\ {\isacharquery}{\kern0pt}set\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ {\isasymin}\ {\isacharquery}{\kern0pt}set{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}lscore\ {\isasymle}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ local{\isachardot}{\kern0pt}finite\ not{\isacharunderscore}{\kern0pt}empty\ Min{\isacharunderscore}{\kern0pt}le\ Min{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ minimax{\isacharunderscore}{\kern0pt}score{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\ minimax{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\ prefer{\isacharunderscore}{\kern0pt}count{\isachardot}{\kern0pt}simps{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymge}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymge}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ prof\ winner\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w\ \ {\isasymge}\ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}minimax{\isacharunderscore}{\kern0pt}score\ minimax{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}without{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prof\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\ finite{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}\ n\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}in\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ n\ \isakeyword{where}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isadigit{6}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffE\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{6}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{6}}\ prof\ winner\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ n\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ prof\ n{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ n{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \ \ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}sym\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{7}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ n\ w\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{3}}{\isadigit{0}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{4}}{\isadigit{0}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{7}}{\isadigit{0}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}wins\ l\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}without{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ wins{\isacharunderscore}{\kern0pt}antisym\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ minimax{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}mmaxscore{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ minimax{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ minimax{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}rating{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ mmax{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}mmax{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}minimax\ A\ p\ {\isasymequiv}\ max{\isacharunderscore}{\kern0pt}eliminator\ minimax{\isacharunderscore}{\kern0pt}score\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}mmaxscore{\isacharunderscore}{\kern0pt}dcc\ mmax{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}mmax\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Minimax_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%41=14%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%53=19%:%
-%:%63=21%:%
-%:%64=21%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=25%:%
-%:%70=26%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%73=29%:%
-%:%74=29%:%
-%:%75=30%:%
-%:%76=31%:%
-%:%77=32%:%
-%:%78=32%:%
-%:%79=33%:%
-%:%86=35%:%
-%:%96=37%:%
-%:%97=37%:%
-%:%98=38%:%
-%:%99=39%:%
-%:%100=40%:%
-%:%101=41%:%
-%:%102=42%:%
-%:%103=43%:%
-%:%110=44%:%
-%:%111=44%:%
-%:%112=45%:%
-%:%113=45%:%
-%:%114=46%:%
-%:%115=47%:%
-%:%116=48%:%
-%:%117=48%:%
-%:%118=49%:%
-%:%119=49%:%
-%:%120=50%:%
-%:%121=50%:%
-%:%122=51%:%
-%:%123=51%:%
-%:%124=52%:%
-%:%125=52%:%
-%:%126=53%:%
-%:%127=53%:%
-%:%128=54%:%
-%:%129=54%:%
-%:%130=55%:%
-%:%131=55%:%
-%:%132=56%:%
-%:%133=56%:%
-%:%134=57%:%
-%:%135=57%:%
-%:%136=58%:%
-%:%137=58%:%
-%:%138=59%:%
-%:%139=59%:%
-%:%140=60%:%
-%:%141=60%:%
-%:%142=61%:%
-%:%143=61%:%
-%:%144=62%:%
-%:%145=62%:%
-%:%146=63%:%
-%:%147=63%:%
-%:%148=64%:%
-%:%148=65%:%
-%:%149=66%:%
-%:%150=66%:%
-%:%151=67%:%
-%:%152=67%:%
-%:%153=68%:%
-%:%154=68%:%
-%:%155=69%:%
-%:%156=69%:%
-%:%157=70%:%
-%:%158=70%:%
-%:%159=71%:%
-%:%160=71%:%
-%:%161=72%:%
-%:%162=72%:%
-%:%163=73%:%
-%:%164=73%:%
-%:%165=74%:%
-%:%180=76%:%
-%:%190=78%:%
-%:%191=78%:%
-%:%198=79%:%
-%:%199=79%:%
-%:%200=80%:%
-%:%201=80%:%
-%:%202=81%:%
-%:%203=82%:%
-%:%204=83%:%
-%:%205=84%:%
-%:%206=85%:%
-%:%207=85%:%
-%:%208=86%:%
-%:%209=87%:%
-%:%210=88%:%
-%:%211=89%:%
-%:%212=89%:%
-%:%213=90%:%
-%:%216=93%:%
-%:%217=94%:%
-%:%218=94%:%
-%:%219=95%:%
-%:%220=95%:%
-%:%221=96%:%
-%:%224=99%:%
-%:%225=100%:%
-%:%226=100%:%
-%:%227=101%:%
-%:%230=104%:%
-%:%231=105%:%
-%:%232=105%:%
-%:%233=106%:%
-%:%234=106%:%
-%:%235=107%:%
-%:%236=108%:%
-%:%237=109%:%
-%:%238=109%:%
-%:%239=110%:%
-%:%240=110%:%
-%:%241=111%:%
-%:%242=111%:%
-%:%243=112%:%
-%:%244=112%:%
-%:%245=113%:%
-%:%246=113%:%
-%:%247=114%:%
-%:%248=114%:%
-%:%249=115%:%
-%:%250=116%:%
-%:%251=116%:%
-%:%252=117%:%
-%:%253=117%:%
-%:%254=135%:%
-%:%255=136%:%
-%:%256=136%:%
-%:%257=137%:%
-%:%258=137%:%
-%:%259=138%:%
-%:%260=138%:%
-%:%261=139%:%
-%:%262=139%:%
-%:%263=140%:%
-%:%264=140%:%
-%:%265=141%:%
-%:%266=141%:%
-%:%267=142%:%
-%:%268=142%:%
-%:%269=143%:%
-%:%270=143%:%
-%:%271=144%:%
-%:%272=144%:%
-%:%273=145%:%
-%:%274=145%:%
-%:%275=146%:%
-%:%276=146%:%
-%:%277=147%:%
-%:%278=147%:%
-%:%279=148%:%
-%:%280=149%:%
-%:%281=150%:%
-%:%282=150%:%
-%:%283=151%:%
-%:%284=151%:%
-%:%285=152%:%
-%:%286=152%:%
-%:%287=152%:%
-%:%288=153%:%
-%:%289=154%:%
-%:%290=155%:%
-%:%291=156%:%
-%:%292=156%:%
-%:%293=157%:%
-%:%294=157%:%
-%:%295=158%:%
-%:%296=158%:%
-%:%297=159%:%
-%:%298=159%:%
-%:%299=160%:%
-%:%300=160%:%
-%:%301=161%:%
-%:%302=161%:%
-%:%303=162%:%
-%:%304=162%:%
-%:%305=163%:%
-%:%306=163%:%
-%:%307=164%:%
-%:%308=164%:%
-%:%309=165%:%
-%:%310=165%:%
-%:%311=166%:%
-%:%312=166%:%
-%:%313=167%:%
-%:%314=167%:%
-%:%315=168%:%
-%:%316=168%:%
-%:%317=169%:%
-%:%318=169%:%
-%:%319=170%:%
-%:%320=170%:%
-%:%321=171%:%
-%:%322=171%:%
-%:%323=172%:%
-%:%324=172%:%
-%:%325=173%:%
-%:%326=174%:%
-%:%327=174%:%
-%:%328=175%:%
-%:%329=175%:%
-%:%330=176%:%
-%:%331=176%:%
-%:%332=177%:%
-%:%333=177%:%
-%:%334=178%:%
-%:%335=178%:%
-%:%336=179%:%
-%:%337=179%:%
-%:%338=180%:%
-%:%339=180%:%
-%:%340=181%:%
-%:%341=181%:%
-%:%342=182%:%
-%:%343=182%:%
-%:%344=183%:%
-%:%345=184%:%
-%:%346=184%:%
-%:%347=185%:%
-%:%348=185%:%
-%:%349=186%:%
-%:%355=186%:%
-%:%358=187%:%
-%:%359=188%:%
-%:%360=188%:%
-%:%367=189%:%
-%:%368=189%:%
-%:%369=190%:%
-%:%370=190%:%
-%:%371=191%:%
-%:%372=192%:%
-%:%373=192%:%
-%:%374=193%:%
-%:%375=193%:%
-%:%376=194%:%
-%:%377=194%:%
-%:%378=195%:%
-%:%379=196%:%
-%:%380=196%:%
-%:%381=197%:%
-%:%382=197%:%
-%:%383=198%:%
-%:%384=198%:%
-%:%385=199%:%
-%:%386=199%:%
-%:%387=200%:%
-%:%388=200%:%
-%:%389=201%:%
-%:%395=201%:%
-%:%400=202%:%
-%:%405=203%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Minimax_Rule.tex b/virage/src/test/resources/old_theories/output/document/Minimax_Rule.tex
deleted file mode 100644
index b55f940..0000000
--- a/virage/src/test/resources/old_theories/output/document/Minimax_Rule.tex
+++ /dev/null
@@ -1,177 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Minimax{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Minimax Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Minimax{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Minimax{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Minimax voting rule. It elects the alternatives with the highest
-Minimax score.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ minimax\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ minimax{\isacharunderscore}{\kern0pt}code\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Condorcet Consistency Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}condorcet{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ minimax{\isacharunderscore}{\kern0pt}rule{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ minimax{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ minimax{\isacharunderscore}{\kern0pt}rule{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Minimax_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%52=18%:%
-%:%62=20%:%
-%:%63=20%:%
-%:%64=21%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%67=23%:%
-%:%68=24%:%
-%:%75=26%:%
-%:%85=28%:%
-%:%86=28%:%
-%:%93=29%:%
-%:%94=29%:%
-%:%95=30%:%
-%:%96=30%:%
-%:%97=31%:%
-%:%98=32%:%
-%:%99=32%:%
-%:%100=33%:%
-%:%101=33%:%
-%:%102=34%:%
-%:%103=34%:%
-%:%104=35%:%
-%:%105=35%:%
-%:%106=36%:%
-%:%107=37%:%
-%:%108=37%:%
-%:%109=38%:%
-%:%115=38%:%
-%:%120=39%:%
-%:%125=40%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Nanson_Baldwin_Rule.tex b/virage/src/test/resources/old_theories/output/document/Nanson_Baldwin_Rule.tex
deleted file mode 100644
index 84a8dab..0000000
--- a/virage/src/test/resources/old_theories/output/document/Nanson_Baldwin_Rule.tex
+++ /dev/null
@@ -1,113 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Nanson{\isacharunderscore}{\kern0pt}Baldwin{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Nanson-Baldwin Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Nanson{\isacharunderscore}{\kern0pt}Baldwin{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Borda{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Nanson-Baldwin voting rule. It excludes alternatives with the
-lowest Borda score from the set of possible winners and then adjusts the
-Borda score to the new (remaining) set of still eligible alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ nanson{\isacharunderscore}{\kern0pt}baldwin{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}nanson{\isacharunderscore}{\kern0pt}baldwin{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ nanson{\isacharunderscore}{\kern0pt}baldwin{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}nanson{\isacharunderscore}{\kern0pt}baldwin{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Nanson_Baldwin_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%53=19%:%
-%:%63=21%:%
-%:%64=21%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=25%:%
-%:%70=26%:%
-%:%71=27%:%
-%:%74=28%:%
-%:%79=29%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Pairwise_Majority_Rule.tex b/virage/src/test/resources/old_theories/output/document/Pairwise_Majority_Rule.tex
deleted file mode 100644
index 23f1c9a..0000000
--- a/virage/src/test/resources/old_theories/output/document/Pairwise_Majority_Rule.tex
+++ /dev/null
@@ -1,197 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Pairwise{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Pairwise Majority Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Pairwise{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the pairwise majority rule, a voting rule that implements the
-Condorcet criterion, i.e., it elects the Condorcet winner if it exists,
-otherwise a tie remains between all alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ condorcet\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ condorcet{\isacharunderscore}{\kern0pt}code\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-{\isachardoublequoteopen}condorcet{\isacharprime}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-{\isachardoublequoteopen}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isacharprime}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ iterelect\ condorcet{\isacharprime}{\kern0pt}\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Condorcet Consistency Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}condorcet{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ condorcet{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Pairwise_Majority_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%53=19%:%
-%:%63=21%:%
-%:%64=21%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=24%:%
-%:%69=25%:%
-%:%70=26%:%
-%:%71=27%:%
-%:%72=27%:%
-%:%73=28%:%
-%:%74=29%:%
-%:%75=30%:%
-%:%76=31%:%
-%:%77=31%:%
-%:%78=32%:%
-%:%85=34%:%
-%:%95=36%:%
-%:%96=36%:%
-%:%103=37%:%
-%:%104=37%:%
-%:%105=38%:%
-%:%106=38%:%
-%:%107=39%:%
-%:%108=40%:%
-%:%109=40%:%
-%:%110=41%:%
-%:%111=41%:%
-%:%112=42%:%
-%:%113=42%:%
-%:%114=43%:%
-%:%115=43%:%
-%:%116=44%:%
-%:%117=45%:%
-%:%118=45%:%
-%:%119=46%:%
-%:%125=46%:%
-%:%130=47%:%
-%:%135=48%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Parallel_Composition.tex b/virage/src/test/resources/old_theories/output/document/Parallel_Composition.tex
deleted file mode 100644
index 1f55cd6..0000000
--- a/virage/src/test/resources/old_theories/output/document/Parallel_Composition.tex
+++ /dev/null
@@ -1,412 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Parallel{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Parallel Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Parallel{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The parallel composition composes a new electoral module from
-two electoral modules combined with an aggregator.
-Therein, the two modules each make a decision and the aggregator combines
-them to a single (aggregated) result.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ parallel{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}parallel{\isacharunderscore}{\kern0pt}composition\ m\ n\ agg\ A\ p\ {\isacharequal}{\kern0pt}\ agg\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ parallel\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{5}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{5}}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymparallel}\isactrlsub a\ n\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ parallel{\isacharunderscore}{\kern0pt}composition\ m\ n\ a{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ agg{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}aggregator\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ aggregator{\isacharunderscore}{\kern0pt}def\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}m\ mod{\isacharunderscore}{\kern0pt}n\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ agg{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Composition Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ conserv{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ conservative{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}aggregator\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ conservative\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}m\ emod{\isacharunderscore}{\kern0pt}n\ agg{\isacharunderscore}{\kern0pt}a\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ c\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ in\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ c\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ conservative\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ emod{\isacharunderscore}{\kern0pt}m\ emod{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}wins\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sup{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}right\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Parallel_Composition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%55=21%:%
-%:%65=23%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%73=29%:%
-%:%74=30%:%
-%:%81=32%:%
-%:%91=34%:%
-%:%92=34%:%
-%:%93=35%:%
-%:%94=36%:%
-%:%95=37%:%
-%:%96=38%:%
-%:%97=39%:%
-%:%100=40%:%
-%:%104=40%:%
-%:%105=40%:%
-%:%106=41%:%
-%:%107=41%:%
-%:%108=42%:%
-%:%109=42%:%
-%:%110=43%:%
-%:%111=44%:%
-%:%112=45%:%
-%:%113=45%:%
-%:%114=46%:%
-%:%115=47%:%
-%:%116=48%:%
-%:%117=48%:%
-%:%118=49%:%
-%:%119=49%:%
-%:%120=50%:%
-%:%121=51%:%
-%:%122=51%:%
-%:%123=52%:%
-%:%124=52%:%
-%:%125=53%:%
-%:%126=53%:%
-%:%127=54%:%
-%:%142=56%:%
-%:%152=62%:%
-%:%153=62%:%
-%:%154=63%:%
-%:%155=64%:%
-%:%156=65%:%
-%:%157=66%:%
-%:%158=67%:%
-%:%161=68%:%
-%:%165=68%:%
-%:%166=68%:%
-%:%167=69%:%
-%:%168=69%:%
-%:%169=70%:%
-%:%170=70%:%
-%:%171=71%:%
-%:%172=71%:%
-%:%173=72%:%
-%:%174=72%:%
-%:%175=73%:%
-%:%176=73%:%
-%:%177=74%:%
-%:%178=74%:%
-%:%179=75%:%
-%:%180=75%:%
-%:%181=76%:%
-%:%182=76%:%
-%:%183=77%:%
-%:%184=77%:%
-%:%185=78%:%
-%:%186=78%:%
-%:%187=79%:%
-%:%188=79%:%
-%:%189=80%:%
-%:%190=80%:%
-%:%191=81%:%
-%:%192=81%:%
-%:%193=82%:%
-%:%194=82%:%
-%:%195=83%:%
-%:%196=83%:%
-%:%197=84%:%
-%:%198=85%:%
-%:%199=86%:%
-%:%200=87%:%
-%:%201=87%:%
-%:%202=88%:%
-%:%203=89%:%
-%:%204=90%:%
-%:%205=91%:%
-%:%206=91%:%
-%:%207=92%:%
-%:%208=92%:%
-%:%209=93%:%
-%:%210=93%:%
-%:%211=94%:%
-%:%212=94%:%
-%:%213=95%:%
-%:%214=95%:%
-%:%215=96%:%
-%:%216=96%:%
-%:%217=97%:%
-%:%218=97%:%
-%:%219=98%:%
-%:%220=99%:%
-%:%221=100%:%
-%:%222=100%:%
-%:%223=101%:%
-%:%224=102%:%
-%:%225=103%:%
-%:%226=103%:%
-%:%227=104%:%
-%:%228=104%:%
-%:%229=105%:%
-%:%230=105%:%
-%:%231=106%:%
-%:%232=106%:%
-%:%233=107%:%
-%:%234=107%:%
-%:%235=108%:%
-%:%236=108%:%
-%:%237=109%:%
-%:%238=110%:%
-%:%239=110%:%
-%:%240=111%:%
-%:%246=111%:%
-%:%251=112%:%
-%:%256=113%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Pass_Module.tex b/virage/src/test/resources/old_theories/output/document/Pass_Module.tex
deleted file mode 100644
index 3985b72..0000000
--- a/virage/src/test/resources/old_theories/output/document/Pass_Module.tex
+++ /dev/null
@@ -1,1466 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Pass{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Pass Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Pass{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is a family of electoral modules. For a natural number n and a
-lexicon (linear order) r of all alternatives, the according pass module
-defers the lexicographically first n alternatives (from A) and rejects
-the rest. It is primarily used as counterpart to the drop module in a
-parallel composition in order to segment the alternatives into two groups.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}mod\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectI\ not{\isacharunderscore}{\kern0pt}less\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}modI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Non-Blocking%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ g{\isadigit{0}}{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}n\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ input{\isacharunderscore}{\kern0pt}fine{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ input{\isacharunderscore}{\kern0pt}fine{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}one{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}singletonI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ is{\isacharunderscore}{\kern0pt}singleton{\isacharunderscore}{\kern0pt}altdef\ leD\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ order{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Non-Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ order{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}dl{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ order\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ A{\isacharunderscore}{\kern0pt}valid{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}valid\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}exists{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}one{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ w\ \isakeyword{where}\ w{\isacharunderscore}{\kern0pt}unique{\isacharunderscore}{\kern0pt}top{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ w\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ w{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ w{\isacharunderscore}{\kern0pt}top{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ w\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ w{\isacharunderscore}{\kern0pt}unique{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}top\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}exists\ w{\isacharunderscore}{\kern0pt}unique{\isacharunderscore}{\kern0pt}top\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ x\ {\isasympreceq}\isactrlsub q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}limit\ above{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}x{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ A{\isacharunderscore}{\kern0pt}valid\ finite{\isacharunderscore}{\kern0pt}subset{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ above{\isacharunderscore}{\kern0pt}finite\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ equals{\isadigit{0}}D\ neq{\isadigit{0}}{\isacharunderscore}{\kern0pt}conv\ x{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}singletonE\ is{\isacharunderscore}{\kern0pt}singleton{\isacharunderscore}{\kern0pt}altdef\ singletonD\ x{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}unique\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\ order\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}two{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}two{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ {\isadigit{2}}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isasymle}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ limitA{\isacharunderscore}{\kern0pt}order{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\isanewline
-\ \ \ \ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\ finA\ profA\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}b\ {\isasymin}\ A{\isachardot}{\kern0pt}\ let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ {\isacharparenleft}{\kern0pt}b\ {\isasympreceq}\isactrlsub q\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limitA{\isacharunderscore}{\kern0pt}order\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}subset\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \ \ \ \ \ \ order\ connex{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}best{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}b\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}b\ {\isasymin}\ A{\isachardot}{\kern0pt}\ a\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ a\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectI\ Suc{\isacharunderscore}{\kern0pt}leI\ not{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}A\ a{\isacharunderscore}{\kern0pt}above\ card{\isacharunderscore}{\kern0pt}UNIV{\isacharunderscore}{\kern0pt}bool\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ empty{\isacharunderscore}{\kern0pt}iff\ finA\isanewline
-\ \ \ \ \ \ \ \ \ \ finite{\isachardot}{\kern0pt}emptyI\ insert{\isacharunderscore}{\kern0pt}iff\ limitA{\isacharunderscore}{\kern0pt}order\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ \ \ \ \ UNIV{\isacharunderscore}{\kern0pt}bool\ nat{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ zero{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}Suc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ finA{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\ Diff{\isacharunderscore}{\kern0pt}empty\ Diff{\isacharunderscore}{\kern0pt}idemp\ Diff{\isacharunderscore}{\kern0pt}insert{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \ \ \ \ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}A\ card{\isachardot}{\kern0pt}insert{\isacharunderscore}{\kern0pt}remove\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ finite{\isachardot}{\kern0pt}emptyI\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ numeral{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}iff\ semiring{\isacharunderscore}{\kern0pt}norm{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isadigit{9}}{\isacharparenright}{\kern0pt}\ card{\isachardot}{\kern0pt}empty\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ limitAa{\isacharunderscore}{\kern0pt}order{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order\ top{\isacharunderscore}{\kern0pt}greatest\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ b\ \isakeyword{where}\ b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ let\ q\ {\isacharequal}{\kern0pt}\ limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r\ in\ {\isacharparenleft}{\kern0pt}c\ {\isasympreceq}\isactrlsub q\ b{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limitAa{\isacharunderscore}{\kern0pt}order\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}subset\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}c{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}best{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}c{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ c{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ c\ {\isasymnotin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ b\ Diff{\isacharunderscore}{\kern0pt}iff\ Diff{\isacharunderscore}{\kern0pt}insert{\isadigit{2}}\ subset{\isacharunderscore}{\kern0pt}UNIV\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}subset\ order\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\ b{\isacharunderscore}{\kern0pt}best\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ order\ insert{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}ab{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}a\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b\ above{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}best\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limitA{\isacharunderscore}{\kern0pt}order\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isachardot}{\kern0pt}\ c\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharcomma}{\kern0pt}\ c{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}above\ b{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharcomma}{\kern0pt}\ c{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffE\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}ab\ card{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ above{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ finA\ finite{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}commute\ numeral{\isacharunderscore}{\kern0pt}{\isadigit{3}}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ finA\ finite{\isacharunderscore}{\kern0pt}subset\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans\ numeral{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}iff\ order{\isacharunderscore}{\kern0pt}refl\ semiring{\isacharunderscore}{\kern0pt}norm{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isadigit{9}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ c\ {\isasymnotin}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ not{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}ab\ card{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Pass_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%55=21%:%
-%:%65=23%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%77=29%:%
-%:%87=31%:%
-%:%88=31%:%
-%:%89=32%:%
-%:%90=33%:%
-%:%97=34%:%
-%:%98=34%:%
-%:%99=35%:%
-%:%100=35%:%
-%:%101=36%:%
-%:%102=36%:%
-%:%103=37%:%
-%:%105=39%:%
-%:%106=40%:%
-%:%107=40%:%
-%:%108=41%:%
-%:%109=41%:%
-%:%110=42%:%
-%:%111=42%:%
-%:%112=43%:%
-%:%114=45%:%
-%:%115=46%:%
-%:%116=46%:%
-%:%117=47%:%
-%:%118=47%:%
-%:%119=48%:%
-%:%120=49%:%
-%:%121=49%:%
-%:%122=50%:%
-%:%123=50%:%
-%:%124=51%:%
-%:%126=53%:%
-%:%127=54%:%
-%:%128=54%:%
-%:%129=55%:%
-%:%130=55%:%
-%:%131=56%:%
-%:%133=58%:%
-%:%134=59%:%
-%:%135=59%:%
-%:%136=60%:%
-%:%137=60%:%
-%:%138=61%:%
-%:%139=62%:%
-%:%140=62%:%
-%:%141=63%:%
-%:%142=63%:%
-%:%143=64%:%
-%:%144=64%:%
-%:%145=65%:%
-%:%146=66%:%
-%:%147=66%:%
-%:%148=67%:%
-%:%149=67%:%
-%:%150=68%:%
-%:%151=69%:%
-%:%152=69%:%
-%:%153=70%:%
-%:%154=70%:%
-%:%155=71%:%
-%:%156=71%:%
-%:%157=72%:%
-%:%158=72%:%
-%:%159=73%:%
-%:%174=75%:%
-%:%184=78%:%
-%:%185=78%:%
-%:%186=79%:%
-%:%187=80%:%
-%:%188=81%:%
-%:%195=82%:%
-%:%196=82%:%
-%:%197=83%:%
-%:%198=83%:%
-%:%199=84%:%
-%:%200=85%:%
-%:%201=85%:%
-%:%202=86%:%
-%:%203=86%:%
-%:%204=87%:%
-%:%205=88%:%
-%:%206=88%:%
-%:%207=89%:%
-%:%208=90%:%
-%:%209=91%:%
-%:%210=91%:%
-%:%211=92%:%
-%:%212=92%:%
-%:%213=93%:%
-%:%214=94%:%
-%:%215=94%:%
-%:%216=95%:%
-%:%217=96%:%
-%:%218=97%:%
-%:%219=97%:%
-%:%220=98%:%
-%:%221=98%:%
-%:%222=99%:%
-%:%223=99%:%
-%:%224=100%:%
-%:%225=100%:%
-%:%226=101%:%
-%:%227=101%:%
-%:%228=101%:%
-%:%229=102%:%
-%:%230=102%:%
-%:%231=103%:%
-%:%232=103%:%
-%:%233=103%:%
-%:%234=104%:%
-%:%235=104%:%
-%:%236=105%:%
-%:%237=105%:%
-%:%238=106%:%
-%:%239=106%:%
-%:%240=106%:%
-%:%241=107%:%
-%:%242=108%:%
-%:%243=109%:%
-%:%244=109%:%
-%:%245=110%:%
-%:%246=110%:%
-%:%247=111%:%
-%:%248=111%:%
-%:%249=112%:%
-%:%250=113%:%
-%:%251=113%:%
-%:%252=114%:%
-%:%253=114%:%
-%:%254=115%:%
-%:%255=115%:%
-%:%256=116%:%
-%:%257=116%:%
-%:%258=117%:%
-%:%259=117%:%
-%:%260=118%:%
-%:%261=118%:%
-%:%262=119%:%
-%:%263=119%:%
-%:%264=120%:%
-%:%265=120%:%
-%:%266=121%:%
-%:%281=123%:%
-%:%291=126%:%
-%:%292=126%:%
-%:%293=127%:%
-%:%294=128%:%
-%:%297=129%:%
-%:%301=129%:%
-%:%302=129%:%
-%:%316=131%:%
-%:%326=134%:%
-%:%327=134%:%
-%:%328=135%:%
-%:%329=136%:%
-%:%332=137%:%
-%:%336=137%:%
-%:%337=137%:%
-%:%342=137%:%
-%:%345=138%:%
-%:%346=144%:%
-%:%347=145%:%
-%:%348=145%:%
-%:%349=146%:%
-%:%350=147%:%
-%:%357=148%:%
-%:%358=148%:%
-%:%359=149%:%
-%:%360=149%:%
-%:%361=150%:%
-%:%362=151%:%
-%:%363=151%:%
-%:%364=152%:%
-%:%365=152%:%
-%:%366=153%:%
-%:%367=154%:%
-%:%368=154%:%
-%:%369=155%:%
-%:%370=156%:%
-%:%371=157%:%
-%:%372=157%:%
-%:%373=158%:%
-%:%374=158%:%
-%:%375=159%:%
-%:%376=160%:%
-%:%377=160%:%
-%:%378=161%:%
-%:%379=162%:%
-%:%380=163%:%
-%:%381=163%:%
-%:%382=164%:%
-%:%383=164%:%
-%:%384=165%:%
-%:%385=166%:%
-%:%386=166%:%
-%:%387=167%:%
-%:%388=167%:%
-%:%389=168%:%
-%:%390=168%:%
-%:%391=168%:%
-%:%392=169%:%
-%:%393=169%:%
-%:%394=170%:%
-%:%395=170%:%
-%:%396=171%:%
-%:%397=171%:%
-%:%398=171%:%
-%:%399=172%:%
-%:%400=173%:%
-%:%401=173%:%
-%:%402=174%:%
-%:%403=174%:%
-%:%404=175%:%
-%:%405=175%:%
-%:%406=175%:%
-%:%407=176%:%
-%:%408=177%:%
-%:%409=178%:%
-%:%410=178%:%
-%:%411=179%:%
-%:%412=179%:%
-%:%413=179%:%
-%:%414=180%:%
-%:%415=181%:%
-%:%416=182%:%
-%:%417=182%:%
-%:%418=183%:%
-%:%419=183%:%
-%:%420=184%:%
-%:%421=184%:%
-%:%422=185%:%
-%:%423=185%:%
-%:%424=186%:%
-%:%425=186%:%
-%:%426=187%:%
-%:%427=188%:%
-%:%428=189%:%
-%:%429=189%:%
-%:%430=190%:%
-%:%431=190%:%
-%:%432=191%:%
-%:%433=191%:%
-%:%434=192%:%
-%:%435=192%:%
-%:%436=193%:%
-%:%437=193%:%
-%:%438=194%:%
-%:%439=194%:%
-%:%440=195%:%
-%:%441=195%:%
-%:%442=195%:%
-%:%443=196%:%
-%:%444=197%:%
-%:%445=197%:%
-%:%446=198%:%
-%:%447=198%:%
-%:%448=199%:%
-%:%449=200%:%
-%:%450=200%:%
-%:%451=201%:%
-%:%452=202%:%
-%:%453=202%:%
-%:%454=203%:%
-%:%455=203%:%
-%:%456=204%:%
-%:%457=204%:%
-%:%458=205%:%
-%:%459=206%:%
-%:%460=206%:%
-%:%461=207%:%
-%:%462=207%:%
-%:%463=208%:%
-%:%464=208%:%
-%:%465=209%:%
-%:%466=209%:%
-%:%467=210%:%
-%:%468=211%:%
-%:%469=211%:%
-%:%470=212%:%
-%:%471=212%:%
-%:%472=213%:%
-%:%473=213%:%
-%:%474=214%:%
-%:%475=214%:%
-%:%476=215%:%
-%:%477=215%:%
-%:%478=216%:%
-%:%479=216%:%
-%:%480=217%:%
-%:%481=217%:%
-%:%482=218%:%
-%:%483=218%:%
-%:%484=219%:%
-%:%485=219%:%
-%:%486=220%:%
-%:%487=220%:%
-%:%488=221%:%
-%:%489=221%:%
-%:%490=222%:%
-%:%491=222%:%
-%:%492=223%:%
-%:%493=223%:%
-%:%494=224%:%
-%:%495=224%:%
-%:%496=224%:%
-%:%497=225%:%
-%:%498=226%:%
-%:%499=226%:%
-%:%500=227%:%
-%:%501=228%:%
-%:%502=228%:%
-%:%503=229%:%
-%:%504=229%:%
-%:%505=229%:%
-%:%506=230%:%
-%:%507=231%:%
-%:%508=231%:%
-%:%509=232%:%
-%:%510=232%:%
-%:%511=233%:%
-%:%512=233%:%
-%:%513=234%:%
-%:%514=234%:%
-%:%515=235%:%
-%:%516=235%:%
-%:%517=236%:%
-%:%518=236%:%
-%:%519=237%:%
-%:%520=237%:%
-%:%521=238%:%
-%:%522=238%:%
-%:%523=239%:%
-%:%524=239%:%
-%:%525=240%:%
-%:%526=240%:%
-%:%527=241%:%
-%:%528=241%:%
-%:%529=241%:%
-%:%530=242%:%
-%:%531=243%:%
-%:%532=243%:%
-%:%533=244%:%
-%:%534=244%:%
-%:%535=245%:%
-%:%536=245%:%
-%:%537=246%:%
-%:%538=246%:%
-%:%539=247%:%
-%:%540=247%:%
-%:%541=248%:%
-%:%542=248%:%
-%:%543=249%:%
-%:%544=249%:%
-%:%545=250%:%
-%:%546=250%:%
-%:%547=251%:%
-%:%548=251%:%
-%:%549=252%:%
-%:%550=252%:%
-%:%551=253%:%
-%:%552=253%:%
-%:%553=254%:%
-%:%554=254%:%
-%:%555=255%:%
-%:%556=255%:%
-%:%557=256%:%
-%:%558=256%:%
-%:%559=257%:%
-%:%565=257%:%
-%:%568=258%:%
-%:%569=259%:%
-%:%570=259%:%
-%:%571=260%:%
-%:%572=261%:%
-%:%575=262%:%
-%:%579=262%:%
-%:%580=262%:%
-%:%581=263%:%
-%:%582=263%:%
-%:%583=264%:%
-%:%584=264%:%
-%:%585=265%:%
-%:%586=265%:%
-%:%587=266%:%
-%:%588=266%:%
-%:%589=267%:%
-%:%590=267%:%
-%:%591=268%:%
-%:%592=268%:%
-%:%593=269%:%
-%:%594=270%:%
-%:%595=271%:%
-%:%596=271%:%
-%:%597=272%:%
-%:%598=273%:%
-%:%599=274%:%
-%:%600=275%:%
-%:%601=275%:%
-%:%602=276%:%
-%:%603=276%:%
-%:%604=277%:%
-%:%605=277%:%
-%:%606=278%:%
-%:%607=278%:%
-%:%608=278%:%
-%:%609=279%:%
-%:%610=280%:%
-%:%611=280%:%
-%:%612=281%:%
-%:%613=281%:%
-%:%614=282%:%
-%:%615=282%:%
-%:%616=282%:%
-%:%617=283%:%
-%:%618=284%:%
-%:%619=284%:%
-%:%620=285%:%
-%:%621=285%:%
-%:%622=286%:%
-%:%623=286%:%
-%:%624=287%:%
-%:%625=287%:%
-%:%626=288%:%
-%:%627=289%:%
-%:%628=290%:%
-%:%629=290%:%
-%:%630=291%:%
-%:%631=291%:%
-%:%632=292%:%
-%:%633=292%:%
-%:%634=293%:%
-%:%635=293%:%
-%:%636=294%:%
-%:%637=294%:%
-%:%638=295%:%
-%:%639=295%:%
-%:%640=295%:%
-%:%641=296%:%
-%:%642=296%:%
-%:%643=297%:%
-%:%644=298%:%
-%:%645=299%:%
-%:%646=300%:%
-%:%647=300%:%
-%:%648=301%:%
-%:%649=301%:%
-%:%650=302%:%
-%:%651=302%:%
-%:%652=303%:%
-%:%653=303%:%
-%:%654=304%:%
-%:%655=304%:%
-%:%656=305%:%
-%:%657=305%:%
-%:%658=305%:%
-%:%659=306%:%
-%:%660=306%:%
-%:%661=307%:%
-%:%662=308%:%
-%:%663=309%:%
-%:%664=310%:%
-%:%665=310%:%
-%:%666=311%:%
-%:%667=311%:%
-%:%668=311%:%
-%:%669=312%:%
-%:%670=313%:%
-%:%671=313%:%
-%:%672=314%:%
-%:%673=314%:%
-%:%674=315%:%
-%:%675=315%:%
-%:%676=315%:%
-%:%677=316%:%
-%:%678=316%:%
-%:%679=317%:%
-%:%680=317%:%
-%:%681=318%:%
-%:%682=318%:%
-%:%683=319%:%
-%:%684=319%:%
-%:%685=320%:%
-%:%686=321%:%
-%:%687=322%:%
-%:%688=322%:%
-%:%689=323%:%
-%:%690=323%:%
-%:%691=324%:%
-%:%692=324%:%
-%:%693=325%:%
-%:%694=325%:%
-%:%695=326%:%
-%:%696=326%:%
-%:%697=327%:%
-%:%698=327%:%
-%:%699=328%:%
-%:%700=328%:%
-%:%701=329%:%
-%:%702=330%:%
-%:%703=330%:%
-%:%704=331%:%
-%:%705=331%:%
-%:%706=331%:%
-%:%707=332%:%
-%:%708=332%:%
-%:%709=333%:%
-%:%710=333%:%
-%:%711=334%:%
-%:%712=334%:%
-%:%713=334%:%
-%:%714=335%:%
-%:%715=335%:%
-%:%716=336%:%
-%:%717=337%:%
-%:%718=337%:%
-%:%719=338%:%
-%:%720=338%:%
-%:%721=338%:%
-%:%722=339%:%
-%:%723=339%:%
-%:%724=340%:%
-%:%725=340%:%
-%:%726=341%:%
-%:%727=341%:%
-%:%728=342%:%
-%:%729=342%:%
-%:%730=343%:%
-%:%731=343%:%
-%:%732=344%:%
-%:%733=344%:%
-%:%734=345%:%
-%:%735=345%:%
-%:%736=346%:%
-%:%737=346%:%
-%:%738=347%:%
-%:%739=347%:%
-%:%740=347%:%
-%:%741=348%:%
-%:%742=349%:%
-%:%743=349%:%
-%:%744=350%:%
-%:%745=350%:%
-%:%746=351%:%
-%:%747=351%:%
-%:%748=352%:%
-%:%749=352%:%
-%:%750=353%:%
-%:%751=353%:%
-%:%752=354%:%
-%:%753=354%:%
-%:%754=355%:%
-%:%755=355%:%
-%:%756=356%:%
-%:%757=356%:%
-%:%758=357%:%
-%:%759=357%:%
-%:%760=358%:%
-%:%761=358%:%
-%:%762=359%:%
-%:%763=359%:%
-%:%764=359%:%
-%:%765=360%:%
-%:%766=360%:%
-%:%767=361%:%
-%:%768=362%:%
-%:%769=363%:%
-%:%770=363%:%
-%:%771=364%:%
-%:%772=364%:%
-%:%773=364%:%
-%:%774=365%:%
-%:%775=366%:%
-%:%776=366%:%
-%:%777=367%:%
-%:%778=367%:%
-%:%779=368%:%
-%:%780=368%:%
-%:%781=369%:%
-%:%782=369%:%
-%:%783=370%:%
-%:%784=370%:%
-%:%785=371%:%
-%:%786=371%:%
-%:%787=372%:%
-%:%788=372%:%
-%:%789=373%:%
-%:%790=373%:%
-%:%791=373%:%
-%:%792=374%:%
-%:%793=374%:%
-%:%794=375%:%
-%:%795=375%:%
-%:%796=375%:%
-%:%797=376%:%
-%:%798=376%:%
-%:%799=377%:%
-%:%800=377%:%
-%:%801=377%:%
-%:%802=378%:%
-%:%803=379%:%
-%:%804=379%:%
-%:%805=380%:%
-%:%806=380%:%
-%:%807=381%:%
-%:%808=381%:%
-%:%809=382%:%
-%:%810=382%:%
-%:%811=383%:%
-%:%817=383%:%
-%:%822=384%:%
-%:%827=385%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Plurality_Module.tex b/virage/src/test/resources/old_theories/output/document/Plurality_Module.tex
deleted file mode 100644
index 95438e8..0000000
--- a/virage/src/test/resources/old_theories/output/document/Plurality_Module.tex
+++ /dev/null
@@ -1,1263 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Plurality{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Plurality Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Plurality{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The plurality module implements the plurality voting rule.
-The plurality rule elects all modules with the maximum amount of top
-preferences among all alternatives, and rejects all the other alternatives.
-It is electing and induces the classical plurality (voting) rule
-from social-choice theory.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ plurality\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}plurality\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}plurality{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ elect\ {\isasyminter}\ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ disjoint{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}elect{\isacharcomma}{\kern0pt}\ reject{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ elect\ {\isasymunion}\ reject\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ unity{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}elect{\isacharcomma}{\kern0pt}\ reject{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint\ unity\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}modI{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}electing{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ allI\ impI\ conjI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ assm{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ max\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}max\ {\isacharequal}{\kern0pt}\ Max{\isacharparenleft}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\ p\ {\isacharbackquote}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharequal}{\kern0pt}\ max\ {\isasymand}\ a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Max{\isacharunderscore}{\kern0pt}in\ assm{\isadigit{0}}\ empty{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}image\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}imageI\ imageE\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ max\ assm{\isadigit{0}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ elect\ plurality\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ assm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}elect\ plurality\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}elem\ assm{\isadigit{1}}\ all{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ plurality{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}electing{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ allI\ impI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}winner{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}subset{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ sizes{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}size\ p\ {\isacharequal}{\kern0pt}\ size\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffE\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ asm{\isadigit{1}}\ insertCI\ insert{\isacharunderscore}{\kern0pt}absorb\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ profile{\isacharunderscore}{\kern0pt}def\ sizes\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}winner\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}QtoP{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isadigit{3}}\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Collect{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}other{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymge}\ win{\isacharunderscore}{\kern0pt}count\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono\ sizes{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ sizes{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}pq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymlongleftrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ this\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ asm{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ profile{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}PtoQ{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}pq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ q\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}cong\ DiffI\ above{\isacharunderscore}{\kern0pt}pq\ above{\isacharunderscore}{\kern0pt}QtoP\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}absorb\ insert{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\ sizes\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isasymnoteq}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}less{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharless}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}less\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}other\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ less{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isacharless}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ antisym\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ y\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ x\ {\isasymnotin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasymnotin}\ elect\ plurality\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ elect\ plurality\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ less\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ q\ a\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ q\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}invariant{\isacharunderscore}{\kern0pt}monotonicity\ plurality{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Plurality_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%55=21%:%
-%:%65=23%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%70=27%:%
-%:%71=28%:%
-%:%72=29%:%
-%:%73=29%:%
-%:%74=30%:%
-%:%84=35%:%
-%:%94=37%:%
-%:%95=37%:%
-%:%102=38%:%
-%:%103=38%:%
-%:%104=39%:%
-%:%105=39%:%
-%:%106=40%:%
-%:%109=43%:%
-%:%110=44%:%
-%:%111=44%:%
-%:%112=45%:%
-%:%113=45%:%
-%:%114=46%:%
-%:%117=49%:%
-%:%118=50%:%
-%:%119=50%:%
-%:%120=51%:%
-%:%121=51%:%
-%:%122=52%:%
-%:%125=55%:%
-%:%126=56%:%
-%:%127=56%:%
-%:%128=57%:%
-%:%129=57%:%
-%:%130=58%:%
-%:%131=58%:%
-%:%132=59%:%
-%:%135=62%:%
-%:%136=63%:%
-%:%137=63%:%
-%:%138=64%:%
-%:%139=64%:%
-%:%140=64%:%
-%:%141=65%:%
-%:%142=65%:%
-%:%143=66%:%
-%:%158=68%:%
-%:%168=70%:%
-%:%169=70%:%
-%:%171=72%:%
-%:%178=73%:%
-%:%179=73%:%
-%:%180=74%:%
-%:%181=74%:%
-%:%182=75%:%
-%:%183=76%:%
-%:%184=77%:%
-%:%185=77%:%
-%:%186=78%:%
-%:%187=79%:%
-%:%188=79%:%
-%:%189=80%:%
-%:%190=81%:%
-%:%191=81%:%
-%:%192=82%:%
-%:%193=82%:%
-%:%194=83%:%
-%:%195=84%:%
-%:%196=84%:%
-%:%197=85%:%
-%:%198=85%:%
-%:%199=85%:%
-%:%200=86%:%
-%:%201=87%:%
-%:%202=87%:%
-%:%203=88%:%
-%:%204=89%:%
-%:%205=89%:%
-%:%206=90%:%
-%:%207=90%:%
-%:%208=91%:%
-%:%209=92%:%
-%:%210=92%:%
-%:%211=93%:%
-%:%212=93%:%
-%:%213=93%:%
-%:%214=94%:%
-%:%215=95%:%
-%:%216=95%:%
-%:%217=96%:%
-%:%218=96%:%
-%:%219=97%:%
-%:%220=97%:%
-%:%221=97%:%
-%:%222=98%:%
-%:%223=99%:%
-%:%224=99%:%
-%:%225=100%:%
-%:%226=100%:%
-%:%227=101%:%
-%:%228=102%:%
-%:%229=102%:%
-%:%230=103%:%
-%:%231=103%:%
-%:%232=104%:%
-%:%233=105%:%
-%:%234=105%:%
-%:%235=106%:%
-%:%236=106%:%
-%:%237=107%:%
-%:%238=107%:%
-%:%239=108%:%
-%:%240=108%:%
-%:%241=109%:%
-%:%247=109%:%
-%:%250=110%:%
-%:%251=111%:%
-%:%252=112%:%
-%:%253=112%:%
-%:%260=113%:%
-%:%261=113%:%
-%:%262=114%:%
-%:%263=114%:%
-%:%264=115%:%
-%:%265=116%:%
-%:%266=116%:%
-%:%267=117%:%
-%:%268=117%:%
-%:%269=118%:%
-%:%270=118%:%
-%:%271=119%:%
-%:%272=119%:%
-%:%273=120%:%
-%:%274=120%:%
-%:%275=121%:%
-%:%276=121%:%
-%:%277=122%:%
-%:%278=122%:%
-%:%279=123%:%
-%:%280=123%:%
-%:%281=124%:%
-%:%282=124%:%
-%:%283=125%:%
-%:%284=125%:%
-%:%285=126%:%
-%:%300=128%:%
-%:%310=130%:%
-%:%311=130%:%
-%:%314=133%:%
-%:%321=134%:%
-%:%322=134%:%
-%:%323=135%:%
-%:%324=135%:%
-%:%325=136%:%
-%:%326=137%:%
-%:%327=138%:%
-%:%328=139%:%
-%:%329=140%:%
-%:%330=140%:%
-%:%331=141%:%
-%:%332=142%:%
-%:%333=143%:%
-%:%334=143%:%
-%:%335=144%:%
-%:%336=145%:%
-%:%337=146%:%
-%:%338=146%:%
-%:%339=147%:%
-%:%340=147%:%
-%:%341=148%:%
-%:%344=151%:%
-%:%345=152%:%
-%:%346=152%:%
-%:%347=153%:%
-%:%348=153%:%
-%:%349=154%:%
-%:%350=154%:%
-%:%351=155%:%
-%:%352=156%:%
-%:%353=157%:%
-%:%354=157%:%
-%:%355=158%:%
-%:%356=158%:%
-%:%357=159%:%
-%:%358=159%:%
-%:%359=160%:%
-%:%360=161%:%
-%:%361=162%:%
-%:%362=162%:%
-%:%363=163%:%
-%:%364=163%:%
-%:%365=163%:%
-%:%366=164%:%
-%:%367=165%:%
-%:%368=165%:%
-%:%369=166%:%
-%:%370=166%:%
-%:%371=167%:%
-%:%372=167%:%
-%:%373=167%:%
-%:%374=168%:%
-%:%375=169%:%
-%:%376=169%:%
-%:%377=170%:%
-%:%378=170%:%
-%:%379=171%:%
-%:%380=172%:%
-%:%381=172%:%
-%:%382=173%:%
-%:%383=173%:%
-%:%384=174%:%
-%:%385=174%:%
-%:%386=175%:%
-%:%388=177%:%
-%:%389=178%:%
-%:%390=178%:%
-%:%391=179%:%
-%:%392=180%:%
-%:%393=181%:%
-%:%394=181%:%
-%:%395=182%:%
-%:%396=182%:%
-%:%397=182%:%
-%:%398=183%:%
-%:%400=185%:%
-%:%401=186%:%
-%:%402=186%:%
-%:%403=187%:%
-%:%404=188%:%
-%:%405=188%:%
-%:%406=189%:%
-%:%407=189%:%
-%:%408=190%:%
-%:%410=192%:%
-%:%411=193%:%
-%:%412=193%:%
-%:%413=194%:%
-%:%414=194%:%
-%:%415=195%:%
-%:%416=196%:%
-%:%417=196%:%
-%:%418=197%:%
-%:%419=197%:%
-%:%420=198%:%
-%:%421=199%:%
-%:%422=200%:%
-%:%423=200%:%
-%:%424=201%:%
-%:%425=201%:%
-%:%426=202%:%
-%:%427=202%:%
-%:%428=203%:%
-%:%429=204%:%
-%:%430=205%:%
-%:%431=205%:%
-%:%432=206%:%
-%:%433=206%:%
-%:%434=206%:%
-%:%435=207%:%
-%:%436=208%:%
-%:%437=208%:%
-%:%438=209%:%
-%:%439=209%:%
-%:%440=209%:%
-%:%441=210%:%
-%:%442=211%:%
-%:%443=212%:%
-%:%444=212%:%
-%:%445=213%:%
-%:%446=213%:%
-%:%447=214%:%
-%:%448=214%:%
-%:%449=215%:%
-%:%450=216%:%
-%:%451=217%:%
-%:%452=217%:%
-%:%453=218%:%
-%:%454=218%:%
-%:%455=218%:%
-%:%456=219%:%
-%:%459=222%:%
-%:%460=223%:%
-%:%461=223%:%
-%:%462=224%:%
-%:%463=224%:%
-%:%464=225%:%
-%:%465=225%:%
-%:%466=225%:%
-%:%467=226%:%
-%:%469=228%:%
-%:%470=229%:%
-%:%471=229%:%
-%:%472=230%:%
-%:%473=230%:%
-%:%474=231%:%
-%:%476=233%:%
-%:%477=234%:%
-%:%478=234%:%
-%:%479=235%:%
-%:%481=237%:%
-%:%482=238%:%
-%:%483=238%:%
-%:%484=239%:%
-%:%485=239%:%
-%:%486=239%:%
-%:%487=239%:%
-%:%488=240%:%
-%:%489=241%:%
-%:%490=241%:%
-%:%491=242%:%
-%:%492=242%:%
-%:%493=243%:%
-%:%494=243%:%
-%:%495=243%:%
-%:%496=243%:%
-%:%497=244%:%
-%:%498=245%:%
-%:%499=245%:%
-%:%500=246%:%
-%:%501=246%:%
-%:%502=246%:%
-%:%503=247%:%
-%:%504=248%:%
-%:%505=248%:%
-%:%506=249%:%
-%:%507=249%:%
-%:%508=250%:%
-%:%509=250%:%
-%:%510=251%:%
-%:%511=251%:%
-%:%512=252%:%
-%:%513=252%:%
-%:%514=253%:%
-%:%515=253%:%
-%:%516=253%:%
-%:%517=254%:%
-%:%519=256%:%
-%:%520=257%:%
-%:%521=257%:%
-%:%522=258%:%
-%:%523=258%:%
-%:%524=259%:%
-%:%526=261%:%
-%:%527=262%:%
-%:%528=262%:%
-%:%529=263%:%
-%:%530=264%:%
-%:%531=264%:%
-%:%532=265%:%
-%:%533=265%:%
-%:%534=266%:%
-%:%535=266%:%
-%:%536=267%:%
-%:%537=267%:%
-%:%538=268%:%
-%:%539=269%:%
-%:%540=270%:%
-%:%541=270%:%
-%:%542=271%:%
-%:%543=271%:%
-%:%544=272%:%
-%:%545=272%:%
-%:%546=273%:%
-%:%547=273%:%
-%:%548=274%:%
-%:%549=274%:%
-%:%550=275%:%
-%:%551=275%:%
-%:%552=276%:%
-%:%553=277%:%
-%:%554=277%:%
-%:%555=278%:%
-%:%556=278%:%
-%:%557=279%:%
-%:%558=279%:%
-%:%559=280%:%
-%:%560=281%:%
-%:%561=281%:%
-%:%562=282%:%
-%:%563=282%:%
-%:%564=283%:%
-%:%565=283%:%
-%:%566=284%:%
-%:%567=284%:%
-%:%568=285%:%
-%:%569=285%:%
-%:%570=286%:%
-%:%571=286%:%
-%:%572=287%:%
-%:%573=288%:%
-%:%574=288%:%
-%:%575=289%:%
-%:%576=290%:%
-%:%577=290%:%
-%:%578=291%:%
-%:%579=291%:%
-%:%580=292%:%
-%:%581=293%:%
-%:%582=293%:%
-%:%583=294%:%
-%:%584=294%:%
-%:%585=295%:%
-%:%586=295%:%
-%:%587=296%:%
-%:%588=297%:%
-%:%589=298%:%
-%:%590=298%:%
-%:%591=299%:%
-%:%592=299%:%
-%:%593=300%:%
-%:%594=301%:%
-%:%595=301%:%
-%:%596=302%:%
-%:%597=302%:%
-%:%598=302%:%
-%:%599=303%:%
-%:%600=304%:%
-%:%601=304%:%
-%:%602=305%:%
-%:%603=305%:%
-%:%604=306%:%
-%:%605=306%:%
-%:%606=307%:%
-%:%607=308%:%
-%:%608=308%:%
-%:%609=309%:%
-%:%610=309%:%
-%:%611=310%:%
-%:%612=310%:%
-%:%613=310%:%
-%:%614=311%:%
-%:%615=312%:%
-%:%616=312%:%
-%:%617=313%:%
-%:%618=313%:%
-%:%619=313%:%
-%:%620=314%:%
-%:%621=315%:%
-%:%622=315%:%
-%:%623=316%:%
-%:%624=316%:%
-%:%625=316%:%
-%:%626=317%:%
-%:%627=318%:%
-%:%628=318%:%
-%:%629=319%:%
-%:%630=319%:%
-%:%631=320%:%
-%:%632=320%:%
-%:%633=320%:%
-%:%634=321%:%
-%:%635=322%:%
-%:%636=323%:%
-%:%637=323%:%
-%:%638=324%:%
-%:%639=324%:%
-%:%640=325%:%
-%:%641=325%:%
-%:%642=326%:%
-%:%643=326%:%
-%:%644=327%:%
-%:%645=327%:%
-%:%646=327%:%
-%:%647=328%:%
-%:%648=329%:%
-%:%649=329%:%
-%:%650=330%:%
-%:%651=330%:%
-%:%652=330%:%
-%:%653=331%:%
-%:%654=331%:%
-%:%655=332%:%
-%:%656=332%:%
-%:%657=333%:%
-%:%658=333%:%
-%:%659=334%:%
-%:%665=334%:%
-%:%668=335%:%
-%:%669=336%:%
-%:%670=337%:%
-%:%671=337%:%
-%:%678=338%:%
-%:%679=338%:%
-%:%680=339%:%
-%:%681=339%:%
-%:%682=340%:%
-%:%686=344%:%
-%:%687=345%:%
-%:%688=345%:%
-%:%689=346%:%
-%:%690=346%:%
-%:%691=347%:%
-%:%692=347%:%
-%:%693=348%:%
-%:%694=348%:%
-%:%695=349%:%
-%:%696=349%:%
-%:%697=350%:%
-%:%699=352%:%
-%:%700=353%:%
-%:%701=353%:%
-%:%702=354%:%
-%:%703=354%:%
-%:%704=355%:%
-%:%705=355%:%
-%:%706=356%:%
-%:%707=356%:%
-%:%708=357%:%
-%:%709=357%:%
-%:%710=358%:%
-%:%716=358%:%
-%:%721=359%:%
-%:%726=360%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Preference_Relation.tex b/virage/src/test/resources/old_theories/output/document/Preference_Relation.tex
deleted file mode 100644
index 737d741..0000000
--- a/virage/src/test/resources/old_theories/output/document/Preference_Relation.tex
+++ /dev/null
@@ -1,4290 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Preference{\isacharunderscore}{\kern0pt}Relation}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupchapter{Social-Choice Types%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Preference Relation%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Preference{\isacharunderscore}{\kern0pt}Relation\isanewline
-\ \ \isakeyword{imports}\ Main\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The very core of the composable modules voting framework: types and
-functions, derivations, lemmata, operations on preference relations, etc.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ rel{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}preferred{\isacharunderscore}{\kern0pt}than\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasympreceq}\isactrlsub {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{5}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{5}}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Ranking%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ rank\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}rank\ r\ x\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ r\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rank{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ refl{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}rank\ r\ x\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ FieldI{\isadigit{2}}\ refl\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ fin\ finite{\isacharunderscore}{\kern0pt}Field{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}\ {\isasymge}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}cong\ FieldI{\isadigit{2}}\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ less{\isacharunderscore}{\kern0pt}one\ not{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less\ rank{\isachardot}{\kern0pt}elims\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Limited Preference%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ limited\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}limited\ A\ r\ {\isasymequiv}\ r\ {\isasymsubseteq}\ A\ {\isasymtimes}\ A{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limitedI{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}x\ y{\isachardot}{\kern0pt}\ {\isasymlbrakk}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ \ x\ {\isasymin}\ A\ {\isasymand}\ y\ {\isasymin}\ A{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ limited\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}dest{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}x\ y{\isachardot}{\kern0pt}\ {\isasymlbrakk}\ x\ {\isasympreceq}\isactrlsub r\ y{\isacharsemicolon}{\kern0pt}\ limited\ A\ r\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ \ x\ {\isasymin}\ A\ {\isasymand}\ y\ {\isasymin}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ limit\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}limit\ A\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ {\isasymand}\ b\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ connex\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}connex\ A\ r\ {\isasymequiv}\ limited\ A\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymor}\ y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}r\ {\isasymsubseteq}\ A\ {\isasymtimes}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}def\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ assms\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}onD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ assms\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}onD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ y\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ assms\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}onD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ refl{\isacharunderscore}{\kern0pt}onD\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}antsym{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}trans{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ connex{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ antisym{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ trans{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ preorder{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}r\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}r\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}r\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}r\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}r\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ y{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymor}\ y\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ connex{\isacharunderscore}{\kern0pt}r\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ connex{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}connex\ S\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}s\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\ b\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasympreceq}\isactrlsub r\ x\ {\isasymor}\ x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ connex\ connex{\isacharunderscore}{\kern0pt}def\ in{\isacharunderscore}{\kern0pt}mono\ subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub {\isacharquery}{\kern0pt}s\ y\ {\isasymor}\ y\ {\isasympreceq}\isactrlsub {\isacharquery}{\kern0pt}s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub {\isacharquery}{\kern0pt}s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}antisym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ antisymmetric{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}antisym\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}def\ antisymmetric\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ transitive{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ \ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}trans\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ z\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{5}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}z\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{4}}\ transE\ transitive\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ S\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}antsym{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}trans{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}antisym\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}trans\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}let\ s\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}C\ {\isasymsubseteq}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit\ C\ r\ {\isacharequal}{\kern0pt}\ limit\ C\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}r\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnot}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ subrelI\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}singleton{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymforall}r{\isachardot}{\kern0pt}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ r\ {\isasymlongrightarrow}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ r\ {\isasymlongrightarrow}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ singletonI\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardot}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ a\ {\isasymand}\ y\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ split{\isacharunderscore}{\kern0pt}beta\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ b\ {\isasymsubseteq}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}mono\ above{\isacharunderscore}{\kern0pt}def\ assms\ transE\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}refl{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}a\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ assms\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ s{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}above\ r\ a\ {\isasymsubseteq}\ above\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ above{\isacharunderscore}{\kern0pt}refl\ insert{\isacharunderscore}{\kern0pt}absorb\isanewline
-\ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ singletonI\ subset{\isacharunderscore}{\kern0pt}singletonD\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}a\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ above{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ b\ {\isasymlongleftrightarrow}\ b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ B\ {\isasymand}\ b\ {\isasymin}\ B{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ assms\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}b\ {\isasymin}\ B{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ n{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ \isakeyword{where}\ n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}plus{\isadigit{1}}\ antisym{\isacharunderscore}{\kern0pt}conv{\isadigit{2}}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ gr{\isadigit{0}}{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}Suc\ le{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymexists}a{\isachardot}{\kern0pt}\ a{\isasymin}A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n\ arbitrary{\isacharcolon}{\kern0pt}\ A\ r{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ {\isadigit{0}}{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ add{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ above{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a{\isasymin}A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ Suc\ n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ B\ \isakeyword{where}\ B{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ B\ {\isacharequal}{\kern0pt}\ n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isasymand}\ B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharunderscore}{\kern0pt}inject\ add{\isacharunderscore}{\kern0pt}Suc\ card{\isachardot}{\kern0pt}insert{\isacharunderscore}{\kern0pt}remove\ finite{\isachardot}{\kern0pt}cases\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}single\ subset{\isacharunderscore}{\kern0pt}insertI\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ B{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}plus{\isadigit{1}}\ add{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}left{\isacharprime}{\kern0pt}\ asm\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset\ finite{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}b{\isasymin}B{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ B\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isachardot}{\kern0pt}IH\ add{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}left{\isacharprime}{\kern0pt}\ asm\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ diff{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}self\ finite{\isacharunderscore}{\kern0pt}subset\ leD\ lessI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ b\ \isakeyword{where}\ b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r\ a\ aa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ refl{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r\ {\isasymor}\ a\ {\isasymin}\ A\ {\isasymand}\ aa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\ {\isasymnot}\ connex\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isasymor}\ refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\ {\isasymnot}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isasymor}\ connex\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ asm\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A\ {\isasymand}\ b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\ b\ {\isacharequal}{\kern0pt}\ a\ {\isasymor}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{5}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ singletonI\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{6}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ A\ Aa{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ A\ {\isacharminus}{\kern0pt}\ Aa\ {\isasymor}\ a\ {\isasymin}\ A\ {\isasymand}\ a\ {\isasymnotin}\ Aa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ ff{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ ff{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ aa\ {\isasymin}\ B\ {\isasymand}\ a\ {\isasymin}\ B{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{5}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isacharunderscore}{\kern0pt}B{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ B\ above{\isacharunderscore}{\kern0pt}def\ f{\isadigit{4}}\ ff{\isadigit{1}}\ ff{\isadigit{2}}\ CollectI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Product{\isacharunderscore}{\kern0pt}Type{\isachardot}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}case{\isacharunderscore}{\kern0pt}prodD\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ a\ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasymin}\ above\ r\ b\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ above{\isacharunderscore}{\kern0pt}def\ antisym{\isacharunderscore}{\kern0pt}def\ asm\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasymin}\ above\ r\ b\ {\isasymlongleftrightarrow}\ x\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ b\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ subsetI\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ r\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}a\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ B\ DiffE\ a\ asm\ b\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ limited{\isacharunderscore}{\kern0pt}dest\ singletonI\ subset{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ g{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r\ Aa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ Aa\ {\isasymsubseteq}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ Aa\ {\isacharparenleft}{\kern0pt}limit\ Aa\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ g{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ B{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ g{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ B\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ total{\isacharunderscore}{\kern0pt}on\ B\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ g{\isadigit{1}}\ B\ asm\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ total{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}aa\ {\isasymnotin}\ A\ {\isasymor}\ a\ {\isacharequal}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ B\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymnotin}\ B\ {\isasymor}\ a\ {\isacharequal}{\kern0pt}\ aa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ g{\isadigit{3}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isadigit{2}}\ B\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ asm\ b\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ singletonI\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ g{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ transE\ b{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ b{\isacharunderscore}{\kern0pt}wins{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ nothing{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a\ asm\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ above{\isacharunderscore}{\kern0pt}connex\ Diff{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasymin}\ above\ r\ a\ {\isasymlongleftrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}def\ asm\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ nothing{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}a\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ CollectD\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ a\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ a\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a{\isasymin}A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}iff\ above{\isacharunderscore}{\kern0pt}trans\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ empty{\isacharunderscore}{\kern0pt}Diff\ insertE\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}if\ insert{\isacharunderscore}{\kern0pt}absorb\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}emp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ above{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ above\ r\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}a\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ b\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isadigit{1}}\ singletonI\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord\ fin{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}emp\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}one{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}dest\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lifting Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a\ {\isasymequiv}\isanewline
-\ \ \ \ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ s\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ lifted\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}lifted\ A\ r\ s\ a\ {\isasymequiv}\isanewline
-\ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ trivial{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ p\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ order{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ lifted\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ s\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub s\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}exist{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub s\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}x\ y{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ imp{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymlbrakk}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharsemicolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ lift{\isacharunderscore}{\kern0pt}ex{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ lift{\isacharunderscore}{\kern0pt}ex\ \isacommand{obtain}\isamarkupfalse%
-\ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ \isakeyword{where}\isanewline
-\ \ \ \ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub r\ y\ {\isasymand}\ y\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ trans{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ y\ z{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ preorder{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ trans{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ f{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ f{\isadigit{2}}\ f{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ insertE\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}a\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ trans\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ imp{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ x{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}a\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ f{\isadigit{2}}\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ transE\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ antisymD\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ DiffD{\isadigit{2}}\ f{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ f{\isadigit{2}}\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\ singletonI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ rest{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ insertE\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ ex{\isacharunderscore}{\kern0pt}lifted\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\ rest{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ trans{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ s\ a\ {\isasymsubseteq}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ rest{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ y\ z{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}s{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ y\ z{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ s\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ refl{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ refl{\isacharunderscore}{\kern0pt}onD{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x\ lifted{\isacharunderscore}{\kern0pt}r\ rest{\isacharunderscore}{\kern0pt}eq\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \ \ trans{\isacharunderscore}{\kern0pt}r\ trans{\isacharunderscore}{\kern0pt}s\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ refl{\isacharunderscore}{\kern0pt}r\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ x\ {\isasymsubseteq}\ above\ s\ x\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}s{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymnotin}\ above\ s\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}z\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ z\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ z{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}z\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}z\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ z\ {\isasymin}\ above\ r\ x\ {\isasymlongleftrightarrow}\ z\ {\isasymin}\ above\ s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasymin}\ above\ r\ x\ {\isasymlongleftrightarrow}\ y\ {\isasymin}\ above\ s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ y{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}s\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ limited{\isacharunderscore}{\kern0pt}dest\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ member{\isacharunderscore}{\kern0pt}remove\isanewline
-\ \ \ \ \ \ \ \ \ \ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ remove{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}r\ y{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}s\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ S\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}limit\ A\ r\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ {\isasymor}\isanewline
-\ \ \ \ \ \ lifted\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ lifted\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ S\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ subset\ \isacommand{have}\isamarkupfalse%
-\ temp{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}rs{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ lifted\ subset\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ a{\isadigit{1}}\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ keep{\isacharunderscore}{\kern0pt}lift{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ a\ {\isasympreceq}\isactrlsub q\ x{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ u\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ in\ x\ {\isasympreceq}\isactrlsub u\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{1}}\ temp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}worse{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub s\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ subset\ lifted{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ subset\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ connex{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\ connex\ A\ r\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limited\ A\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymin}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymin}\ A\ {\isasymlongrightarrow}\ a\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub r\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Ball{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}raw\ connex{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ limit{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ aa\ {\isasymnotin}\ A\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r\ {\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex\ connex{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ limit{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ aa\ A\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ limit\ A\ r\ {\isasymor}\ a\ {\isasympreceq}\isactrlsub r\ aa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ in\ a\ {\isasympreceq}\isactrlsub q\ aa\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub q\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ connex{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ s\ {\isasymor}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub s\ x{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\ a{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}s{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isadigit{2}}\ limit{\isadigit{1}}\ connex{\isadigit{2}}\ a{\isadigit{1}}\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ not{\isacharunderscore}{\kern0pt}worse\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ a\ {\isasympreceq}\isactrlsub q\ x{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ in\ a\ {\isasympreceq}\isactrlsub q\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{1}}\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ not{\isacharunderscore}{\kern0pt}worse\ DiffD{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\ connex{\isadigit{2}}\ limit{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{1}}\ connex\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ y{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}rs\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}rs\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits\ limited{\isacharunderscore}{\kern0pt}dest\ subrelI\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ change{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ notInA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit\ A\ r\ {\isacharequal}{\kern0pt}\ limit\ A\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ notInA\ subset\ subset{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}insert{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ change\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ in{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ above{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ s\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}one\ lifted{\isacharunderscore}{\kern0pt}a\ above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}above\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ s\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ s\ x\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ y\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ aa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ imp{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymlongrightarrow}\ aa\ {\isasymnotin}\ A\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ singletonD\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}connex\ A\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymexists}aa{\isachardot}{\kern0pt}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}aa{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ {\isasymnot}\ a\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymand}\ aa\ {\isasymin}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\ a\ {\isasymin}\ A{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ limited\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasympreceq}\isactrlsub r\ a\ {\isasymor}\ a\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymor}\ aa\ {\isasymnotin}\ A{\isacharparenright}{\kern0pt}\ {\isasymor}\ a\ {\isasymnotin}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\ limited\ A\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ connex\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}exc{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}aa\ {\isasymnotin}\ A\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ above{\isacharunderscore}{\kern0pt}x\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ limited{\isacharunderscore}{\kern0pt}dest\ insertCI\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ eq{\isacharunderscore}{\kern0pt}exc{\isacharunderscore}{\kern0pt}a\ above{\isacharunderscore}{\kern0pt}one\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\ above{\isacharunderscore}{\kern0pt}x\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ mk{\isacharunderscore}{\kern0pt}disjoint{\isacharunderscore}{\kern0pt}insert\ insertE\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\ asm{\isadigit{1}}\ assms\ calculation\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ member{\isacharunderscore}{\kern0pt}remove\ remove{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}absorb\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasympreceq}\isactrlsub s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ lifted{\isacharunderscore}{\kern0pt}a\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}others{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ above\ s\ y\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{2}}\ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ all{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}conv\ lifted{\isacharunderscore}{\kern0pt}a\ fin{\isacharunderscore}{\kern0pt}A\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one\ singleton{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ above{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ s\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ x\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ y\ \isakeyword{where}\ y{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ y\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\ fin{\isacharunderscore}{\kern0pt}A\ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ s\ y\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\ fin{\isacharunderscore}{\kern0pt}A\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}b{\isachardot}{\kern0pt}\ above\ s\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ b\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ all{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}conv\ lifted{\isacharunderscore}{\kern0pt}a\ above{\isacharunderscore}{\kern0pt}x\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ y\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Component_Types/Social_Choice_Types/Preference_Relation.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%18=10%:%
-%:%34=12%:%
-%:%35=12%:%
-%:%36=13%:%
-%:%37=14%:%
-%:%46=17%:%
-%:%47=18%:%
-%:%56=20%:%
-%:%66=26%:%
-%:%67=26%:%
-%:%68=27%:%
-%:%69=28%:%
-%:%70=28%:%
-%:%71=29%:%
-%:%72=30%:%
-%:%73=31%:%
-%:%74=32%:%
-%:%75=32%:%
-%:%76=33%:%
-%:%77=34%:%
-%:%80=35%:%
-%:%84=35%:%
-%:%85=35%:%
-%:%86=36%:%
-%:%87=36%:%
-%:%92=36%:%
-%:%95=37%:%
-%:%96=38%:%
-%:%97=38%:%
-%:%98=39%:%
-%:%99=40%:%
-%:%102=41%:%
-%:%106=41%:%
-%:%107=41%:%
-%:%108=42%:%
-%:%109=42%:%
-%:%123=44%:%
-%:%133=46%:%
-%:%134=46%:%
-%:%135=47%:%
-%:%136=48%:%
-%:%137=49%:%
-%:%138=49%:%
-%:%139=50%:%
-%:%140=51%:%
-%:%141=52%:%
-%:%142=53%:%
-%:%149=54%:%
-%:%150=54%:%
-%:%151=55%:%
-%:%152=55%:%
-%:%153=56%:%
-%:%154=56%:%
-%:%155=57%:%
-%:%156=57%:%
-%:%157=58%:%
-%:%158=58%:%
-%:%159=59%:%
-%:%160=59%:%
-%:%161=60%:%
-%:%162=60%:%
-%:%163=61%:%
-%:%164=61%:%
-%:%165=62%:%
-%:%166=62%:%
-%:%167=62%:%
-%:%168=63%:%
-%:%169=63%:%
-%:%170=64%:%
-%:%171=64%:%
-%:%172=65%:%
-%:%173=65%:%
-%:%174=65%:%
-%:%175=66%:%
-%:%176=66%:%
-%:%177=67%:%
-%:%178=68%:%
-%:%179=68%:%
-%:%180=69%:%
-%:%195=71%:%
-%:%205=73%:%
-%:%206=73%:%
-%:%207=74%:%
-%:%208=75%:%
-%:%209=76%:%
-%:%210=76%:%
-%:%211=77%:%
-%:%214=78%:%
-%:%218=78%:%
-%:%219=78%:%
-%:%220=79%:%
-%:%221=79%:%
-%:%226=79%:%
-%:%229=80%:%
-%:%230=81%:%
-%:%231=81%:%
-%:%232=82%:%
-%:%235=83%:%
-%:%239=83%:%
-%:%240=83%:%
-%:%241=84%:%
-%:%242=84%:%
-%:%247=84%:%
-%:%250=85%:%
-%:%251=86%:%
-%:%252=86%:%
-%:%253=87%:%
-%:%254=88%:%
-%:%255=89%:%
-%:%256=89%:%
-%:%257=90%:%
-%:%258=91%:%
-%:%259=92%:%
-%:%260=92%:%
-%:%261=93%:%
-%:%262=94%:%
-%:%269=95%:%
-%:%270=95%:%
-%:%271=96%:%
-%:%272=96%:%
-%:%273=97%:%
-%:%274=97%:%
-%:%275=98%:%
-%:%276=98%:%
-%:%277=99%:%
-%:%278=99%:%
-%:%279=100%:%
-%:%280=100%:%
-%:%281=101%:%
-%:%282=102%:%
-%:%283=102%:%
-%:%284=103%:%
-%:%285=104%:%
-%:%286=104%:%
-%:%287=105%:%
-%:%288=105%:%
-%:%289=106%:%
-%:%290=106%:%
-%:%291=107%:%
-%:%292=107%:%
-%:%293=108%:%
-%:%294=108%:%
-%:%295=109%:%
-%:%301=109%:%
-%:%304=110%:%
-%:%305=111%:%
-%:%306=111%:%
-%:%307=112%:%
-%:%308=113%:%
-%:%311=114%:%
-%:%315=114%:%
-%:%316=114%:%
-%:%317=115%:%
-%:%318=115%:%
-%:%319=116%:%
-%:%320=116%:%
-%:%321=117%:%
-%:%322=118%:%
-%:%323=119%:%
-%:%324=119%:%
-%:%325=120%:%
-%:%326=121%:%
-%:%327=121%:%
-%:%328=122%:%
-%:%329=122%:%
-%:%330=123%:%
-%:%331=124%:%
-%:%332=124%:%
-%:%333=125%:%
-%:%334=125%:%
-%:%335=126%:%
-%:%336=126%:%
-%:%337=127%:%
-%:%338=128%:%
-%:%339=129%:%
-%:%340=129%:%
-%:%341=130%:%
-%:%342=131%:%
-%:%343=131%:%
-%:%344=132%:%
-%:%345=132%:%
-%:%346=133%:%
-%:%347=134%:%
-%:%348=134%:%
-%:%349=135%:%
-%:%350=135%:%
-%:%351=136%:%
-%:%352=136%:%
-%:%353=137%:%
-%:%354=138%:%
-%:%355=139%:%
-%:%356=139%:%
-%:%357=140%:%
-%:%358=141%:%
-%:%359=142%:%
-%:%360=143%:%
-%:%361=143%:%
-%:%362=144%:%
-%:%363=144%:%
-%:%364=145%:%
-%:%365=145%:%
-%:%366=146%:%
-%:%367=146%:%
-%:%368=147%:%
-%:%369=147%:%
-%:%370=148%:%
-%:%371=149%:%
-%:%372=149%:%
-%:%373=150%:%
-%:%374=150%:%
-%:%375=151%:%
-%:%376=151%:%
-%:%377=152%:%
-%:%383=152%:%
-%:%386=153%:%
-%:%387=154%:%
-%:%388=154%:%
-%:%389=155%:%
-%:%390=156%:%
-%:%391=157%:%
-%:%392=158%:%
-%:%393=159%:%
-%:%396=160%:%
-%:%400=160%:%
-%:%401=160%:%
-%:%402=161%:%
-%:%403=162%:%
-%:%404=162%:%
-%:%405=163%:%
-%:%406=163%:%
-%:%407=164%:%
-%:%408=165%:%
-%:%409=166%:%
-%:%410=166%:%
-%:%411=167%:%
-%:%412=168%:%
-%:%413=168%:%
-%:%414=169%:%
-%:%415=169%:%
-%:%416=170%:%
-%:%417=170%:%
-%:%418=171%:%
-%:%419=171%:%
-%:%420=172%:%
-%:%421=172%:%
-%:%422=173%:%
-%:%423=174%:%
-%:%424=175%:%
-%:%425=175%:%
-%:%426=176%:%
-%:%427=177%:%
-%:%428=177%:%
-%:%429=178%:%
-%:%430=178%:%
-%:%431=179%:%
-%:%432=179%:%
-%:%433=180%:%
-%:%434=180%:%
-%:%435=181%:%
-%:%436=181%:%
-%:%437=182%:%
-%:%438=183%:%
-%:%439=183%:%
-%:%440=184%:%
-%:%441=185%:%
-%:%442=185%:%
-%:%443=186%:%
-%:%444=186%:%
-%:%445=187%:%
-%:%446=187%:%
-%:%447=188%:%
-%:%448=188%:%
-%:%449=189%:%
-%:%450=189%:%
-%:%451=190%:%
-%:%452=190%:%
-%:%453=191%:%
-%:%454=191%:%
-%:%455=192%:%
-%:%456=192%:%
-%:%457=193%:%
-%:%458=193%:%
-%:%459=194%:%
-%:%460=194%:%
-%:%461=195%:%
-%:%462=195%:%
-%:%463=196%:%
-%:%464=196%:%
-%:%465=197%:%
-%:%466=197%:%
-%:%467=198%:%
-%:%468=199%:%
-%:%469=200%:%
-%:%470=200%:%
-%:%471=201%:%
-%:%472=202%:%
-%:%473=203%:%
-%:%474=204%:%
-%:%475=205%:%
-%:%476=205%:%
-%:%477=206%:%
-%:%478=206%:%
-%:%479=207%:%
-%:%480=207%:%
-%:%481=208%:%
-%:%482=208%:%
-%:%483=209%:%
-%:%484=209%:%
-%:%485=210%:%
-%:%486=210%:%
-%:%487=211%:%
-%:%488=211%:%
-%:%489=212%:%
-%:%490=212%:%
-%:%491=213%:%
-%:%497=213%:%
-%:%500=214%:%
-%:%501=215%:%
-%:%502=215%:%
-%:%505=216%:%
-%:%509=216%:%
-%:%510=216%:%
-%:%511=217%:%
-%:%512=217%:%
-%:%517=217%:%
-%:%520=218%:%
-%:%521=219%:%
-%:%522=219%:%
-%:%523=220%:%
-%:%524=221%:%
-%:%525=222%:%
-%:%526=223%:%
-%:%529=224%:%
-%:%533=224%:%
-%:%534=224%:%
-%:%535=225%:%
-%:%536=225%:%
-%:%537=226%:%
-%:%538=226%:%
-%:%539=227%:%
-%:%540=227%:%
-%:%541=228%:%
-%:%542=229%:%
-%:%543=230%:%
-%:%544=231%:%
-%:%545=232%:%
-%:%546=232%:%
-%:%547=233%:%
-%:%548=234%:%
-%:%549=235%:%
-%:%550=236%:%
-%:%551=236%:%
-%:%552=237%:%
-%:%553=237%:%
-%:%554=238%:%
-%:%555=238%:%
-%:%556=239%:%
-%:%557=239%:%
-%:%558=240%:%
-%:%559=241%:%
-%:%560=241%:%
-%:%561=242%:%
-%:%562=242%:%
-%:%563=243%:%
-%:%564=243%:%
-%:%565=244%:%
-%:%566=244%:%
-%:%567=245%:%
-%:%568=245%:%
-%:%569=246%:%
-%:%570=246%:%
-%:%571=247%:%
-%:%572=247%:%
-%:%573=248%:%
-%:%579=248%:%
-%:%582=249%:%
-%:%583=250%:%
-%:%584=250%:%
-%:%585=251%:%
-%:%586=252%:%
-%:%587=253%:%
-%:%588=254%:%
-%:%591=255%:%
-%:%595=255%:%
-%:%596=255%:%
-%:%597=256%:%
-%:%598=256%:%
-%:%603=256%:%
-%:%606=257%:%
-%:%607=258%:%
-%:%608=258%:%
-%:%609=259%:%
-%:%610=260%:%
-%:%611=261%:%
-%:%612=262%:%
-%:%615=263%:%
-%:%619=263%:%
-%:%620=263%:%
-%:%621=264%:%
-%:%622=264%:%
-%:%623=265%:%
-%:%624=265%:%
-%:%625=266%:%
-%:%626=267%:%
-%:%627=268%:%
-%:%628=269%:%
-%:%629=269%:%
-%:%630=270%:%
-%:%631=271%:%
-%:%632=272%:%
-%:%633=273%:%
-%:%634=274%:%
-%:%635=275%:%
-%:%636=275%:%
-%:%637=276%:%
-%:%638=276%:%
-%:%639=277%:%
-%:%640=277%:%
-%:%641=278%:%
-%:%647=278%:%
-%:%650=279%:%
-%:%651=280%:%
-%:%652=280%:%
-%:%653=281%:%
-%:%654=282%:%
-%:%655=283%:%
-%:%656=284%:%
-%:%659=285%:%
-%:%663=285%:%
-%:%664=285%:%
-%:%665=286%:%
-%:%666=287%:%
-%:%667=288%:%
-%:%668=289%:%
-%:%669=290%:%
-%:%670=290%:%
-%:%675=290%:%
-%:%678=291%:%
-%:%679=292%:%
-%:%680=292%:%
-%:%681=293%:%
-%:%682=294%:%
-%:%683=295%:%
-%:%684=296%:%
-%:%685=297%:%
-%:%688=298%:%
-%:%692=298%:%
-%:%693=298%:%
-%:%694=299%:%
-%:%695=299%:%
-%:%700=299%:%
-%:%703=300%:%
-%:%704=301%:%
-%:%705=301%:%
-%:%706=302%:%
-%:%707=303%:%
-%:%710=304%:%
-%:%714=304%:%
-%:%715=304%:%
-%:%716=305%:%
-%:%717=305%:%
-%:%722=305%:%
-%:%725=306%:%
-%:%726=307%:%
-%:%727=307%:%
-%:%728=308%:%
-%:%729=309%:%
-%:%730=310%:%
-%:%731=311%:%
-%:%732=312%:%
-%:%735=313%:%
-%:%739=313%:%
-%:%740=313%:%
-%:%741=314%:%
-%:%742=314%:%
-%:%747=314%:%
-%:%750=315%:%
-%:%751=316%:%
-%:%752=316%:%
-%:%753=317%:%
-%:%754=318%:%
-%:%757=319%:%
-%:%761=319%:%
-%:%762=319%:%
-%:%763=320%:%
-%:%764=321%:%
-%:%765=321%:%
-%:%770=321%:%
-%:%773=322%:%
-%:%774=323%:%
-%:%775=323%:%
-%:%776=324%:%
-%:%783=325%:%
-%:%784=325%:%
-%:%785=326%:%
-%:%786=326%:%
-%:%787=327%:%
-%:%788=327%:%
-%:%789=328%:%
-%:%790=328%:%
-%:%791=329%:%
-%:%792=329%:%
-%:%793=330%:%
-%:%794=330%:%
-%:%795=331%:%
-%:%796=331%:%
-%:%797=332%:%
-%:%798=332%:%
-%:%799=333%:%
-%:%800=333%:%
-%:%801=333%:%
-%:%802=334%:%
-%:%803=334%:%
-%:%804=335%:%
-%:%805=336%:%
-%:%806=336%:%
-%:%807=337%:%
-%:%808=337%:%
-%:%809=337%:%
-%:%810=338%:%
-%:%811=338%:%
-%:%812=339%:%
-%:%813=339%:%
-%:%814=340%:%
-%:%829=342%:%
-%:%839=344%:%
-%:%840=344%:%
-%:%841=345%:%
-%:%842=346%:%
-%:%843=347%:%
-%:%844=348%:%
-%:%847=349%:%
-%:%851=349%:%
-%:%852=349%:%
-%:%853=350%:%
-%:%854=350%:%
-%:%859=350%:%
-%:%862=351%:%
-%:%863=352%:%
-%:%864=352%:%
-%:%865=353%:%
-%:%866=354%:%
-%:%867=355%:%
-%:%868=356%:%
-%:%871=357%:%
-%:%875=357%:%
-%:%876=357%:%
-%:%877=358%:%
-%:%878=358%:%
-%:%883=358%:%
-%:%886=359%:%
-%:%887=360%:%
-%:%888=360%:%
-%:%889=361%:%
-%:%890=362%:%
-%:%891=363%:%
-%:%892=364%:%
-%:%893=365%:%
-%:%896=366%:%
-%:%900=366%:%
-%:%901=366%:%
-%:%902=367%:%
-%:%903=368%:%
-%:%904=369%:%
-%:%905=369%:%
-%:%910=369%:%
-%:%913=370%:%
-%:%914=371%:%
-%:%915=371%:%
-%:%916=372%:%
-%:%917=373%:%
-%:%918=374%:%
-%:%919=375%:%
-%:%922=376%:%
-%:%926=376%:%
-%:%927=376%:%
-%:%928=377%:%
-%:%929=377%:%
-%:%934=377%:%
-%:%937=378%:%
-%:%938=379%:%
-%:%939=379%:%
-%:%942=380%:%
-%:%946=380%:%
-%:%947=380%:%
-%:%952=380%:%
-%:%955=381%:%
-%:%956=382%:%
-%:%957=382%:%
-%:%958=383%:%
-%:%959=384%:%
-%:%960=385%:%
-%:%960=386%:%
-%:%961=387%:%
-%:%962=388%:%
-%:%965=389%:%
-%:%969=389%:%
-%:%970=389%:%
-%:%971=390%:%
-%:%972=390%:%
-%:%977=390%:%
-%:%980=391%:%
-%:%981=392%:%
-%:%982=392%:%
-%:%983=393%:%
-%:%984=394%:%
-%:%985=395%:%
-%:%986=396%:%
-%:%987=397%:%
-%:%988=398%:%
-%:%989=399%:%
-%:%992=400%:%
-%:%996=400%:%
-%:%997=400%:%
-%:%998=401%:%
-%:%999=401%:%
-%:%1000=402%:%
-%:%1001=403%:%
-%:%1002=403%:%
-%:%1007=403%:%
-%:%1010=404%:%
-%:%1011=405%:%
-%:%1012=405%:%
-%:%1013=406%:%
-%:%1014=407%:%
-%:%1015=408%:%
-%:%1016=409%:%
-%:%1023=410%:%
-%:%1024=410%:%
-%:%1025=411%:%
-%:%1026=411%:%
-%:%1027=412%:%
-%:%1028=412%:%
-%:%1029=413%:%
-%:%1030=414%:%
-%:%1031=414%:%
-%:%1032=415%:%
-%:%1033=415%:%
-%:%1034=416%:%
-%:%1035=417%:%
-%:%1036=418%:%
-%:%1037=418%:%
-%:%1038=419%:%
-%:%1039=419%:%
-%:%1040=420%:%
-%:%1041=420%:%
-%:%1042=421%:%
-%:%1043=421%:%
-%:%1044=422%:%
-%:%1045=422%:%
-%:%1046=423%:%
-%:%1047=423%:%
-%:%1048=423%:%
-%:%1049=424%:%
-%:%1050=424%:%
-%:%1051=425%:%
-%:%1052=425%:%
-%:%1053=426%:%
-%:%1054=426%:%
-%:%1055=427%:%
-%:%1056=427%:%
-%:%1057=428%:%
-%:%1058=429%:%
-%:%1059=429%:%
-%:%1060=430%:%
-%:%1061=430%:%
-%:%1062=431%:%
-%:%1063=431%:%
-%:%1064=432%:%
-%:%1065=432%:%
-%:%1066=433%:%
-%:%1067=433%:%
-%:%1068=434%:%
-%:%1069=434%:%
-%:%1070=435%:%
-%:%1071=435%:%
-%:%1072=436%:%
-%:%1073=436%:%
-%:%1074=437%:%
-%:%1075=437%:%
-%:%1076=438%:%
-%:%1077=439%:%
-%:%1078=439%:%
-%:%1079=439%:%
-%:%1080=440%:%
-%:%1081=440%:%
-%:%1082=441%:%
-%:%1083=442%:%
-%:%1084=442%:%
-%:%1085=443%:%
-%:%1086=443%:%
-%:%1087=443%:%
-%:%1088=444%:%
-%:%1089=444%:%
-%:%1090=445%:%
-%:%1091=446%:%
-%:%1092=446%:%
-%:%1093=447%:%
-%:%1094=447%:%
-%:%1095=448%:%
-%:%1096=448%:%
-%:%1097=449%:%
-%:%1098=450%:%
-%:%1099=451%:%
-%:%1100=451%:%
-%:%1101=452%:%
-%:%1102=452%:%
-%:%1103=452%:%
-%:%1104=453%:%
-%:%1105=453%:%
-%:%1106=454%:%
-%:%1107=454%:%
-%:%1108=455%:%
-%:%1109=455%:%
-%:%1110=456%:%
-%:%1111=456%:%
-%:%1112=457%:%
-%:%1113=458%:%
-%:%1114=458%:%
-%:%1115=459%:%
-%:%1116=460%:%
-%:%1117=461%:%
-%:%1118=461%:%
-%:%1119=462%:%
-%:%1120=462%:%
-%:%1121=463%:%
-%:%1122=463%:%
-%:%1123=464%:%
-%:%1124=465%:%
-%:%1125=465%:%
-%:%1126=466%:%
-%:%1127=466%:%
-%:%1128=467%:%
-%:%1129=467%:%
-%:%1130=468%:%
-%:%1131=469%:%
-%:%1132=469%:%
-%:%1133=470%:%
-%:%1134=470%:%
-%:%1135=471%:%
-%:%1136=471%:%
-%:%1137=472%:%
-%:%1138=472%:%
-%:%1139=473%:%
-%:%1140=473%:%
-%:%1141=474%:%
-%:%1142=474%:%
-%:%1143=475%:%
-%:%1144=475%:%
-%:%1145=476%:%
-%:%1146=476%:%
-%:%1147=477%:%
-%:%1148=478%:%
-%:%1149=478%:%
-%:%1150=479%:%
-%:%1151=479%:%
-%:%1152=480%:%
-%:%1153=480%:%
-%:%1154=481%:%
-%:%1155=482%:%
-%:%1156=482%:%
-%:%1157=483%:%
-%:%1158=483%:%
-%:%1159=484%:%
-%:%1160=484%:%
-%:%1161=485%:%
-%:%1162=486%:%
-%:%1163=486%:%
-%:%1164=487%:%
-%:%1165=487%:%
-%:%1166=488%:%
-%:%1167=489%:%
-%:%1168=489%:%
-%:%1169=490%:%
-%:%1170=490%:%
-%:%1171=491%:%
-%:%1172=491%:%
-%:%1173=492%:%
-%:%1174=493%:%
-%:%1175=493%:%
-%:%1176=494%:%
-%:%1177=494%:%
-%:%1178=495%:%
-%:%1179=495%:%
-%:%1180=495%:%
-%:%1181=496%:%
-%:%1182=497%:%
-%:%1183=497%:%
-%:%1184=498%:%
-%:%1185=499%:%
-%:%1186=499%:%
-%:%1187=500%:%
-%:%1188=500%:%
-%:%1189=500%:%
-%:%1190=501%:%
-%:%1191=501%:%
-%:%1192=502%:%
-%:%1193=502%:%
-%:%1194=503%:%
-%:%1195=503%:%
-%:%1196=503%:%
-%:%1197=504%:%
-%:%1198=505%:%
-%:%1199=505%:%
-%:%1200=506%:%
-%:%1201=506%:%
-%:%1202=507%:%
-%:%1203=507%:%
-%:%1204=508%:%
-%:%1205=508%:%
-%:%1206=509%:%
-%:%1207=509%:%
-%:%1208=510%:%
-%:%1209=510%:%
-%:%1210=511%:%
-%:%1211=511%:%
-%:%1212=512%:%
-%:%1213=512%:%
-%:%1214=513%:%
-%:%1215=513%:%
-%:%1216=513%:%
-%:%1217=514%:%
-%:%1218=514%:%
-%:%1219=515%:%
-%:%1220=516%:%
-%:%1221=516%:%
-%:%1222=517%:%
-%:%1223=517%:%
-%:%1224=517%:%
-%:%1225=518%:%
-%:%1226=518%:%
-%:%1227=519%:%
-%:%1228=519%:%
-%:%1229=520%:%
-%:%1230=520%:%
-%:%1231=521%:%
-%:%1232=521%:%
-%:%1233=522%:%
-%:%1234=522%:%
-%:%1235=523%:%
-%:%1236=523%:%
-%:%1237=524%:%
-%:%1238=524%:%
-%:%1239=525%:%
-%:%1240=525%:%
-%:%1241=526%:%
-%:%1242=526%:%
-%:%1243=527%:%
-%:%1244=528%:%
-%:%1245=529%:%
-%:%1246=529%:%
-%:%1247=530%:%
-%:%1248=530%:%
-%:%1249=531%:%
-%:%1250=531%:%
-%:%1251=532%:%
-%:%1252=532%:%
-%:%1253=533%:%
-%:%1256=536%:%
-%:%1257=537%:%
-%:%1258=537%:%
-%:%1259=538%:%
-%:%1260=538%:%
-%:%1261=539%:%
-%:%1262=539%:%
-%:%1263=540%:%
-%:%1264=541%:%
-%:%1265=541%:%
-%:%1266=542%:%
-%:%1267=542%:%
-%:%1268=543%:%
-%:%1269=543%:%
-%:%1270=544%:%
-%:%1271=544%:%
-%:%1272=545%:%
-%:%1273=545%:%
-%:%1274=546%:%
-%:%1275=547%:%
-%:%1276=547%:%
-%:%1277=548%:%
-%:%1278=548%:%
-%:%1279=549%:%
-%:%1280=549%:%
-%:%1281=550%:%
-%:%1283=552%:%
-%:%1284=553%:%
-%:%1285=553%:%
-%:%1286=554%:%
-%:%1287=554%:%
-%:%1288=555%:%
-%:%1289=555%:%
-%:%1290=556%:%
-%:%1292=558%:%
-%:%1293=559%:%
-%:%1294=559%:%
-%:%1295=560%:%
-%:%1296=560%:%
-%:%1297=561%:%
-%:%1298=561%:%
-%:%1299=562%:%
-%:%1300=562%:%
-%:%1301=563%:%
-%:%1302=564%:%
-%:%1303=565%:%
-%:%1304=565%:%
-%:%1305=566%:%
-%:%1306=566%:%
-%:%1307=567%:%
-%:%1308=568%:%
-%:%1309=568%:%
-%:%1310=569%:%
-%:%1311=569%:%
-%:%1312=570%:%
-%:%1313=571%:%
-%:%1314=571%:%
-%:%1315=572%:%
-%:%1316=572%:%
-%:%1317=573%:%
-%:%1318=573%:%
-%:%1319=574%:%
-%:%1320=574%:%
-%:%1321=575%:%
-%:%1322=575%:%
-%:%1323=576%:%
-%:%1324=576%:%
-%:%1325=577%:%
-%:%1326=577%:%
-%:%1327=578%:%
-%:%1328=578%:%
-%:%1329=579%:%
-%:%1330=579%:%
-%:%1331=580%:%
-%:%1332=580%:%
-%:%1333=581%:%
-%:%1334=581%:%
-%:%1335=582%:%
-%:%1336=583%:%
-%:%1337=583%:%
-%:%1338=584%:%
-%:%1339=584%:%
-%:%1340=585%:%
-%:%1341=585%:%
-%:%1342=586%:%
-%:%1343=587%:%
-%:%1344=588%:%
-%:%1345=588%:%
-%:%1346=589%:%
-%:%1347=589%:%
-%:%1348=589%:%
-%:%1349=590%:%
-%:%1350=590%:%
-%:%1351=591%:%
-%:%1352=592%:%
-%:%1353=592%:%
-%:%1354=593%:%
-%:%1355=593%:%
-%:%1356=593%:%
-%:%1357=594%:%
-%:%1358=594%:%
-%:%1359=595%:%
-%:%1360=595%:%
-%:%1361=596%:%
-%:%1362=596%:%
-%:%1363=597%:%
-%:%1364=597%:%
-%:%1365=598%:%
-%:%1366=598%:%
-%:%1367=599%:%
-%:%1368=599%:%
-%:%1369=600%:%
-%:%1370=600%:%
-%:%1371=601%:%
-%:%1372=601%:%
-%:%1373=602%:%
-%:%1374=602%:%
-%:%1375=603%:%
-%:%1376=603%:%
-%:%1377=604%:%
-%:%1378=604%:%
-%:%1379=605%:%
-%:%1380=605%:%
-%:%1381=606%:%
-%:%1382=606%:%
-%:%1383=607%:%
-%:%1384=608%:%
-%:%1385=609%:%
-%:%1386=609%:%
-%:%1387=610%:%
-%:%1393=610%:%
-%:%1396=611%:%
-%:%1397=612%:%
-%:%1398=612%:%
-%:%1399=613%:%
-%:%1400=614%:%
-%:%1401=615%:%
-%:%1402=616%:%
-%:%1403=617%:%
-%:%1410=618%:%
-%:%1411=618%:%
-%:%1412=619%:%
-%:%1413=619%:%
-%:%1414=620%:%
-%:%1415=620%:%
-%:%1416=621%:%
-%:%1417=621%:%
-%:%1418=622%:%
-%:%1419=622%:%
-%:%1420=622%:%
-%:%1421=623%:%
-%:%1422=624%:%
-%:%1423=625%:%
-%:%1424=625%:%
-%:%1425=626%:%
-%:%1426=626%:%
-%:%1427=627%:%
-%:%1428=627%:%
-%:%1429=627%:%
-%:%1430=628%:%
-%:%1431=628%:%
-%:%1432=629%:%
-%:%1433=629%:%
-%:%1434=630%:%
-%:%1435=630%:%
-%:%1436=630%:%
-%:%1437=631%:%
-%:%1438=631%:%
-%:%1439=632%:%
-%:%1440=632%:%
-%:%1441=633%:%
-%:%1447=633%:%
-%:%1450=634%:%
-%:%1451=635%:%
-%:%1452=635%:%
-%:%1453=636%:%
-%:%1454=637%:%
-%:%1457=638%:%
-%:%1461=638%:%
-%:%1462=638%:%
-%:%1463=639%:%
-%:%1464=639%:%
-%:%1478=641%:%
-%:%1488=643%:%
-%:%1489=643%:%
-%:%1490=644%:%
-%:%1491=645%:%
-%:%1493=647%:%
-%:%1494=648%:%
-%:%1495=649%:%
-%:%1496=649%:%
-%:%1497=650%:%
-%:%1498=651%:%
-%:%1499=652%:%
-%:%1500=653%:%
-%:%1501=654%:%
-%:%1502=654%:%
-%:%1503=655%:%
-%:%1504=656%:%
-%:%1507=657%:%
-%:%1511=657%:%
-%:%1512=657%:%
-%:%1517=657%:%
-%:%1520=658%:%
-%:%1521=659%:%
-%:%1522=659%:%
-%:%1523=660%:%
-%:%1524=661%:%
-%:%1531=662%:%
-%:%1532=662%:%
-%:%1533=663%:%
-%:%1534=663%:%
-%:%1535=663%:%
-%:%1536=664%:%
-%:%1537=665%:%
-%:%1538=666%:%
-%:%1539=666%:%
-%:%1540=667%:%
-%:%1541=667%:%
-%:%1542=668%:%
-%:%1543=668%:%
-%:%1544=669%:%
-%:%1550=669%:%
-%:%1553=670%:%
-%:%1554=671%:%
-%:%1555=671%:%
-%:%1556=672%:%
-%:%1557=673%:%
-%:%1564=674%:%
-%:%1565=674%:%
-%:%1566=675%:%
-%:%1567=675%:%
-%:%1568=676%:%
-%:%1569=677%:%
-%:%1570=677%:%
-%:%1571=678%:%
-%:%1572=679%:%
-%:%1573=680%:%
-%:%1574=681%:%
-%:%1575=682%:%
-%:%1576=683%:%
-%:%1577=683%:%
-%:%1578=684%:%
-%:%1579=684%:%
-%:%1580=685%:%
-%:%1581=685%:%
-%:%1582=686%:%
-%:%1583=686%:%
-%:%1584=687%:%
-%:%1585=687%:%
-%:%1586=688%:%
-%:%1587=688%:%
-%:%1588=689%:%
-%:%1589=689%:%
-%:%1590=690%:%
-%:%1591=690%:%
-%:%1592=691%:%
-%:%1593=692%:%
-%:%1594=693%:%
-%:%1595=693%:%
-%:%1596=694%:%
-%:%1597=694%:%
-%:%1598=695%:%
-%:%1599=696%:%
-%:%1600=696%:%
-%:%1601=697%:%
-%:%1602=697%:%
-%:%1603=698%:%
-%:%1604=698%:%
-%:%1605=699%:%
-%:%1606=700%:%
-%:%1607=700%:%
-%:%1608=701%:%
-%:%1609=701%:%
-%:%1610=702%:%
-%:%1611=702%:%
-%:%1612=703%:%
-%:%1613=703%:%
-%:%1614=704%:%
-%:%1615=704%:%
-%:%1616=704%:%
-%:%1617=705%:%
-%:%1618=706%:%
-%:%1619=706%:%
-%:%1620=707%:%
-%:%1621=707%:%
-%:%1622=708%:%
-%:%1623=709%:%
-%:%1624=709%:%
-%:%1625=710%:%
-%:%1626=710%:%
-%:%1627=711%:%
-%:%1628=712%:%
-%:%1629=712%:%
-%:%1630=713%:%
-%:%1631=713%:%
-%:%1632=714%:%
-%:%1633=714%:%
-%:%1634=715%:%
-%:%1635=716%:%
-%:%1636=716%:%
-%:%1637=717%:%
-%:%1638=717%:%
-%:%1639=718%:%
-%:%1640=718%:%
-%:%1641=719%:%
-%:%1642=720%:%
-%:%1643=720%:%
-%:%1644=721%:%
-%:%1645=721%:%
-%:%1646=722%:%
-%:%1647=722%:%
-%:%1648=723%:%
-%:%1649=724%:%
-%:%1650=724%:%
-%:%1651=725%:%
-%:%1652=725%:%
-%:%1653=726%:%
-%:%1654=726%:%
-%:%1655=727%:%
-%:%1656=728%:%
-%:%1657=728%:%
-%:%1658=729%:%
-%:%1659=729%:%
-%:%1660=730%:%
-%:%1661=730%:%
-%:%1662=731%:%
-%:%1663=732%:%
-%:%1664=733%:%
-%:%1665=733%:%
-%:%1666=734%:%
-%:%1667=734%:%
-%:%1668=735%:%
-%:%1669=735%:%
-%:%1670=736%:%
-%:%1671=737%:%
-%:%1672=737%:%
-%:%1673=738%:%
-%:%1679=738%:%
-%:%1682=739%:%
-%:%1683=740%:%
-%:%1684=740%:%
-%:%1685=741%:%
-%:%1686=742%:%
-%:%1687=743%:%
-%:%1688=744%:%
-%:%1695=745%:%
-%:%1696=745%:%
-%:%1697=746%:%
-%:%1698=746%:%
-%:%1699=747%:%
-%:%1700=747%:%
-%:%1701=748%:%
-%:%1702=748%:%
-%:%1703=749%:%
-%:%1704=749%:%
-%:%1705=750%:%
-%:%1706=750%:%
-%:%1707=751%:%
-%:%1708=752%:%
-%:%1709=753%:%
-%:%1710=753%:%
-%:%1711=754%:%
-%:%1712=754%:%
-%:%1713=755%:%
-%:%1714=755%:%
-%:%1715=756%:%
-%:%1716=756%:%
-%:%1717=757%:%
-%:%1718=757%:%
-%:%1719=758%:%
-%:%1720=759%:%
-%:%1721=759%:%
-%:%1722=760%:%
-%:%1723=760%:%
-%:%1724=761%:%
-%:%1725=761%:%
-%:%1726=762%:%
-%:%1727=762%:%
-%:%1728=763%:%
-%:%1729=763%:%
-%:%1730=764%:%
-%:%1731=765%:%
-%:%1732=765%:%
-%:%1733=766%:%
-%:%1734=766%:%
-%:%1735=767%:%
-%:%1736=767%:%
-%:%1737=768%:%
-%:%1738=768%:%
-%:%1739=769%:%
-%:%1740=769%:%
-%:%1741=770%:%
-%:%1742=770%:%
-%:%1743=771%:%
-%:%1744=772%:%
-%:%1745=772%:%
-%:%1746=773%:%
-%:%1747=773%:%
-%:%1748=774%:%
-%:%1749=774%:%
-%:%1750=775%:%
-%:%1751=775%:%
-%:%1752=776%:%
-%:%1753=776%:%
-%:%1754=777%:%
-%:%1755=778%:%
-%:%1756=779%:%
-%:%1757=780%:%
-%:%1758=780%:%
-%:%1759=781%:%
-%:%1760=781%:%
-%:%1761=782%:%
-%:%1767=782%:%
-%:%1770=783%:%
-%:%1771=784%:%
-%:%1772=784%:%
-%:%1773=785%:%
-%:%1774=786%:%
-%:%1777=787%:%
-%:%1781=787%:%
-%:%1782=787%:%
-%:%1783=788%:%
-%:%1784=788%:%
-%:%1785=789%:%
-%:%1786=789%:%
-%:%1787=790%:%
-%:%1788=791%:%
-%:%1789=791%:%
-%:%1790=792%:%
-%:%1791=793%:%
-%:%1792=793%:%
-%:%1793=794%:%
-%:%1794=794%:%
-%:%1795=795%:%
-%:%1796=795%:%
-%:%1797=796%:%
-%:%1798=796%:%
-%:%1799=797%:%
-%:%1800=798%:%
-%:%1801=798%:%
-%:%1802=799%:%
-%:%1803=799%:%
-%:%1804=800%:%
-%:%1805=800%:%
-%:%1806=801%:%
-%:%1807=801%:%
-%:%1808=802%:%
-%:%1809=802%:%
-%:%1810=803%:%
-%:%1811=804%:%
-%:%1812=804%:%
-%:%1813=805%:%
-%:%1814=805%:%
-%:%1815=806%:%
-%:%1816=807%:%
-%:%1817=807%:%
-%:%1818=808%:%
-%:%1819=809%:%
-%:%1820=809%:%
-%:%1821=810%:%
-%:%1822=810%:%
-%:%1823=811%:%
-%:%1824=812%:%
-%:%1825=812%:%
-%:%1826=813%:%
-%:%1827=814%:%
-%:%1828=814%:%
-%:%1829=815%:%
-%:%1830=815%:%
-%:%1831=816%:%
-%:%1832=817%:%
-%:%1833=817%:%
-%:%1834=818%:%
-%:%1835=819%:%
-%:%1836=819%:%
-%:%1837=820%:%
-%:%1838=820%:%
-%:%1839=821%:%
-%:%1840=821%:%
-%:%1841=822%:%
-%:%1842=823%:%
-%:%1843=823%:%
-%:%1844=824%:%
-%:%1845=824%:%
-%:%1846=825%:%
-%:%1847=825%:%
-%:%1848=826%:%
-%:%1849=827%:%
-%:%1850=827%:%
-%:%1851=828%:%
-%:%1857=828%:%
-%:%1860=829%:%
-%:%1861=830%:%
-%:%1862=830%:%
-%:%1863=831%:%
-%:%1864=832%:%
-%:%1865=833%:%
-%:%1866=834%:%
-%:%1873=835%:%
-%:%1874=835%:%
-%:%1875=836%:%
-%:%1876=836%:%
-%:%1877=837%:%
-%:%1878=837%:%
-%:%1879=838%:%
-%:%1880=839%:%
-%:%1881=840%:%
-%:%1882=840%:%
-%:%1883=841%:%
-%:%1884=841%:%
-%:%1885=842%:%
-%:%1886=842%:%
-%:%1887=843%:%
-%:%1888=843%:%
-%:%1889=844%:%
-%:%1890=844%:%
-%:%1891=845%:%
-%:%1892=845%:%
-%:%1893=846%:%
-%:%1894=846%:%
-%:%1895=847%:%
-%:%1896=847%:%
-%:%1897=848%:%
-%:%1898=848%:%
-%:%1899=849%:%
-%:%1900=850%:%
-%:%1901=851%:%
-%:%1902=852%:%
-%:%1903=852%:%
-%:%1904=853%:%
-%:%1905=853%:%
-%:%1906=854%:%
-%:%1907=854%:%
-%:%1908=855%:%
-%:%1909=855%:%
-%:%1910=856%:%
-%:%1916=856%:%
-%:%1919=857%:%
-%:%1920=858%:%
-%:%1921=858%:%
-%:%1922=859%:%
-%:%1923=860%:%
-%:%1924=861%:%
-%:%1925=862%:%
-%:%1926=863%:%
-%:%1927=864%:%
-%:%1934=865%:%
-%:%1935=865%:%
-%:%1936=866%:%
-%:%1937=866%:%
-%:%1938=866%:%
-%:%1939=867%:%
-%:%1940=868%:%
-%:%1941=868%:%
-%:%1942=869%:%
-%:%1943=869%:%
-%:%1944=869%:%
-%:%1945=870%:%
-%:%1946=871%:%
-%:%1947=871%:%
-%:%1948=872%:%
-%:%1949=872%:%
-%:%1950=873%:%
-%:%1951=874%:%
-%:%1952=875%:%
-%:%1953=875%:%
-%:%1954=876%:%
-%:%1955=876%:%
-%:%1956=877%:%
-%:%1957=877%:%
-%:%1958=878%:%
-%:%1959=878%:%
-%:%1960=879%:%
-%:%1961=879%:%
-%:%1962=880%:%
-%:%1963=880%:%
-%:%1964=881%:%
-%:%1965=881%:%
-%:%1966=882%:%
-%:%1967=883%:%
-%:%1968=883%:%
-%:%1969=884%:%
-%:%1970=884%:%
-%:%1971=884%:%
-%:%1972=885%:%
-%:%1973=886%:%
-%:%1974=886%:%
-%:%1975=887%:%
-%:%1976=887%:%
-%:%1977=888%:%
-%:%1978=888%:%
-%:%1979=888%:%
-%:%1980=888%:%
-%:%1981=889%:%
-%:%1982=890%:%
-%:%1983=891%:%
-%:%1984=891%:%
-%:%1985=892%:%
-%:%1986=892%:%
-%:%1987=893%:%
-%:%1988=893%:%
-%:%1989=893%:%
-%:%1990=894%:%
-%:%1991=894%:%
-%:%1992=895%:%
-%:%1993=895%:%
-%:%1994=896%:%
-%:%1995=896%:%
-%:%1996=897%:%
-%:%1997=897%:%
-%:%1998=898%:%
-%:%1999=899%:%
-%:%2000=899%:%
-%:%2001=900%:%
-%:%2002=901%:%
-%:%2003=901%:%
-%:%2004=902%:%
-%:%2005=902%:%
-%:%2006=902%:%
-%:%2007=903%:%
-%:%2008=904%:%
-%:%2009=904%:%
-%:%2010=905%:%
-%:%2011=905%:%
-%:%2012=906%:%
-%:%2013=906%:%
-%:%2014=906%:%
-%:%2015=907%:%
-%:%2016=908%:%
-%:%2017=908%:%
-%:%2018=909%:%
-%:%2019=910%:%
-%:%2020=910%:%
-%:%2021=911%:%
-%:%2022=911%:%
-%:%2023=911%:%
-%:%2024=912%:%
-%:%2026=914%:%
-%:%2027=915%:%
-%:%2028=915%:%
-%:%2029=916%:%
-%:%2030=916%:%
-%:%2031=917%:%
-%:%2035=921%:%
-%:%2036=922%:%
-%:%2037=922%:%
-%:%2038=923%:%
-%:%2039=923%:%
-%:%2040=924%:%
-%:%2041=924%:%
-%:%2042=925%:%
-%:%2043=926%:%
-%:%2044=926%:%
-%:%2045=927%:%
-%:%2046=927%:%
-%:%2047=928%:%
-%:%2048=928%:%
-%:%2049=929%:%
-%:%2052=932%:%
-%:%2053=933%:%
-%:%2054=933%:%
-%:%2055=934%:%
-%:%2056=934%:%
-%:%2057=935%:%
-%:%2058=935%:%
-%:%2059=936%:%
-%:%2062=939%:%
-%:%2063=940%:%
-%:%2064=940%:%
-%:%2065=941%:%
-%:%2066=941%:%
-%:%2067=941%:%
-%:%2068=942%:%
-%:%2069=943%:%
-%:%2070=943%:%
-%:%2071=944%:%
-%:%2072=944%:%
-%:%2073=945%:%
-%:%2074=945%:%
-%:%2075=946%:%
-%:%2076=947%:%
-%:%2077=947%:%
-%:%2078=948%:%
-%:%2079=948%:%
-%:%2080=949%:%
-%:%2081=950%:%
-%:%2082=950%:%
-%:%2083=951%:%
-%:%2084=951%:%
-%:%2085=952%:%
-%:%2086=952%:%
-%:%2087=953%:%
-%:%2088=954%:%
-%:%2089=955%:%
-%:%2090=955%:%
-%:%2091=956%:%
-%:%2092=956%:%
-%:%2093=956%:%
-%:%2094=957%:%
-%:%2095=958%:%
-%:%2096=958%:%
-%:%2097=959%:%
-%:%2098=959%:%
-%:%2099=960%:%
-%:%2100=960%:%
-%:%2101=960%:%
-%:%2102=961%:%
-%:%2103=962%:%
-%:%2104=962%:%
-%:%2105=963%:%
-%:%2106=963%:%
-%:%2107=964%:%
-%:%2108=964%:%
-%:%2109=964%:%
-%:%2110=965%:%
-%:%2111=966%:%
-%:%2112=966%:%
-%:%2113=967%:%
-%:%2114=967%:%
-%:%2115=968%:%
-%:%2116=968%:%
-%:%2117=968%:%
-%:%2118=969%:%
-%:%2119=970%:%
-%:%2120=970%:%
-%:%2121=971%:%
-%:%2122=971%:%
-%:%2123=972%:%
-%:%2124=972%:%
-%:%2125=973%:%
-%:%2126=973%:%
-%:%2127=974%:%
-%:%2128=974%:%
-%:%2129=975%:%
-%:%2130=975%:%
-%:%2131=976%:%
-%:%2132=976%:%
-%:%2133=977%:%
-%:%2134=977%:%
-%:%2135=977%:%
-%:%2136=978%:%
-%:%2137=979%:%
-%:%2138=979%:%
-%:%2139=980%:%
-%:%2140=980%:%
-%:%2141=981%:%
-%:%2142=981%:%
-%:%2143=982%:%
-%:%2144=982%:%
-%:%2145=983%:%
-%:%2146=983%:%
-%:%2147=984%:%
-%:%2153=984%:%
-%:%2156=985%:%
-%:%2157=986%:%
-%:%2158=986%:%
-%:%2159=987%:%
-%:%2160=988%:%
-%:%2161=989%:%
-%:%2162=990%:%
-%:%2163=991%:%
-%:%2170=992%:%
-%:%2171=992%:%
-%:%2172=993%:%
-%:%2173=993%:%
-%:%2174=994%:%
-%:%2175=994%:%
-%:%2176=995%:%
-%:%2177=995%:%
-%:%2178=996%:%
-%:%2179=996%:%
-%:%2180=997%:%
-%:%2181=997%:%
-%:%2182=998%:%
-%:%2183=998%:%
-%:%2184=999%:%
-%:%2190=999%:%
-%:%2193=1000%:%
-%:%2194=1001%:%
-%:%2195=1001%:%
-%:%2196=1002%:%
-%:%2197=1003%:%
-%:%2198=1004%:%
-%:%2199=1005%:%
-%:%2200=1006%:%
-%:%2207=1007%:%
-%:%2208=1007%:%
-%:%2209=1008%:%
-%:%2210=1008%:%
-%:%2211=1009%:%
-%:%2212=1009%:%
-%:%2213=1010%:%
-%:%2214=1010%:%
-%:%2215=1011%:%
-%:%2216=1012%:%
-%:%2217=1012%:%
-%:%2218=1013%:%
-%:%2219=1013%:%
-%:%2220=1014%:%
-%:%2221=1014%:%
-%:%2222=1015%:%
-%:%2223=1015%:%
-%:%2224=1016%:%
-%:%2225=1016%:%
-%:%2226=1017%:%
-%:%2227=1017%:%
-%:%2228=1018%:%
-%:%2229=1018%:%
-%:%2230=1019%:%
-%:%2231=1019%:%
-%:%2232=1020%:%
-%:%2233=1020%:%
-%:%2234=1021%:%
-%:%2235=1021%:%
-%:%2236=1022%:%
-%:%2237=1022%:%
-%:%2238=1023%:%
-%:%2239=1023%:%
-%:%2240=1024%:%
-%:%2241=1024%:%
-%:%2242=1025%:%
-%:%2243=1025%:%
-%:%2244=1026%:%
-%:%2245=1026%:%
-%:%2246=1027%:%
-%:%2247=1027%:%
-%:%2248=1028%:%
-%:%2249=1028%:%
-%:%2250=1028%:%
-%:%2251=1029%:%
-%:%2256=1034%:%
-%:%2257=1035%:%
-%:%2258=1035%:%
-%:%2259=1036%:%
-%:%2260=1036%:%
-%:%2261=1037%:%
-%:%2262=1037%:%
-%:%2263=1037%:%
-%:%2264=1038%:%
-%:%2265=1039%:%
-%:%2266=1039%:%
-%:%2267=1040%:%
-%:%2268=1040%:%
-%:%2269=1041%:%
-%:%2270=1041%:%
-%:%2271=1041%:%
-%:%2272=1042%:%
-%:%2273=1042%:%
-%:%2274=1043%:%
-%:%2275=1044%:%
-%:%2276=1044%:%
-%:%2277=1045%:%
-%:%2278=1045%:%
-%:%2279=1046%:%
-%:%2280=1046%:%
-%:%2281=1047%:%
-%:%2282=1048%:%
-%:%2283=1049%:%
-%:%2284=1049%:%
-%:%2285=1050%:%
-%:%2286=1050%:%
-%:%2287=1051%:%
-%:%2288=1051%:%
-%:%2289=1051%:%
-%:%2290=1052%:%
-%:%2291=1052%:%
-%:%2292=1053%:%
-%:%2293=1053%:%
-%:%2294=1054%:%
-%:%2295=1054%:%
-%:%2296=1054%:%
-%:%2297=1055%:%
-%:%2298=1055%:%
-%:%2299=1056%:%
-%:%2300=1057%:%
-%:%2301=1058%:%
-%:%2302=1058%:%
-%:%2303=1059%:%
-%:%2304=1059%:%
-%:%2305=1059%:%
-%:%2306=1060%:%
-%:%2307=1060%:%
-%:%2308=1061%:%
-%:%2309=1061%:%
-%:%2310=1062%:%
-%:%2311=1062%:%
-%:%2312=1063%:%
-%:%2313=1063%:%
-%:%2314=1064%:%
-%:%2315=1064%:%
-%:%2316=1065%:%
-%:%2317=1065%:%
-%:%2318=1066%:%
-%:%2319=1066%:%
-%:%2320=1067%:%
-%:%2321=1068%:%
-%:%2322=1068%:%
-%:%2323=1069%:%
-%:%2324=1069%:%
-%:%2325=1070%:%
-%:%2326=1070%:%
-%:%2327=1071%:%
-%:%2328=1071%:%
-%:%2329=1072%:%
-%:%2335=1072%:%
-%:%2338=1073%:%
-%:%2339=1074%:%
-%:%2340=1074%:%
-%:%2341=1075%:%
-%:%2342=1076%:%
-%:%2343=1077%:%
-%:%2344=1078%:%
-%:%2345=1079%:%
-%:%2348=1080%:%
-%:%2352=1080%:%
-%:%2353=1080%:%
-%:%2354=1081%:%
-%:%2355=1081%:%
-%:%2360=1081%:%
-%:%2363=1082%:%
-%:%2364=1083%:%
-%:%2365=1083%:%
-%:%2366=1084%:%
-%:%2367=1085%:%
-%:%2368=1086%:%
-%:%2369=1087%:%
-%:%2370=1088%:%
-%:%2371=1089%:%
-%:%2378=1090%:%
-%:%2379=1090%:%
-%:%2380=1091%:%
-%:%2381=1091%:%
-%:%2382=1092%:%
-%:%2383=1092%:%
-%:%2384=1092%:%
-%:%2385=1093%:%
-%:%2386=1093%:%
-%:%2387=1094%:%
-%:%2388=1095%:%
-%:%2389=1095%:%
-%:%2390=1096%:%
-%:%2391=1096%:%
-%:%2392=1097%:%
-%:%2393=1097%:%
-%:%2394=1098%:%
-%:%2395=1098%:%
-%:%2396=1099%:%
-%:%2397=1099%:%
-%:%2398=1099%:%
-%:%2399=1100%:%
-%:%2400=1100%:%
-%:%2401=1101%:%
-%:%2402=1102%:%
-%:%2403=1102%:%
-%:%2404=1103%:%
-%:%2405=1103%:%
-%:%2406=1103%:%
-%:%2407=1104%:%
-%:%2408=1104%:%
-%:%2409=1105%:%
-%:%2410=1105%:%
-%:%2411=1106%:%
-%:%2412=1106%:%
-%:%2413=1106%:%
-%:%2414=1107%:%
-%:%2415=1107%:%
-%:%2416=1108%:%
-%:%2417=1108%:%
-%:%2418=1109%:%
-%:%2419=1109%:%
-%:%2420=1109%:%
-%:%2421=1110%:%
-%:%2422=1110%:%
-%:%2423=1111%:%
-%:%2429=1111%:%
-%:%2434=1112%:%
-%:%2439=1113%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Profile.tex b/virage/src/test/resources/old_theories/output/document/Profile.tex
deleted file mode 100644
index b6f1213..0000000
--- a/virage/src/test/resources/old_theories/output/document/Profile.tex
+++ /dev/null
@@ -1,1906 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Profile}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-\isanewline
-%
-\isamarkupsection{Preference Profile%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Profile\isanewline
-\ \ \isakeyword{imports}\ Preference{\isacharunderscore}{\kern0pt}Relation\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-Preference profiles denote the decisions made by the individual voters on
-the eligible alternatives. They are represented in the form of one preference
-relation (e.g., selected on a ballot) per voter, collectively captured in a
-list of such preference relations.
-Unlike a the common preference profiles in the social-choice sense, the
-profiles described here considers only the (sub-)set of alternatives that are
-received.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Profile\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isacharparenright}{\kern0pt}\ list{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ profile\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}profile\ A\ p\ {\isasymequiv}\ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ profile{\isacharunderscore}{\kern0pt}set\ {\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymforall}b\ {\isasymin}\ {\isacharparenleft}{\kern0pt}set\ p{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ b{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ all{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}conv{\isacharunderscore}{\kern0pt}all{\isacharunderscore}{\kern0pt}nth\ profile{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymequiv}\ finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Preference Counts and Comparisons%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ Nil\ a\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}above\ p\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ a{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ prefer{\isacharunderscore}{\kern0pt}count\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ Nil\ x\ y\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ y\ {\isasympreceq}\isactrlsub p\ x\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ x\ y{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}compr{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}\ f\ x\ {\isacharbar}{\kern0pt}\ x\ {\isachardot}{\kern0pt}\ x\ {\isasymin}\ S\ {\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f\ {\isacharbackquote}{\kern0pt}\ S{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}compr{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharbar}{\kern0pt}\ y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x{\isacharparenright}{\kern0pt}\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ neq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ y{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}size\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ y\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ size\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prof\ profile{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ connex\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ antisym\ \ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisymD\ neq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Collect{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{3}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card{\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card{\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{3}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ size\ p\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ y\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{2}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{3}}{\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}sym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{assumes}\ p{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ a\ x\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ neq{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ neq{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ prof\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ neq{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ a\ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}size\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ prof\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ neq{\isadigit{2}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ b\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}size\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}size\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}\ {\isasymge}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}size\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ calculation\ p{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}size\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}size\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ diff{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}eq\ diff{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}self\ neq{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ pref{\isacharunderscore}{\kern0pt}count\ prof\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\ calculation\ p{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ empty{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}p\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}\ x\ y{\isachardot}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}incr{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}y\ {\isasympreceq}\isactrlsub p\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isacharplus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}constant{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub p\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ wins\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}wins\ x\ p\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ y\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ wins{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}wins{\isacharunderscore}{\kern0pt}code\ x\ p\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ y\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ wins{\isacharunderscore}{\kern0pt}antisym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}wins\ a\ p\ b{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ b\ p\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ wins{\isacharunderscore}{\kern0pt}irreflex{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ w\ p\ w{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ wins{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Condorcet Winner%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}code\ A\ p\ w\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}code\ w\ p\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharunderscore}{\kern0pt}c{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ c{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ winner{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}w\ {\isacharequal}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ assumption{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnoteq}\ c{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}wins\ w\ p\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assumption\ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}iff\ winner{\isacharunderscore}{\kern0pt}c\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}\ wins\ c\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}c\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ c{\isacharunderscore}{\kern0pt}wins{\isacharunderscore}{\kern0pt}against{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}wins\ c\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ assumption\isanewline
-\ \ \ \ \ \ \ \ \ \ singletonD\ winner{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnot}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}w\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique\ winner\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}xa\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ xa{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}xa{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\ assm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ assm\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ assm\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}wins\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}x\ x{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}w\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ xa{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}xa{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}wins\ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Limited Profile%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}C\ {\isasymsubseteq}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ C\ p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ C\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ B\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ length{\isacharunderscore}{\kern0pt}map\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \ \ \ \ \ \ profile{\isacharunderscore}{\kern0pt}def\ subset\ infinite{\isacharunderscore}{\kern0pt}super\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}size{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ subset{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}size\ p\ {\isacharequal}{\kern0pt}\ size\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lifting Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\ size\ p\ {\isacharequal}{\kern0pt}\ size\ q\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ lifted\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}lifted\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\ size\ p\ {\isacharequal}{\kern0pt}\ size\ q\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymand}\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ i\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}ok{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}i\ {\isacharless}{\kern0pt}\ size\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ i{\isacharunderscore}{\kern0pt}ok\ lifted\ profile{\isacharunderscore}{\kern0pt}def\ trivial{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ change{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ p\ q\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ notInA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ change\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ notInA\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit\ subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ change\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ length{\isacharunderscore}{\kern0pt}map\ nth{\isacharunderscore}{\kern0pt}equalityI\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ S\ p\ q\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ lifted\ A\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ q{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ inA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ S\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ one{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted\ subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ length{\isacharunderscore}{\kern0pt}map\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}equalityI\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ assm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}q\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}profile\ A\ {\isacharquery}{\kern0pt}p\ {\isasymand}\ profile\ A\ {\isacharquery}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound\ subset\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}size\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}p\ {\isasymand}\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm\ lifted{\isacharunderscore}{\kern0pt}def\ length{\isacharunderscore}{\kern0pt}map\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}map\ one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}i\ {\isacharless}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}p\ {\isasymand}\ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ length{\isacharunderscore}{\kern0pt}map\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}map\ one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}lifted\ A\ {\isacharquery}{\kern0pt}p\ {\isacharquery}{\kern0pt}q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ inA\ lifted\ rev{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}subset\ subset\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharunderscore}{\kern0pt}prof\ subset\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Component_Types/Social_Choice_Types/Profile.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%13=6%:%
-%:%15=9%:%
-%:%31=11%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%48=21%:%
-%:%49=22%:%
-%:%58=24%:%
-%:%68=27%:%
-%:%69=27%:%
-%:%70=28%:%
-%:%71=32%:%
-%:%72=33%:%
-%:%73=33%:%
-%:%74=34%:%
-%:%75=35%:%
-%:%76=36%:%
-%:%77=36%:%
-%:%80=37%:%
-%:%84=37%:%
-%:%85=37%:%
-%:%90=37%:%
-%:%93=38%:%
-%:%94=39%:%
-%:%95=39%:%
-%:%96=40%:%
-%:%103=42%:%
-%:%113=48%:%
-%:%114=48%:%
-%:%115=49%:%
-%:%116=50%:%
-%:%117=51%:%
-%:%118=52%:%
-%:%119=52%:%
-%:%120=53%:%
-%:%121=54%:%
-%:%122=55%:%
-%:%123=56%:%
-%:%124=57%:%
-%:%125=57%:%
-%:%126=58%:%
-%:%127=59%:%
-%:%128=60%:%
-%:%129=61%:%
-%:%130=61%:%
-%:%131=62%:%
-%:%132=63%:%
-%:%133=64%:%
-%:%134=65%:%
-%:%135=66%:%
-%:%136=66%:%
-%:%139=67%:%
-%:%143=67%:%
-%:%144=67%:%
-%:%149=67%:%
-%:%152=68%:%
-%:%153=69%:%
-%:%154=69%:%
-%:%155=70%:%
-%:%158=71%:%
-%:%162=71%:%
-%:%163=71%:%
-%:%168=71%:%
-%:%171=72%:%
-%:%172=73%:%
-%:%173=73%:%
-%:%174=74%:%
-%:%175=75%:%
-%:%176=76%:%
-%:%177=77%:%
-%:%178=78%:%
-%:%185=79%:%
-%:%186=79%:%
-%:%187=80%:%
-%:%188=80%:%
-%:%189=81%:%
-%:%190=81%:%
-%:%191=82%:%
-%:%192=82%:%
-%:%193=83%:%
-%:%195=85%:%
-%:%196=86%:%
-%:%197=86%:%
-%:%198=87%:%
-%:%199=87%:%
-%:%200=88%:%
-%:%201=88%:%
-%:%202=89%:%
-%:%203=89%:%
-%:%204=90%:%
-%:%205=90%:%
-%:%206=91%:%
-%:%207=91%:%
-%:%208=92%:%
-%:%209=92%:%
-%:%210=93%:%
-%:%211=94%:%
-%:%212=94%:%
-%:%213=95%:%
-%:%214=95%:%
-%:%215=96%:%
-%:%216=96%:%
-%:%217=96%:%
-%:%218=97%:%
-%:%219=98%:%
-%:%220=98%:%
-%:%221=99%:%
-%:%222=99%:%
-%:%223=100%:%
-%:%224=100%:%
-%:%225=101%:%
-%:%226=101%:%
-%:%227=102%:%
-%:%228=102%:%
-%:%229=103%:%
-%:%230=103%:%
-%:%231=104%:%
-%:%232=105%:%
-%:%233=105%:%
-%:%234=106%:%
-%:%235=106%:%
-%:%236=106%:%
-%:%237=107%:%
-%:%238=108%:%
-%:%239=109%:%
-%:%240=109%:%
-%:%241=110%:%
-%:%242=110%:%
-%:%243=111%:%
-%:%244=112%:%
-%:%245=113%:%
-%:%246=113%:%
-%:%247=114%:%
-%:%248=114%:%
-%:%249=115%:%
-%:%250=116%:%
-%:%251=117%:%
-%:%252=117%:%
-%:%253=118%:%
-%:%254=118%:%
-%:%255=119%:%
-%:%256=119%:%
-%:%257=120%:%
-%:%258=120%:%
-%:%259=121%:%
-%:%260=121%:%
-%:%261=122%:%
-%:%264=125%:%
-%:%265=126%:%
-%:%266=126%:%
-%:%267=127%:%
-%:%268=127%:%
-%:%269=128%:%
-%:%270=129%:%
-%:%271=129%:%
-%:%272=130%:%
-%:%273=130%:%
-%:%274=130%:%
-%:%275=131%:%
-%:%276=132%:%
-%:%277=133%:%
-%:%278=133%:%
-%:%279=134%:%
-%:%280=134%:%
-%:%281=135%:%
-%:%282=135%:%
-%:%283=135%:%
-%:%284=136%:%
-%:%285=137%:%
-%:%286=138%:%
-%:%287=138%:%
-%:%288=139%:%
-%:%289=139%:%
-%:%290=140%:%
-%:%291=140%:%
-%:%292=140%:%
-%:%293=141%:%
-%:%294=142%:%
-%:%295=142%:%
-%:%296=143%:%
-%:%297=143%:%
-%:%298=143%:%
-%:%299=144%:%
-%:%300=144%:%
-%:%301=145%:%
-%:%307=145%:%
-%:%310=146%:%
-%:%311=147%:%
-%:%312=147%:%
-%:%313=148%:%
-%:%314=149%:%
-%:%315=150%:%
-%:%316=151%:%
-%:%317=152%:%
-%:%318=153%:%
-%:%319=154%:%
-%:%320=155%:%
-%:%327=156%:%
-%:%328=156%:%
-%:%329=157%:%
-%:%330=157%:%
-%:%331=157%:%
-%:%332=158%:%
-%:%333=159%:%
-%:%334=159%:%
-%:%335=160%:%
-%:%336=160%:%
-%:%337=161%:%
-%:%338=161%:%
-%:%339=161%:%
-%:%340=161%:%
-%:%341=162%:%
-%:%342=163%:%
-%:%343=163%:%
-%:%344=164%:%
-%:%345=164%:%
-%:%346=165%:%
-%:%347=165%:%
-%:%348=166%:%
-%:%349=167%:%
-%:%350=167%:%
-%:%351=168%:%
-%:%352=168%:%
-%:%353=169%:%
-%:%354=169%:%
-%:%355=170%:%
-%:%356=171%:%
-%:%357=171%:%
-%:%358=172%:%
-%:%359=173%:%
-%:%360=173%:%
-%:%361=174%:%
-%:%362=174%:%
-%:%363=175%:%
-%:%364=175%:%
-%:%365=176%:%
-%:%366=176%:%
-%:%367=177%:%
-%:%368=177%:%
-%:%369=178%:%
-%:%370=178%:%
-%:%371=179%:%
-%:%377=179%:%
-%:%380=180%:%
-%:%381=181%:%
-%:%382=181%:%
-%:%383=182%:%
-%:%384=183%:%
-%:%387=184%:%
-%:%391=184%:%
-%:%392=184%:%
-%:%393=185%:%
-%:%394=185%:%
-%:%399=185%:%
-%:%402=186%:%
-%:%403=187%:%
-%:%404=187%:%
-%:%405=188%:%
-%:%406=189%:%
-%:%407=190%:%
-%:%408=191%:%
-%:%411=192%:%
-%:%415=192%:%
-%:%416=192%:%
-%:%417=193%:%
-%:%418=193%:%
-%:%423=193%:%
-%:%426=194%:%
-%:%427=195%:%
-%:%428=195%:%
-%:%429=196%:%
-%:%430=197%:%
-%:%431=198%:%
-%:%432=199%:%
-%:%435=200%:%
-%:%439=200%:%
-%:%440=200%:%
-%:%441=201%:%
-%:%442=201%:%
-%:%447=201%:%
-%:%450=202%:%
-%:%451=203%:%
-%:%452=203%:%
-%:%453=204%:%
-%:%454=205%:%
-%:%455=206%:%
-%:%456=207%:%
-%:%457=207%:%
-%:%458=208%:%
-%:%459=209%:%
-%:%460=210%:%
-%:%461=211%:%
-%:%462=212%:%
-%:%463=212%:%
-%:%464=213%:%
-%:%465=214%:%
-%:%468=215%:%
-%:%472=215%:%
-%:%473=215%:%
-%:%474=216%:%
-%:%475=216%:%
-%:%480=216%:%
-%:%483=217%:%
-%:%484=218%:%
-%:%485=218%:%
-%:%488=219%:%
-%:%492=219%:%
-%:%493=219%:%
-%:%494=220%:%
-%:%495=220%:%
-%:%509=222%:%
-%:%519=224%:%
-%:%520=224%:%
-%:%521=225%:%
-%:%522=226%:%
-%:%523=227%:%
-%:%524=228%:%
-%:%525=228%:%
-%:%526=229%:%
-%:%528=231%:%
-%:%529=232%:%
-%:%530=233%:%
-%:%531=233%:%
-%:%532=234%:%
-%:%533=235%:%
-%:%534=236%:%
-%:%541=237%:%
-%:%542=237%:%
-%:%543=238%:%
-%:%544=238%:%
-%:%545=239%:%
-%:%546=240%:%
-%:%547=240%:%
-%:%548=241%:%
-%:%549=241%:%
-%:%550=242%:%
-%:%551=242%:%
-%:%552=243%:%
-%:%553=243%:%
-%:%554=244%:%
-%:%555=244%:%
-%:%556=245%:%
-%:%557=245%:%
-%:%558=246%:%
-%:%559=246%:%
-%:%560=246%:%
-%:%561=247%:%
-%:%562=247%:%
-%:%563=248%:%
-%:%564=249%:%
-%:%565=249%:%
-%:%566=250%:%
-%:%567=251%:%
-%:%568=251%:%
-%:%569=252%:%
-%:%570=252%:%
-%:%571=252%:%
-%:%572=253%:%
-%:%573=253%:%
-%:%574=254%:%
-%:%580=254%:%
-%:%583=255%:%
-%:%584=256%:%
-%:%585=256%:%
-%:%586=257%:%
-%:%587=258%:%
-%:%588=259%:%
-%:%589=260%:%
-%:%592=261%:%
-%:%596=261%:%
-%:%597=261%:%
-%:%598=262%:%
-%:%599=262%:%
-%:%604=262%:%
-%:%607=263%:%
-%:%608=264%:%
-%:%609=264%:%
-%:%610=265%:%
-%:%611=266%:%
-%:%618=267%:%
-%:%619=267%:%
-%:%620=268%:%
-%:%621=268%:%
-%:%622=269%:%
-%:%623=270%:%
-%:%624=270%:%
-%:%625=271%:%
-%:%626=272%:%
-%:%627=273%:%
-%:%628=274%:%
-%:%629=275%:%
-%:%631=277%:%
-%:%632=278%:%
-%:%633=278%:%
-%:%634=278%:%
-%:%635=279%:%
-%:%638=282%:%
-%:%639=283%:%
-%:%640=283%:%
-%:%641=284%:%
-%:%642=284%:%
-%:%643=285%:%
-%:%645=287%:%
-%:%646=288%:%
-%:%647=288%:%
-%:%648=289%:%
-%:%649=289%:%
-%:%650=290%:%
-%:%652=292%:%
-%:%653=293%:%
-%:%654=293%:%
-%:%655=294%:%
-%:%656=294%:%
-%:%657=295%:%
-%:%658=295%:%
-%:%659=295%:%
-%:%660=295%:%
-%:%661=296%:%
-%:%662=297%:%
-%:%663=297%:%
-%:%664=298%:%
-%:%665=298%:%
-%:%666=298%:%
-%:%667=298%:%
-%:%668=299%:%
-%:%669=300%:%
-%:%670=300%:%
-%:%671=301%:%
-%:%672=301%:%
-%:%673=302%:%
-%:%675=304%:%
-%:%676=305%:%
-%:%677=305%:%
-%:%678=306%:%
-%:%679=306%:%
-%:%680=307%:%
-%:%681=307%:%
-%:%682=307%:%
-%:%683=308%:%
-%:%684=309%:%
-%:%685=309%:%
-%:%686=310%:%
-%:%687=310%:%
-%:%688=311%:%
-%:%689=311%:%
-%:%690=312%:%
-%:%691=313%:%
-%:%692=313%:%
-%:%693=313%:%
-%:%694=314%:%
-%:%695=314%:%
-%:%696=315%:%
-%:%697=315%:%
-%:%698=316%:%
-%:%699=316%:%
-%:%700=317%:%
-%:%701=318%:%
-%:%702=318%:%
-%:%703=318%:%
-%:%704=319%:%
-%:%705=319%:%
-%:%706=320%:%
-%:%707=320%:%
-%:%708=321%:%
-%:%709=321%:%
-%:%710=322%:%
-%:%711=323%:%
-%:%712=323%:%
-%:%713=323%:%
-%:%714=324%:%
-%:%715=324%:%
-%:%716=325%:%
-%:%717=325%:%
-%:%718=326%:%
-%:%719=326%:%
-%:%720=327%:%
-%:%721=328%:%
-%:%722=328%:%
-%:%723=328%:%
-%:%724=329%:%
-%:%725=329%:%
-%:%726=330%:%
-%:%727=330%:%
-%:%728=331%:%
-%:%729=331%:%
-%:%730=332%:%
-%:%731=333%:%
-%:%732=334%:%
-%:%733=334%:%
-%:%734=335%:%
-%:%735=336%:%
-%:%736=337%:%
-%:%737=338%:%
-%:%738=339%:%
-%:%739=339%:%
-%:%740=339%:%
-%:%741=340%:%
-%:%744=343%:%
-%:%745=344%:%
-%:%746=344%:%
-%:%747=345%:%
-%:%748=345%:%
-%:%749=346%:%
-%:%750=346%:%
-%:%751=347%:%
-%:%752=347%:%
-%:%753=348%:%
-%:%768=350%:%
-%:%778=356%:%
-%:%779=356%:%
-%:%780=357%:%
-%:%781=358%:%
-%:%782=359%:%
-%:%783=359%:%
-%:%784=360%:%
-%:%785=361%:%
-%:%786=362%:%
-%:%787=363%:%
-%:%788=364%:%
-%:%791=365%:%
-%:%795=365%:%
-%:%796=365%:%
-%:%797=366%:%
-%:%798=366%:%
-%:%803=366%:%
-%:%806=367%:%
-%:%807=368%:%
-%:%808=368%:%
-%:%809=369%:%
-%:%810=370%:%
-%:%811=371%:%
-%:%812=372%:%
-%:%819=373%:%
-%:%820=373%:%
-%:%821=374%:%
-%:%822=374%:%
-%:%823=375%:%
-%:%824=375%:%
-%:%825=376%:%
-%:%826=376%:%
-%:%827=377%:%
-%:%828=378%:%
-%:%829=378%:%
-%:%830=379%:%
-%:%836=379%:%
-%:%839=380%:%
-%:%840=381%:%
-%:%841=381%:%
-%:%842=382%:%
-%:%843=383%:%
-%:%844=384%:%
-%:%847=385%:%
-%:%851=385%:%
-%:%852=385%:%
-%:%866=387%:%
-%:%876=389%:%
-%:%877=389%:%
-%:%878=390%:%
-%:%879=391%:%
-%:%883=395%:%
-%:%884=396%:%
-%:%885=400%:%
-%:%886=401%:%
-%:%887=401%:%
-%:%888=402%:%
-%:%893=407%:%
-%:%894=408%:%
-%:%895=409%:%
-%:%896=409%:%
-%:%897=410%:%
-%:%898=411%:%
-%:%905=412%:%
-%:%906=412%:%
-%:%907=413%:%
-%:%908=413%:%
-%:%909=414%:%
-%:%910=415%:%
-%:%911=416%:%
-%:%912=416%:%
-%:%913=417%:%
-%:%914=417%:%
-%:%915=418%:%
-%:%916=418%:%
-%:%917=419%:%
-%:%918=420%:%
-%:%919=421%:%
-%:%920=421%:%
-%:%921=422%:%
-%:%922=422%:%
-%:%923=423%:%
-%:%924=423%:%
-%:%925=424%:%
-%:%926=424%:%
-%:%927=425%:%
-%:%928=426%:%
-%:%929=426%:%
-%:%930=427%:%
-%:%931=427%:%
-%:%932=428%:%
-%:%933=428%:%
-%:%934=429%:%
-%:%935=429%:%
-%:%936=430%:%
-%:%937=430%:%
-%:%938=431%:%
-%:%939=431%:%
-%:%940=432%:%
-%:%946=432%:%
-%:%949=433%:%
-%:%950=434%:%
-%:%951=434%:%
-%:%952=435%:%
-%:%953=436%:%
-%:%954=437%:%
-%:%955=438%:%
-%:%956=439%:%
-%:%963=440%:%
-%:%964=440%:%
-%:%965=441%:%
-%:%966=441%:%
-%:%967=442%:%
-%:%968=443%:%
-%:%969=444%:%
-%:%970=444%:%
-%:%971=445%:%
-%:%972=445%:%
-%:%973=446%:%
-%:%974=446%:%
-%:%975=447%:%
-%:%976=447%:%
-%:%977=448%:%
-%:%978=448%:%
-%:%979=449%:%
-%:%980=449%:%
-%:%981=450%:%
-%:%982=450%:%
-%:%983=451%:%
-%:%984=452%:%
-%:%985=452%:%
-%:%986=453%:%
-%:%987=453%:%
-%:%988=454%:%
-%:%989=454%:%
-%:%990=455%:%
-%:%996=455%:%
-%:%999=456%:%
-%:%1000=457%:%
-%:%1001=457%:%
-%:%1002=458%:%
-%:%1003=459%:%
-%:%1004=460%:%
-%:%1005=461%:%
-%:%1006=462%:%
-%:%1007=463%:%
-%:%1014=464%:%
-%:%1015=464%:%
-%:%1016=465%:%
-%:%1017=465%:%
-%:%1018=466%:%
-%:%1019=466%:%
-%:%1020=467%:%
-%:%1021=468%:%
-%:%1022=469%:%
-%:%1023=469%:%
-%:%1024=470%:%
-%:%1025=470%:%
-%:%1026=471%:%
-%:%1027=471%:%
-%:%1028=472%:%
-%:%1030=474%:%
-%:%1031=475%:%
-%:%1032=475%:%
-%:%1033=476%:%
-%:%1034=476%:%
-%:%1035=477%:%
-%:%1036=477%:%
-%:%1037=478%:%
-%:%1038=478%:%
-%:%1039=479%:%
-%:%1040=479%:%
-%:%1041=480%:%
-%:%1042=480%:%
-%:%1043=481%:%
-%:%1044=481%:%
-%:%1045=482%:%
-%:%1046=483%:%
-%:%1047=483%:%
-%:%1048=484%:%
-%:%1049=484%:%
-%:%1050=485%:%
-%:%1051=485%:%
-%:%1052=486%:%
-%:%1053=487%:%
-%:%1054=487%:%
-%:%1055=488%:%
-%:%1056=488%:%
-%:%1057=489%:%
-%:%1058=489%:%
-%:%1059=490%:%
-%:%1060=490%:%
-%:%1061=491%:%
-%:%1062=491%:%
-%:%1063=492%:%
-%:%1064=492%:%
-%:%1065=492%:%
-%:%1066=493%:%
-%:%1067=493%:%
-%:%1068=494%:%
-%:%1069=494%:%
-%:%1070=495%:%
-%:%1071=495%:%
-%:%1072=495%:%
-%:%1073=496%:%
-%:%1074=497%:%
-%:%1075=497%:%
-%:%1076=498%:%
-%:%1077=499%:%
-%:%1078=499%:%
-%:%1079=500%:%
-%:%1080=500%:%
-%:%1081=500%:%
-%:%1082=501%:%
-%:%1084=503%:%
-%:%1085=504%:%
-%:%1086=504%:%
-%:%1087=505%:%
-%:%1088=506%:%
-%:%1089=506%:%
-%:%1090=507%:%
-%:%1091=507%:%
-%:%1092=507%:%
-%:%1093=508%:%
-%:%1094=508%:%
-%:%1095=509%:%
-%:%1096=509%:%
-%:%1097=510%:%
-%:%1098=510%:%
-%:%1099=511%:%
-%:%1100=511%:%
-%:%1101=512%:%
-%:%1102=512%:%
-%:%1103=513%:%
-%:%1104=513%:%
-%:%1105=514%:%
-%:%1106=514%:%
-%:%1107=515%:%
-%:%1108=515%:%
-%:%1109=516%:%
-%:%1110=516%:%
-%:%1111=517%:%
-%:%1112=518%:%
-%:%1113=518%:%
-%:%1114=519%:%
-%:%1120=519%:%
-%:%1125=520%:%
-%:%1130=521%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Result.tex b/virage/src/test/resources/old_theories/output/document/Result.tex
deleted file mode 100644
index c2899fe..0000000
--- a/virage/src/test/resources/old_theories/output/document/Result.tex
+++ /dev/null
@@ -1,469 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Result}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Electoral Result%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Result\isanewline
-\ \ \isakeyword{imports}\ Main\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-An electoral result is the principal result type of the composable modules
-voting framework, as it is a generalization of the set of winning alternatives
-from social choice functions. Electoral results are selections of the received
-(possibly empty) set of alternatives into the three disjoint groups of elected,
-rejected and deferred alternatives.
-Any of those sets, e.g., the set of winning (elected) alternatives, may also
-be left empty, as long as they collectively still hold all the received
-alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Result\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isacharasterisk}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ set\ {\isacharasterisk}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Functions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ disjoint{\isadigit{3}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ well{\isacharunderscore}{\kern0pt}formed\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ result\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}disjoint{\isadigit{3}}\ result\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ result{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elect{\isacharunderscore}{\kern0pt}r\ r\ {\isasymequiv}\ fst\ r{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}reject{\isacharunderscore}{\kern0pt}r\ r\ {\isasymequiv}\ fst\ {\isacharparenleft}{\kern0pt}snd\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}r\ r\ {\isasymequiv}\ snd\ {\isacharparenleft}{\kern0pt}snd\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej\ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}elec{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}rej\ x{\isacharunderscore}{\kern0pt}elec\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}rej\ x{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\ card\ e\ {\isacharplus}{\kern0pt}\ card\ r\ {\isacharplus}{\kern0pt}\ card\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ disj\ set{\isacharunderscore}{\kern0pt}partit\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ finite{\isacharunderscore}{\kern0pt}Un\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}disjoint\ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}right{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Component_Types/Social_Choice_Types/Result.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%48=21%:%
-%:%49=22%:%
-%:%58=24%:%
-%:%68=30%:%
-%:%69=30%:%
-%:%76=32%:%
-%:%86=38%:%
-%:%87=38%:%
-%:%88=39%:%
-%:%91=42%:%
-%:%92=43%:%
-%:%93=44%:%
-%:%94=44%:%
-%:%95=45%:%
-%:%96=46%:%
-%:%97=47%:%
-%:%98=47%:%
-%:%99=48%:%
-%:%100=49%:%
-%:%101=50%:%
-%:%102=51%:%
-%:%103=51%:%
-%:%104=52%:%
-%:%105=53%:%
-%:%106=54%:%
-%:%107=54%:%
-%:%108=55%:%
-%:%109=56%:%
-%:%110=57%:%
-%:%111=57%:%
-%:%112=58%:%
-%:%119=60%:%
-%:%129=62%:%
-%:%130=62%:%
-%:%131=63%:%
-%:%132=64%:%
-%:%139=65%:%
-%:%140=65%:%
-%:%141=66%:%
-%:%142=66%:%
-%:%143=67%:%
-%:%144=68%:%
-%:%145=68%:%
-%:%146=69%:%
-%:%147=70%:%
-%:%148=71%:%
-%:%149=72%:%
-%:%150=72%:%
-%:%151=72%:%
-%:%152=73%:%
-%:%153=74%:%
-%:%154=75%:%
-%:%155=75%:%
-%:%156=76%:%
-%:%157=76%:%
-%:%158=77%:%
-%:%159=77%:%
-%:%160=78%:%
-%:%161=78%:%
-%:%162=79%:%
-%:%163=79%:%
-%:%164=80%:%
-%:%165=80%:%
-%:%166=81%:%
-%:%167=82%:%
-%:%168=82%:%
-%:%169=83%:%
-%:%170=84%:%
-%:%171=84%:%
-%:%172=84%:%
-%:%173=85%:%
-%:%174=86%:%
-%:%175=87%:%
-%:%176=87%:%
-%:%177=88%:%
-%:%178=88%:%
-%:%179=89%:%
-%:%180=89%:%
-%:%181=90%:%
-%:%182=90%:%
-%:%183=91%:%
-%:%184=91%:%
-%:%185=92%:%
-%:%186=92%:%
-%:%187=93%:%
-%:%188=94%:%
-%:%189=94%:%
-%:%190=95%:%
-%:%191=96%:%
-%:%192=97%:%
-%:%193=97%:%
-%:%194=97%:%
-%:%195=98%:%
-%:%196=99%:%
-%:%197=100%:%
-%:%198=100%:%
-%:%199=101%:%
-%:%200=101%:%
-%:%201=102%:%
-%:%202=102%:%
-%:%203=103%:%
-%:%204=103%:%
-%:%205=104%:%
-%:%206=104%:%
-%:%207=105%:%
-%:%208=105%:%
-%:%209=106%:%
-%:%210=107%:%
-%:%211=107%:%
-%:%212=108%:%
-%:%213=109%:%
-%:%214=110%:%
-%:%215=110%:%
-%:%216=110%:%
-%:%217=111%:%
-%:%218=112%:%
-%:%219=113%:%
-%:%220=113%:%
-%:%221=114%:%
-%:%222=114%:%
-%:%223=115%:%
-%:%224=115%:%
-%:%225=116%:%
-%:%226=116%:%
-%:%227=117%:%
-%:%233=117%:%
-%:%236=118%:%
-%:%237=119%:%
-%:%238=119%:%
-%:%239=120%:%
-%:%240=121%:%
-%:%241=122%:%
-%:%242=123%:%
-%:%249=124%:%
-%:%250=124%:%
-%:%251=125%:%
-%:%252=125%:%
-%:%253=125%:%
-%:%254=126%:%
-%:%255=127%:%
-%:%256=127%:%
-%:%257=128%:%
-%:%258=128%:%
-%:%259=128%:%
-%:%260=129%:%
-%:%261=130%:%
-%:%262=130%:%
-%:%263=131%:%
-%:%264=131%:%
-%:%265=132%:%
-%:%266=132%:%
-%:%267=133%:%
-%:%268=134%:%
-%:%269=134%:%
-%:%270=135%:%
-%:%276=135%:%
-%:%281=136%:%
-%:%286=137%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Revision_Composition.tex b/virage/src/test/resources/old_theories/output/document/Revision_Composition.tex
deleted file mode 100644
index d0a613b..0000000
--- a/virage/src/test/resources/old_theories/output/document/Revision_Composition.tex
+++ /dev/null
@@ -1,470 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Revision{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Revision Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Revision{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-A revised electoral module rejects all originally rejected or deferred
-alternatives, and defers the originally elected alternatives.
-It does not elect any alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ revision{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}revision{\isacharunderscore}{\kern0pt}composition\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ rev\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-{\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}{\isasymdown}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m{\isasymdown}\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ revision{\isacharunderscore}{\kern0pt}composition\ m{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}revision{\isacharunderscore}{\kern0pt}composition\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ unity{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}revision{\isacharunderscore}{\kern0pt}composition\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ disjoint{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}revision{\isacharunderscore}{\kern0pt}composition\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ unity\ disjoint\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}modI{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Composition Rules%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ electing{\isacharunderscore}{\kern0pt}def\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ no{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ no{\isacharunderscore}{\kern0pt}elect\ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ fin{\isacharunderscore}{\kern0pt}A\ electing{\isacharunderscore}{\kern0pt}def\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ Diff{\isacharunderscore}{\kern0pt}disjoint\ Int{\isacharunderscore}{\kern0pt}absorb{\isadigit{2}}\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elect\ assms\ electing{\isacharunderscore}{\kern0pt}def\ empty{\isacharunderscore}{\kern0pt}iff\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}w\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymor}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}w\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymor}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Revision_Composition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%53=19%:%
-%:%63=21%:%
-%:%64=21%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=24%:%
-%:%69=25%:%
-%:%70=26%:%
-%:%77=28%:%
-%:%87=30%:%
-%:%88=30%:%
-%:%89=31%:%
-%:%90=32%:%
-%:%97=33%:%
-%:%98=33%:%
-%:%99=34%:%
-%:%100=34%:%
-%:%101=34%:%
-%:%102=35%:%
-%:%103=35%:%
-%:%104=36%:%
-%:%105=36%:%
-%:%106=37%:%
-%:%107=37%:%
-%:%108=38%:%
-%:%109=38%:%
-%:%110=39%:%
-%:%111=39%:%
-%:%112=40%:%
-%:%113=41%:%
-%:%114=42%:%
-%:%115=42%:%
-%:%116=43%:%
-%:%117=43%:%
-%:%118=44%:%
-%:%119=44%:%
-%:%120=45%:%
-%:%121=45%:%
-%:%122=46%:%
-%:%123=47%:%
-%:%124=47%:%
-%:%125=48%:%
-%:%126=48%:%
-%:%127=48%:%
-%:%128=49%:%
-%:%129=49%:%
-%:%130=50%:%
-%:%145=52%:%
-%:%155=55%:%
-%:%156=55%:%
-%:%157=56%:%
-%:%158=57%:%
-%:%161=58%:%
-%:%165=58%:%
-%:%166=58%:%
-%:%171=58%:%
-%:%174=59%:%
-%:%175=63%:%
-%:%176=64%:%
-%:%177=64%:%
-%:%178=65%:%
-%:%179=66%:%
-%:%182=67%:%
-%:%186=67%:%
-%:%187=67%:%
-%:%188=68%:%
-%:%189=68%:%
-%:%190=69%:%
-%:%191=69%:%
-%:%192=70%:%
-%:%193=70%:%
-%:%194=71%:%
-%:%195=71%:%
-%:%196=72%:%
-%:%197=72%:%
-%:%198=73%:%
-%:%199=73%:%
-%:%200=74%:%
-%:%201=75%:%
-%:%202=76%:%
-%:%203=77%:%
-%:%204=77%:%
-%:%205=78%:%
-%:%206=79%:%
-%:%207=80%:%
-%:%208=81%:%
-%:%209=82%:%
-%:%210=82%:%
-%:%211=82%:%
-%:%212=83%:%
-%:%213=84%:%
-%:%214=84%:%
-%:%215=85%:%
-%:%216=86%:%
-%:%217=86%:%
-%:%218=87%:%
-%:%219=87%:%
-%:%220=88%:%
-%:%221=88%:%
-%:%222=89%:%
-%:%223=90%:%
-%:%224=90%:%
-%:%225=91%:%
-%:%231=91%:%
-%:%234=92%:%
-%:%235=96%:%
-%:%236=97%:%
-%:%237=97%:%
-%:%238=98%:%
-%:%239=99%:%
-%:%246=100%:%
-%:%247=100%:%
-%:%248=101%:%
-%:%249=101%:%
-%:%250=102%:%
-%:%251=103%:%
-%:%252=103%:%
-%:%253=104%:%
-%:%254=104%:%
-%:%255=105%:%
-%:%256=105%:%
-%:%257=105%:%
-%:%258=106%:%
-%:%259=106%:%
-%:%260=107%:%
-%:%261=107%:%
-%:%262=108%:%
-%:%263=108%:%
-%:%264=108%:%
-%:%265=109%:%
-%:%266=109%:%
-%:%267=110%:%
-%:%268=110%:%
-%:%269=111%:%
-%:%270=111%:%
-%:%271=111%:%
-%:%273=113%:%
-%:%274=114%:%
-%:%275=114%:%
-%:%276=115%:%
-%:%277=115%:%
-%:%278=116%:%
-%:%279=116%:%
-%:%280=117%:%
-%:%281=117%:%
-%:%282=118%:%
-%:%288=118%:%
-%:%293=119%:%
-%:%298=120%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Schwartz_Rule.tex b/virage/src/test/resources/old_theories/output/document/Schwartz_Rule.tex
deleted file mode 100644
index 404ef4d..0000000
--- a/virage/src/test/resources/old_theories/output/document/Schwartz_Rule.tex
+++ /dev/null
@@ -1,115 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Schwartz{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Schwartz Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Schwartz{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Borda{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Schwartz voting rule. Confusingly, it is sometimes also referred as
-Nanson's rule. The Schwartz rule proceeds as in the classic Nanson's rule, but
-excludes alternatives with a Borda score that is strictly less than the
-average Borda score.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ schwartz{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}schwartz{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ schwartz{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}schwartz{\isacharunderscore}{\kern0pt}rule{\isacharunderscore}{\kern0pt}code\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}code{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Schwartz_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%54=20%:%
-%:%64=22%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=26%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%75=29%:%
-%:%80=30%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Sequential_Composition.tex b/virage/src/test/resources/old_theories/output/document/Sequential_Composition.tex
deleted file mode 100644
index 2619cb1..0000000
--- a/virage/src/test/resources/old_theories/output/document/Sequential_Composition.tex
+++ /dev/null
@@ -1,3359 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Sequential{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Sequential Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Sequential{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Component{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The sequential composition creates a new electoral module from
-two electoral modules. In a sequential composition, the second
-electoral module makes decisions over alternatives deferred by
-the first electoral module.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}sequential{\isacharunderscore}{\kern0pt}composition\ m\ n\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}let\ new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ new{\isacharunderscore}{\kern0pt}A\ p\ in\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ new{\isacharunderscore}{\kern0pt}A\ new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ n\ new{\isacharunderscore}{\kern0pt}A\ new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ defer\ n\ new{\isacharunderscore}{\kern0pt}A\ new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ sequence\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ {\isacharparenleft}{\kern0pt}\isakeyword{infix}\ {\isachardoublequoteopen}{\isasymtriangleright}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymtriangleright}\ n\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ sequential{\isacharunderscore}{\kern0pt}composition\ m\ n{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ well{\isacharunderscore}{\kern0pt}formed{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ well{\isacharunderscore}{\kern0pt}formed{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasyminter}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ subset{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasyminter}\ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\isanewline
-\ \ \ \ \ \ \ \ \ \ rev{\isacharunderscore}{\kern0pt}subsetD\ result{\isacharunderscore}{\kern0pt}disj\ distrib{\isacharunderscore}{\kern0pt}imp{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}distrib{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}neutr{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ sup{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}right\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasyminter}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}mp\ result{\isacharunderscore}{\kern0pt}disj\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ Un{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}Int\ boolean{\isacharunderscore}{\kern0pt}algebra{\isacharunderscore}{\kern0pt}cancel{\isachardot}{\kern0pt}inf{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ inf{\isachardot}{\kern0pt}order{\isacharunderscore}{\kern0pt}iff\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}aci{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ subsetD\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasyminter}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ set{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}mp\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\ Int{\isacharunderscore}{\kern0pt}commute\ boolean{\isacharunderscore}{\kern0pt}algebra{\isacharunderscore}{\kern0pt}cancel{\isachardot}{\kern0pt}inf{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ inf{\isachardot}{\kern0pt}commute\ inf{\isachardot}{\kern0pt}orderE\ inf{\isacharunderscore}{\kern0pt}commute\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}aci{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\ sup{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}absorb\ sup{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}distrib{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ distrib{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ sup{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\ disjoint{\isacharunderscore}{\kern0pt}n\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ Un{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\ f{\isacharunderscore}{\kern0pt}prof\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ distrib{\isacharunderscore}{\kern0pt}imp{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ subset{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}distrib{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymunion}\ reject\ m\ A\ p\ {\isasymunion}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\ module{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharparenleft}{\kern0pt}n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ well{\isacharunderscore}{\kern0pt}formed{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}r{\isachardot}{\kern0pt}\ well{\isacharunderscore}{\kern0pt}formed\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}disjoint{\isadigit{3}}\ r\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ empty{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ \ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}lower{\isadigit{1}}\ Un{\isacharunderscore}{\kern0pt}absorb{\isadigit{2}}\ bot{\isachardot}{\kern0pt}extremum{\isacharunderscore}{\kern0pt}uniqueI\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ empty{\isacharunderscore}{\kern0pt}defer\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ prod{\isachardot}{\kern0pt}collapse\isanewline
-\ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}n\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bot{\isachardot}{\kern0pt}extremum{\isacharunderscore}{\kern0pt}uniqueI\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ assm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ ele{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assm\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ finite\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ singletonI\ subsetCE\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ ele\ def\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\ rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m\ defers{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ ele\ rej\ def\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ res{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ elect\ n\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card\ def\ electing{\isacharunderscore}{\kern0pt}n\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}alt\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prod{\isachardot}{\kern0pt}sel\isanewline
-\ \ \ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ def\ def{\isacharunderscore}{\kern0pt}card\ electing{\isacharunderscore}{\kern0pt}n\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ res{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}disjoint\ Diff{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}absorb\ Int{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}right\isanewline
-\ \ \ \ \ \ \ \ \ \ Un{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}Int\ electing{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}alt\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prod{\isachardot}{\kern0pt}sel\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ singletonI\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}bounded{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ elect\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}commute\ fst{\isacharunderscore}{\kern0pt}conv\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ enough{\isacharunderscore}{\kern0pt}leftover{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}leftover\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ a\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ in{\isacharunderscore}{\kern0pt}mono\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Composition Rules%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ reject\ m\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ A\ {\isacharminus}{\kern0pt}\ reject\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}iff\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ elect\ m\ A\ p\ {\isasymunion}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ elec{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isasymunion}\ defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymor}\ defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ Diff{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}conv\ Un{\isacharunderscore}{\kern0pt}empty\ fst{\isacharunderscore}{\kern0pt}conv\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ inf{\isachardot}{\kern0pt}absorb{\isadigit{1}}\ sup{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}right\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}empty\ assms\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prod{\isachardot}{\kern0pt}sel\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isadigit{1}}{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ electing{\isacharunderscore}{\kern0pt}m{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ m{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electing\ {\isacharparenleft}{\kern0pt}m{\isadigit{1}}\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m{\isadigit{1}}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ m{\isadigit{1}}\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ zero{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ electing{\isacharunderscore}{\kern0pt}m{\isadigit{2}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}help{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq\ q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ modules{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ profile{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ p\ q\ a\ {\isasymlongrightarrow}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ mono{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ modules\ profile{\isacharunderscore}{\kern0pt}p\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharequal}{\kern0pt}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ mono{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}eq\ mono{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ modules\ new{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}eq\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ modules\ profile{\isacharunderscore}{\kern0pt}p\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}subset\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ \isacommand{have}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}lengths{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}size\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{2}}\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ modules\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isasymle}\ i\ {\isasymand}\ i\ {\isacharless}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}size\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ modules\ mono{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isasymle}\ i\ {\isasymand}\ i\ {\isacharless}{\kern0pt}\ size\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}size\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{0}}\ eql{\isacharunderscore}{\kern0pt}lengths\ mono{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ leI\ not{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}zero\ nth{\isacharunderscore}{\kern0pt}equalityI\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ mono{\isacharunderscore}{\kern0pt}m\ mono{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}help\isanewline
-\ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ pos{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isasymle}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ pos{\isacharunderscore}{\kern0pt}card\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ \isacommand{have}\isamarkupfalse%
-\ m{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ {\isasymand}\ a\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pos{\isacharunderscore}{\kern0pt}card\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subset{\isacharunderscore}{\kern0pt}antisym\ subset{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ subsetI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ empty{\isacharunderscore}{\kern0pt}iff\ pos{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ pos{\isacharunderscore}{\kern0pt}card\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}fun{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n\ defers{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}A\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}n\ f{\isachardot}{\kern0pt}\ defers\ n\ f\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymnot}\ n\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ {\isasymor}\ infinite\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ profile\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}fun\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}compat{\isacharunderscore}{\kern0pt}seq{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ module{\isacharunderscore}{\kern0pt}m{\isadigit{2}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}S{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ S{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ modules{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ module{\isacharunderscore}{\kern0pt}m{\isadigit{2}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ A\ \isakeyword{where}\ A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}S\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ p\ q{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ A\ {\isasymand}\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ p\ q\ a\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer\ m\ S\ p\ {\isacharequal}{\kern0pt}\ defer\ m\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ a\ b\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ a\ b\ \isacommand{have}\isamarkupfalse%
-\ profiles{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ q{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ DiffD{\isadigit{2}}\ a\ b\ compatible\ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ eq{\isacharunderscore}{\kern0pt}def\ profiles\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}def\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}m{\isadigit{2}}\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ m{\isadigit{2}}\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ q{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}m\ S\ p\ {\isacharequal}{\kern0pt}\ m\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ a\ b\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ UnI{\isadigit{1}}\ prod{\isachardot}{\kern0pt}sel\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\ modules\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}monotonicity\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ elect{\isacharunderscore}{\kern0pt}w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Profile{\isachardot}{\kern0pt}lifted\ A\ p\ q\ w{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}w\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}p\ lifted{\isacharunderscore}{\kern0pt}w\ def{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m\ electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ lifted{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Profile{\isachardot}{\kern0pt}lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}unchanged{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ q\ {\isasymnoteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}single{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ q\ {\isasymlongrightarrow}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\ \isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ a{\isacharunderscore}{\kern0pt}single{\isacharunderscore}{\kern0pt}defer\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ empty{\isacharunderscore}{\kern0pt}iff\ finite{\isachardot}{\kern0pt}emptyI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\ order{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\ \isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}mem{\isacharunderscore}{\kern0pt}eq\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ empty{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ insertCI\ subset{\isacharunderscore}{\kern0pt}singletonD\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ defers{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}subset\ Diff{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}absorb\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ card{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}eq\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}singleton\ finite{\isachardot}{\kern0pt}emptyI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ finite{\isacharunderscore}{\kern0pt}Diff\ insertCI\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\ \isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ calculation\ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}different{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}defer\ m\ A\ q\ {\isasymnoteq}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}elect\ m\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}equal{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ elect\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}different{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}a\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}profile{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ results{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ same{\isacharunderscore}{\kern0pt}alternatives{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ results{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}equal\ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}profile\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ still{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}lifted\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ equals{\isadigit{0}}D\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}p\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}subset\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ still{\isacharunderscore}{\kern0pt}lifted\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}n\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}p\ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}q\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ same{\isacharunderscore}{\kern0pt}alternatives\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}q\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Sequential_Composition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%54=20%:%
-%:%64=22%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%72=29%:%
-%:%73=30%:%
-%:%74=31%:%
-%:%75=31%:%
-%:%76=32%:%
-%:%77=33%:%
-%:%78=34%:%
-%:%79=35%:%
-%:%80=36%:%
-%:%81=36%:%
-%:%82=37%:%
-%:%83=38%:%
-%:%84=39%:%
-%:%85=40%:%
-%:%92=41%:%
-%:%93=41%:%
-%:%94=42%:%
-%:%95=42%:%
-%:%96=43%:%
-%:%97=43%:%
-%:%98=44%:%
-%:%99=44%:%
-%:%100=44%:%
-%:%101=45%:%
-%:%102=45%:%
-%:%103=46%:%
-%:%104=46%:%
-%:%105=47%:%
-%:%106=47%:%
-%:%107=47%:%
-%:%108=48%:%
-%:%109=49%:%
-%:%110=49%:%
-%:%111=50%:%
-%:%112=50%:%
-%:%113=51%:%
-%:%114=51%:%
-%:%115=51%:%
-%:%116=52%:%
-%:%117=53%:%
-%:%118=53%:%
-%:%119=54%:%
-%:%120=55%:%
-%:%121=55%:%
-%:%122=56%:%
-%:%123=56%:%
-%:%124=57%:%
-%:%125=57%:%
-%:%126=58%:%
-%:%127=59%:%
-%:%128=59%:%
-%:%129=60%:%
-%:%130=61%:%
-%:%131=62%:%
-%:%132=63%:%
-%:%133=64%:%
-%:%134=64%:%
-%:%135=65%:%
-%:%136=65%:%
-%:%137=66%:%
-%:%138=66%:%
-%:%139=67%:%
-%:%140=68%:%
-%:%141=68%:%
-%:%142=69%:%
-%:%143=70%:%
-%:%144=71%:%
-%:%145=72%:%
-%:%146=72%:%
-%:%147=73%:%
-%:%148=73%:%
-%:%149=74%:%
-%:%150=74%:%
-%:%151=75%:%
-%:%152=76%:%
-%:%153=76%:%
-%:%154=77%:%
-%:%155=78%:%
-%:%156=79%:%
-%:%157=79%:%
-%:%158=80%:%
-%:%159=80%:%
-%:%160=80%:%
-%:%161=81%:%
-%:%162=82%:%
-%:%163=83%:%
-%:%164=83%:%
-%:%165=84%:%
-%:%166=85%:%
-%:%167=86%:%
-%:%168=86%:%
-%:%169=87%:%
-%:%170=87%:%
-%:%171=87%:%
-%:%172=87%:%
-%:%173=88%:%
-%:%174=89%:%
-%:%175=90%:%
-%:%176=90%:%
-%:%177=91%:%
-%:%178=91%:%
-%:%179=92%:%
-%:%180=92%:%
-%:%181=92%:%
-%:%182=93%:%
-%:%183=93%:%
-%:%184=94%:%
-%:%185=95%:%
-%:%186=96%:%
-%:%187=96%:%
-%:%188=97%:%
-%:%189=98%:%
-%:%190=99%:%
-%:%191=99%:%
-%:%192=100%:%
-%:%193=100%:%
-%:%194=100%:%
-%:%195=101%:%
-%:%197=103%:%
-%:%198=104%:%
-%:%199=104%:%
-%:%200=105%:%
-%:%201=105%:%
-%:%202=106%:%
-%:%203=106%:%
-%:%204=107%:%
-%:%205=107%:%
-%:%206=108%:%
-%:%212=108%:%
-%:%215=109%:%
-%:%216=110%:%
-%:%217=110%:%
-%:%218=111%:%
-%:%219=112%:%
-%:%220=113%:%
-%:%221=114%:%
-%:%228=115%:%
-%:%229=115%:%
-%:%230=116%:%
-%:%231=116%:%
-%:%232=117%:%
-%:%233=117%:%
-%:%234=118%:%
-%:%235=118%:%
-%:%236=118%:%
-%:%237=119%:%
-%:%238=119%:%
-%:%239=120%:%
-%:%240=120%:%
-%:%241=120%:%
-%:%242=121%:%
-%:%243=122%:%
-%:%244=122%:%
-%:%245=123%:%
-%:%246=123%:%
-%:%247=123%:%
-%:%248=124%:%
-%:%249=125%:%
-%:%250=125%:%
-%:%251=126%:%
-%:%252=126%:%
-%:%253=127%:%
-%:%254=127%:%
-%:%255=127%:%
-%:%256=128%:%
-%:%258=130%:%
-%:%259=131%:%
-%:%260=131%:%
-%:%261=132%:%
-%:%262=132%:%
-%:%263=133%:%
-%:%264=133%:%
-%:%265=133%:%
-%:%266=134%:%
-%:%268=136%:%
-%:%269=137%:%
-%:%270=137%:%
-%:%271=138%:%
-%:%272=138%:%
-%:%273=139%:%
-%:%276=142%:%
-%:%277=143%:%
-%:%278=143%:%
-%:%279=144%:%
-%:%280=144%:%
-%:%281=145%:%
-%:%282=145%:%
-%:%283=146%:%
-%:%284=146%:%
-%:%285=147%:%
-%:%300=149%:%
-%:%310=151%:%
-%:%311=151%:%
-%:%312=152%:%
-%:%313=153%:%
-%:%314=154%:%
-%:%317=155%:%
-%:%321=155%:%
-%:%322=155%:%
-%:%323=156%:%
-%:%324=156%:%
-%:%325=157%:%
-%:%326=157%:%
-%:%327=158%:%
-%:%328=159%:%
-%:%329=160%:%
-%:%330=160%:%
-%:%331=161%:%
-%:%332=162%:%
-%:%333=163%:%
-%:%334=163%:%
-%:%335=164%:%
-%:%336=165%:%
-%:%337=165%:%
-%:%338=166%:%
-%:%339=166%:%
-%:%340=167%:%
-%:%341=167%:%
-%:%342=168%:%
-%:%343=169%:%
-%:%344=169%:%
-%:%345=170%:%
-%:%360=172%:%
-%:%370=174%:%
-%:%371=174%:%
-%:%372=175%:%
-%:%373=176%:%
-%:%374=177%:%
-%:%375=178%:%
-%:%376=179%:%
-%:%377=180%:%
-%:%380=181%:%
-%:%384=181%:%
-%:%385=181%:%
-%:%386=182%:%
-%:%387=183%:%
-%:%388=184%:%
-%:%389=185%:%
-%:%390=185%:%
-%:%395=185%:%
-%:%398=186%:%
-%:%399=187%:%
-%:%400=187%:%
-%:%401=188%:%
-%:%402=189%:%
-%:%403=190%:%
-%:%404=191%:%
-%:%405=192%:%
-%:%406=193%:%
-%:%413=194%:%
-%:%414=194%:%
-%:%415=195%:%
-%:%416=195%:%
-%:%417=196%:%
-%:%418=196%:%
-%:%419=196%:%
-%:%420=197%:%
-%:%421=197%:%
-%:%422=198%:%
-%:%423=199%:%
-%:%424=199%:%
-%:%425=200%:%
-%:%426=200%:%
-%:%427=201%:%
-%:%428=201%:%
-%:%429=202%:%
-%:%430=202%:%
-%:%431=202%:%
-%:%432=203%:%
-%:%433=203%:%
-%:%434=204%:%
-%:%435=204%:%
-%:%436=205%:%
-%:%437=205%:%
-%:%438=205%:%
-%:%439=206%:%
-%:%440=207%:%
-%:%441=207%:%
-%:%442=208%:%
-%:%443=208%:%
-%:%444=208%:%
-%:%445=209%:%
-%:%446=210%:%
-%:%447=210%:%
-%:%448=211%:%
-%:%449=212%:%
-%:%450=212%:%
-%:%451=213%:%
-%:%452=213%:%
-%:%453=213%:%
-%:%454=214%:%
-%:%455=215%:%
-%:%456=215%:%
-%:%457=216%:%
-%:%458=216%:%
-%:%459=217%:%
-%:%460=217%:%
-%:%461=217%:%
-%:%462=218%:%
-%:%463=219%:%
-%:%464=219%:%
-%:%465=220%:%
-%:%466=221%:%
-%:%467=221%:%
-%:%468=222%:%
-%:%469=222%:%
-%:%470=223%:%
-%:%471=224%:%
-%:%472=225%:%
-%:%473=225%:%
-%:%474=226%:%
-%:%475=227%:%
-%:%476=227%:%
-%:%477=228%:%
-%:%478=228%:%
-%:%479=228%:%
-%:%480=229%:%
-%:%481=230%:%
-%:%482=230%:%
-%:%483=231%:%
-%:%484=232%:%
-%:%485=233%:%
-%:%486=233%:%
-%:%487=234%:%
-%:%488=234%:%
-%:%489=235%:%
-%:%490=235%:%
-%:%491=236%:%
-%:%492=236%:%
-%:%493=237%:%
-%:%494=238%:%
-%:%495=239%:%
-%:%496=240%:%
-%:%497=240%:%
-%:%498=241%:%
-%:%504=241%:%
-%:%507=242%:%
-%:%508=243%:%
-%:%509=243%:%
-%:%510=244%:%
-%:%511=245%:%
-%:%512=246%:%
-%:%513=247%:%
-%:%514=248%:%
-%:%517=249%:%
-%:%521=249%:%
-%:%522=249%:%
-%:%523=250%:%
-%:%524=251%:%
-%:%525=251%:%
-%:%530=251%:%
-%:%533=252%:%
-%:%534=253%:%
-%:%535=253%:%
-%:%536=254%:%
-%:%537=255%:%
-%:%538=256%:%
-%:%539=257%:%
-%:%540=258%:%
-%:%543=259%:%
-%:%547=259%:%
-%:%548=259%:%
-%:%549=260%:%
-%:%550=261%:%
-%:%551=261%:%
-%:%556=261%:%
-%:%559=262%:%
-%:%560=263%:%
-%:%561=263%:%
-%:%562=264%:%
-%:%563=265%:%
-%:%564=266%:%
-%:%565=267%:%
-%:%566=268%:%
-%:%567=269%:%
-%:%568=270%:%
-%:%571=271%:%
-%:%575=271%:%
-%:%576=271%:%
-%:%577=272%:%
-%:%578=272%:%
-%:%583=272%:%
-%:%586=273%:%
-%:%587=274%:%
-%:%588=274%:%
-%:%589=275%:%
-%:%590=276%:%
-%:%591=277%:%
-%:%592=278%:%
-%:%593=279%:%
-%:%594=280%:%
-%:%596=282%:%
-%:%599=283%:%
-%:%603=283%:%
-%:%604=283%:%
-%:%605=284%:%
-%:%606=284%:%
-%:%611=284%:%
-%:%614=285%:%
-%:%615=286%:%
-%:%616=286%:%
-%:%617=287%:%
-%:%618=288%:%
-%:%619=289%:%
-%:%620=290%:%
-%:%621=291%:%
-%:%622=292%:%
-%:%625=293%:%
-%:%629=293%:%
-%:%630=293%:%
-%:%631=294%:%
-%:%632=295%:%
-%:%633=296%:%
-%:%634=296%:%
-%:%639=296%:%
-%:%642=297%:%
-%:%643=298%:%
-%:%644=298%:%
-%:%645=299%:%
-%:%646=300%:%
-%:%647=301%:%
-%:%648=302%:%
-%:%649=303%:%
-%:%656=304%:%
-%:%657=304%:%
-%:%658=305%:%
-%:%659=305%:%
-%:%660=306%:%
-%:%661=306%:%
-%:%662=307%:%
-%:%663=307%:%
-%:%664=308%:%
-%:%665=308%:%
-%:%666=309%:%
-%:%668=311%:%
-%:%669=312%:%
-%:%670=312%:%
-%:%671=313%:%
-%:%672=313%:%
-%:%673=314%:%
-%:%674=315%:%
-%:%675=316%:%
-%:%676=316%:%
-%:%677=317%:%
-%:%678=317%:%
-%:%679=318%:%
-%:%680=318%:%
-%:%681=319%:%
-%:%682=319%:%
-%:%683=320%:%
-%:%684=320%:%
-%:%685=321%:%
-%:%691=321%:%
-%:%694=322%:%
-%:%695=323%:%
-%:%696=323%:%
-%:%697=324%:%
-%:%698=325%:%
-%:%699=326%:%
-%:%700=327%:%
-%:%701=328%:%
-%:%702=329%:%
-%:%704=331%:%
-%:%707=332%:%
-%:%711=332%:%
-%:%712=332%:%
-%:%713=333%:%
-%:%714=334%:%
-%:%715=334%:%
-%:%729=336%:%
-%:%739=339%:%
-%:%740=339%:%
-%:%741=340%:%
-%:%742=341%:%
-%:%743=342%:%
-%:%744=343%:%
-%:%751=344%:%
-%:%752=344%:%
-%:%753=345%:%
-%:%754=345%:%
-%:%755=346%:%
-%:%756=347%:%
-%:%757=348%:%
-%:%758=348%:%
-%:%759=349%:%
-%:%760=349%:%
-%:%761=349%:%
-%:%762=350%:%
-%:%763=351%:%
-%:%764=351%:%
-%:%765=352%:%
-%:%766=352%:%
-%:%767=352%:%
-%:%768=353%:%
-%:%769=354%:%
-%:%770=354%:%
-%:%771=355%:%
-%:%772=356%:%
-%:%773=356%:%
-%:%774=357%:%
-%:%775=357%:%
-%:%776=357%:%
-%:%777=358%:%
-%:%778=359%:%
-%:%779=359%:%
-%:%780=360%:%
-%:%781=360%:%
-%:%782=361%:%
-%:%783=362%:%
-%:%784=363%:%
-%:%785=363%:%
-%:%786=364%:%
-%:%787=364%:%
-%:%788=365%:%
-%:%789=365%:%
-%:%790=365%:%
-%:%791=366%:%
-%:%792=367%:%
-%:%793=367%:%
-%:%794=368%:%
-%:%795=368%:%
-%:%796=369%:%
-%:%797=369%:%
-%:%798=370%:%
-%:%799=370%:%
-%:%800=371%:%
-%:%801=371%:%
-%:%802=372%:%
-%:%803=372%:%
-%:%804=373%:%
-%:%805=374%:%
-%:%806=375%:%
-%:%807=376%:%
-%:%808=376%:%
-%:%809=377%:%
-%:%815=377%:%
-%:%818=378%:%
-%:%819=379%:%
-%:%820=380%:%
-%:%821=380%:%
-%:%822=381%:%
-%:%823=382%:%
-%:%824=383%:%
-%:%825=384%:%
-%:%828=385%:%
-%:%832=385%:%
-%:%833=385%:%
-%:%834=386%:%
-%:%835=387%:%
-%:%836=387%:%
-%:%841=387%:%
-%:%844=388%:%
-%:%845=393%:%
-%:%846=394%:%
-%:%847=394%:%
-%:%848=395%:%
-%:%849=396%:%
-%:%850=397%:%
-%:%857=398%:%
-%:%858=398%:%
-%:%859=399%:%
-%:%860=399%:%
-%:%861=400%:%
-%:%862=401%:%
-%:%863=402%:%
-%:%864=402%:%
-%:%865=403%:%
-%:%866=403%:%
-%:%867=404%:%
-%:%868=404%:%
-%:%869=405%:%
-%:%870=406%:%
-%:%871=407%:%
-%:%872=407%:%
-%:%873=408%:%
-%:%874=409%:%
-%:%875=409%:%
-%:%876=410%:%
-%:%877=410%:%
-%:%878=411%:%
-%:%879=411%:%
-%:%880=412%:%
-%:%881=413%:%
-%:%882=414%:%
-%:%883=414%:%
-%:%884=415%:%
-%:%890=415%:%
-%:%893=416%:%
-%:%894=417%:%
-%:%895=417%:%
-%:%896=418%:%
-%:%897=419%:%
-%:%898=420%:%
-%:%899=421%:%
-%:%900=422%:%
-%:%907=423%:%
-%:%908=423%:%
-%:%909=424%:%
-%:%910=424%:%
-%:%911=425%:%
-%:%912=425%:%
-%:%913=426%:%
-%:%914=426%:%
-%:%915=427%:%
-%:%916=427%:%
-%:%917=428%:%
-%:%918=428%:%
-%:%919=428%:%
-%:%920=429%:%
-%:%921=430%:%
-%:%922=430%:%
-%:%923=431%:%
-%:%924=431%:%
-%:%925=432%:%
-%:%926=432%:%
-%:%927=433%:%
-%:%928=433%:%
-%:%929=434%:%
-%:%930=434%:%
-%:%931=434%:%
-%:%932=435%:%
-%:%933=435%:%
-%:%934=436%:%
-%:%935=436%:%
-%:%936=436%:%
-%:%937=437%:%
-%:%938=437%:%
-%:%939=438%:%
-%:%940=438%:%
-%:%941=439%:%
-%:%942=439%:%
-%:%943=440%:%
-%:%944=440%:%
-%:%945=441%:%
-%:%946=442%:%
-%:%947=442%:%
-%:%948=443%:%
-%:%949=443%:%
-%:%950=444%:%
-%:%951=444%:%
-%:%952=445%:%
-%:%953=445%:%
-%:%954=446%:%
-%:%955=447%:%
-%:%956=447%:%
-%:%957=448%:%
-%:%958=448%:%
-%:%959=449%:%
-%:%960=449%:%
-%:%961=450%:%
-%:%962=451%:%
-%:%963=451%:%
-%:%964=452%:%
-%:%965=452%:%
-%:%966=453%:%
-%:%967=453%:%
-%:%968=454%:%
-%:%969=454%:%
-%:%970=455%:%
-%:%971=456%:%
-%:%972=456%:%
-%:%973=457%:%
-%:%974=457%:%
-%:%975=458%:%
-%:%976=458%:%
-%:%977=459%:%
-%:%978=459%:%
-%:%979=459%:%
-%:%980=460%:%
-%:%981=460%:%
-%:%982=461%:%
-%:%983=461%:%
-%:%984=461%:%
-%:%985=462%:%
-%:%986=463%:%
-%:%987=463%:%
-%:%988=464%:%
-%:%989=464%:%
-%:%990=465%:%
-%:%991=465%:%
-%:%992=465%:%
-%:%993=466%:%
-%:%994=466%:%
-%:%995=467%:%
-%:%996=468%:%
-%:%997=468%:%
-%:%998=469%:%
-%:%999=469%:%
-%:%1000=470%:%
-%:%1001=470%:%
-%:%1002=470%:%
-%:%1003=471%:%
-%:%1004=471%:%
-%:%1005=472%:%
-%:%1006=472%:%
-%:%1007=473%:%
-%:%1008=473%:%
-%:%1009=473%:%
-%:%1010=473%:%
-%:%1011=474%:%
-%:%1012=475%:%
-%:%1013=475%:%
-%:%1014=476%:%
-%:%1015=476%:%
-%:%1016=476%:%
-%:%1017=477%:%
-%:%1021=481%:%
-%:%1022=482%:%
-%:%1023=482%:%
-%:%1024=483%:%
-%:%1025=483%:%
-%:%1026=484%:%
-%:%1027=484%:%
-%:%1028=484%:%
-%:%1029=485%:%
-%:%1030=486%:%
-%:%1031=487%:%
-%:%1032=487%:%
-%:%1033=488%:%
-%:%1034=488%:%
-%:%1035=489%:%
-%:%1036=489%:%
-%:%1037=489%:%
-%:%1038=490%:%
-%:%1040=492%:%
-%:%1041=493%:%
-%:%1042=493%:%
-%:%1043=494%:%
-%:%1044=495%:%
-%:%1045=496%:%
-%:%1046=496%:%
-%:%1047=497%:%
-%:%1048=497%:%
-%:%1049=498%:%
-%:%1050=498%:%
-%:%1051=499%:%
-%:%1052=499%:%
-%:%1053=500%:%
-%:%1054=500%:%
-%:%1055=501%:%
-%:%1056=501%:%
-%:%1057=502%:%
-%:%1058=502%:%
-%:%1059=503%:%
-%:%1060=503%:%
-%:%1061=504%:%
-%:%1062=504%:%
-%:%1063=505%:%
-%:%1064=505%:%
-%:%1065=506%:%
-%:%1071=506%:%
-%:%1074=507%:%
-%:%1075=508%:%
-%:%1076=509%:%
-%:%1077=509%:%
-%:%1078=510%:%
-%:%1079=511%:%
-%:%1080=512%:%
-%:%1081=513%:%
-%:%1084=514%:%
-%:%1088=514%:%
-%:%1089=514%:%
-%:%1090=515%:%
-%:%1091=516%:%
-%:%1092=516%:%
-%:%1097=516%:%
-%:%1100=517%:%
-%:%1101=523%:%
-%:%1102=524%:%
-%:%1103=524%:%
-%:%1104=525%:%
-%:%1105=526%:%
-%:%1106=527%:%
-%:%1107=528%:%
-%:%1108=529%:%
-%:%1111=530%:%
-%:%1115=530%:%
-%:%1116=530%:%
-%:%1117=531%:%
-%:%1118=531%:%
-%:%1119=532%:%
-%:%1120=532%:%
-%:%1121=533%:%
-%:%1122=533%:%
-%:%1123=534%:%
-%:%1124=534%:%
-%:%1125=535%:%
-%:%1126=535%:%
-%:%1127=536%:%
-%:%1128=536%:%
-%:%1129=537%:%
-%:%1130=537%:%
-%:%1131=538%:%
-%:%1132=538%:%
-%:%1133=539%:%
-%:%1134=539%:%
-%:%1135=540%:%
-%:%1136=540%:%
-%:%1137=541%:%
-%:%1138=541%:%
-%:%1139=542%:%
-%:%1140=542%:%
-%:%1141=543%:%
-%:%1142=544%:%
-%:%1143=545%:%
-%:%1144=545%:%
-%:%1145=546%:%
-%:%1146=547%:%
-%:%1147=548%:%
-%:%1148=549%:%
-%:%1149=549%:%
-%:%1150=549%:%
-%:%1151=550%:%
-%:%1152=551%:%
-%:%1153=551%:%
-%:%1154=552%:%
-%:%1155=552%:%
-%:%1156=552%:%
-%:%1157=553%:%
-%:%1158=554%:%
-%:%1159=554%:%
-%:%1160=555%:%
-%:%1161=555%:%
-%:%1162=556%:%
-%:%1163=556%:%
-%:%1164=557%:%
-%:%1165=558%:%
-%:%1166=558%:%
-%:%1167=559%:%
-%:%1168=560%:%
-%:%1169=561%:%
-%:%1170=561%:%
-%:%1171=562%:%
-%:%1172=562%:%
-%:%1173=563%:%
-%:%1174=563%:%
-%:%1175=564%:%
-%:%1176=565%:%
-%:%1177=565%:%
-%:%1178=566%:%
-%:%1179=566%:%
-%:%1180=567%:%
-%:%1181=568%:%
-%:%1182=568%:%
-%:%1183=569%:%
-%:%1184=570%:%
-%:%1185=570%:%
-%:%1186=571%:%
-%:%1187=571%:%
-%:%1188=572%:%
-%:%1189=573%:%
-%:%1190=574%:%
-%:%1191=574%:%
-%:%1192=575%:%
-%:%1193=576%:%
-%:%1194=576%:%
-%:%1195=577%:%
-%:%1196=577%:%
-%:%1197=578%:%
-%:%1202=583%:%
-%:%1203=584%:%
-%:%1204=584%:%
-%:%1205=585%:%
-%:%1206=585%:%
-%:%1207=586%:%
-%:%1208=586%:%
-%:%1209=587%:%
-%:%1210=588%:%
-%:%1211=589%:%
-%:%1212=589%:%
-%:%1213=590%:%
-%:%1214=591%:%
-%:%1215=592%:%
-%:%1216=592%:%
-%:%1217=593%:%
-%:%1218=593%:%
-%:%1219=594%:%
-%:%1220=594%:%
-%:%1221=595%:%
-%:%1222=595%:%
-%:%1223=596%:%
-%:%1229=596%:%
-%:%1232=597%:%
-%:%1233=601%:%
-%:%1234=602%:%
-%:%1235=602%:%
-%:%1236=603%:%
-%:%1237=604%:%
-%:%1238=605%:%
-%:%1239=606%:%
-%:%1242=607%:%
-%:%1246=607%:%
-%:%1247=607%:%
-%:%1248=608%:%
-%:%1249=608%:%
-%:%1250=609%:%
-%:%1251=609%:%
-%:%1252=610%:%
-%:%1253=610%:%
-%:%1254=611%:%
-%:%1255=611%:%
-%:%1256=612%:%
-%:%1257=612%:%
-%:%1258=613%:%
-%:%1259=613%:%
-%:%1260=614%:%
-%:%1261=614%:%
-%:%1262=615%:%
-%:%1263=615%:%
-%:%1264=616%:%
-%:%1265=616%:%
-%:%1266=617%:%
-%:%1267=617%:%
-%:%1268=618%:%
-%:%1269=619%:%
-%:%1270=619%:%
-%:%1271=620%:%
-%:%1272=621%:%
-%:%1273=621%:%
-%:%1274=622%:%
-%:%1275=623%:%
-%:%1276=623%:%
-%:%1277=624%:%
-%:%1278=624%:%
-%:%1279=625%:%
-%:%1280=625%:%
-%:%1281=626%:%
-%:%1285=630%:%
-%:%1286=631%:%
-%:%1287=631%:%
-%:%1288=632%:%
-%:%1289=632%:%
-%:%1290=633%:%
-%:%1291=633%:%
-%:%1292=634%:%
-%:%1296=638%:%
-%:%1297=639%:%
-%:%1298=639%:%
-%:%1299=640%:%
-%:%1300=640%:%
-%:%1301=641%:%
-%:%1303=643%:%
-%:%1304=644%:%
-%:%1305=644%:%
-%:%1306=645%:%
-%:%1307=645%:%
-%:%1308=646%:%
-%:%1309=647%:%
-%:%1310=648%:%
-%:%1311=649%:%
-%:%1312=649%:%
-%:%1313=650%:%
-%:%1314=651%:%
-%:%1315=652%:%
-%:%1316=652%:%
-%:%1317=653%:%
-%:%1318=654%:%
-%:%1319=654%:%
-%:%1320=655%:%
-%:%1321=655%:%
-%:%1322=656%:%
-%:%1323=656%:%
-%:%1324=656%:%
-%:%1325=657%:%
-%:%1326=658%:%
-%:%1327=658%:%
-%:%1328=659%:%
-%:%1329=659%:%
-%:%1330=660%:%
-%:%1331=660%:%
-%:%1332=661%:%
-%:%1333=661%:%
-%:%1334=662%:%
-%:%1335=662%:%
-%:%1336=663%:%
-%:%1337=663%:%
-%:%1338=664%:%
-%:%1339=665%:%
-%:%1340=666%:%
-%:%1341=666%:%
-%:%1342=667%:%
-%:%1343=668%:%
-%:%1344=669%:%
-%:%1345=669%:%
-%:%1346=670%:%
-%:%1347=670%:%
-%:%1348=670%:%
-%:%1349=671%:%
-%:%1350=672%:%
-%:%1351=673%:%
-%:%1352=673%:%
-%:%1353=674%:%
-%:%1354=674%:%
-%:%1355=674%:%
-%:%1356=675%:%
-%:%1357=675%:%
-%:%1358=676%:%
-%:%1359=676%:%
-%:%1360=677%:%
-%:%1361=677%:%
-%:%1362=677%:%
-%:%1363=678%:%
-%:%1364=679%:%
-%:%1365=679%:%
-%:%1366=680%:%
-%:%1367=680%:%
-%:%1368=681%:%
-%:%1369=681%:%
-%:%1370=682%:%
-%:%1371=682%:%
-%:%1372=682%:%
-%:%1373=683%:%
-%:%1374=684%:%
-%:%1375=684%:%
-%:%1376=685%:%
-%:%1377=685%:%
-%:%1378=686%:%
-%:%1379=686%:%
-%:%1380=686%:%
-%:%1381=687%:%
-%:%1385=691%:%
-%:%1386=692%:%
-%:%1387=692%:%
-%:%1388=693%:%
-%:%1389=693%:%
-%:%1390=694%:%
-%:%1391=694%:%
-%:%1392=695%:%
-%:%1398=695%:%
-%:%1401=696%:%
-%:%1402=702%:%
-%:%1403=703%:%
-%:%1404=703%:%
-%:%1405=704%:%
-%:%1406=705%:%
-%:%1407=706%:%
-%:%1408=707%:%
-%:%1409=708%:%
-%:%1410=709%:%
-%:%1413=710%:%
-%:%1417=710%:%
-%:%1418=710%:%
-%:%1419=711%:%
-%:%1420=711%:%
-%:%1421=712%:%
-%:%1422=712%:%
-%:%1423=713%:%
-%:%1424=713%:%
-%:%1425=714%:%
-%:%1426=714%:%
-%:%1427=715%:%
-%:%1428=715%:%
-%:%1429=716%:%
-%:%1430=716%:%
-%:%1431=717%:%
-%:%1432=717%:%
-%:%1433=718%:%
-%:%1434=718%:%
-%:%1435=719%:%
-%:%1436=719%:%
-%:%1437=720%:%
-%:%1438=720%:%
-%:%1439=721%:%
-%:%1440=721%:%
-%:%1441=722%:%
-%:%1442=722%:%
-%:%1443=723%:%
-%:%1444=724%:%
-%:%1445=725%:%
-%:%1446=726%:%
-%:%1447=727%:%
-%:%1448=727%:%
-%:%1449=728%:%
-%:%1450=729%:%
-%:%1451=730%:%
-%:%1452=731%:%
-%:%1453=731%:%
-%:%1454=732%:%
-%:%1455=733%:%
-%:%1456=733%:%
-%:%1457=734%:%
-%:%1458=734%:%
-%:%1459=735%:%
-%:%1460=735%:%
-%:%1461=736%:%
-%:%1462=736%:%
-%:%1463=737%:%
-%:%1464=738%:%
-%:%1465=739%:%
-%:%1466=739%:%
-%:%1467=740%:%
-%:%1473=740%:%
-%:%1476=741%:%
-%:%1477=746%:%
-%:%1478=747%:%
-%:%1479=747%:%
-%:%1480=748%:%
-%:%1481=749%:%
-%:%1482=750%:%
-%:%1483=751%:%
-%:%1484=752%:%
-%:%1485=753%:%
-%:%1488=754%:%
-%:%1492=754%:%
-%:%1493=754%:%
-%:%1494=755%:%
-%:%1495=755%:%
-%:%1496=756%:%
-%:%1497=756%:%
-%:%1498=757%:%
-%:%1499=757%:%
-%:%1500=758%:%
-%:%1501=759%:%
-%:%1502=759%:%
-%:%1503=760%:%
-%:%1504=760%:%
-%:%1505=761%:%
-%:%1506=761%:%
-%:%1507=762%:%
-%:%1508=762%:%
-%:%1509=763%:%
-%:%1510=763%:%
-%:%1511=764%:%
-%:%1512=764%:%
-%:%1513=765%:%
-%:%1514=765%:%
-%:%1515=766%:%
-%:%1516=766%:%
-%:%1517=767%:%
-%:%1518=767%:%
-%:%1519=768%:%
-%:%1520=769%:%
-%:%1521=770%:%
-%:%1522=771%:%
-%:%1523=772%:%
-%:%1524=772%:%
-%:%1525=773%:%
-%:%1526=774%:%
-%:%1527=775%:%
-%:%1528=775%:%
-%:%1529=776%:%
-%:%1530=776%:%
-%:%1531=777%:%
-%:%1532=777%:%
-%:%1533=778%:%
-%:%1534=778%:%
-%:%1535=779%:%
-%:%1536=779%:%
-%:%1537=780%:%
-%:%1538=780%:%
-%:%1539=781%:%
-%:%1540=781%:%
-%:%1541=782%:%
-%:%1542=782%:%
-%:%1543=783%:%
-%:%1544=783%:%
-%:%1545=784%:%
-%:%1546=784%:%
-%:%1547=785%:%
-%:%1548=785%:%
-%:%1549=786%:%
-%:%1550=786%:%
-%:%1551=787%:%
-%:%1552=787%:%
-%:%1553=788%:%
-%:%1554=788%:%
-%:%1555=789%:%
-%:%1556=789%:%
-%:%1557=790%:%
-%:%1558=790%:%
-%:%1559=791%:%
-%:%1560=791%:%
-%:%1561=792%:%
-%:%1562=792%:%
-%:%1563=793%:%
-%:%1564=793%:%
-%:%1565=794%:%
-%:%1566=794%:%
-%:%1567=795%:%
-%:%1568=796%:%
-%:%1569=797%:%
-%:%1570=798%:%
-%:%1571=798%:%
-%:%1572=799%:%
-%:%1573=799%:%
-%:%1574=799%:%
-%:%1575=800%:%
-%:%1576=801%:%
-%:%1577=801%:%
-%:%1578=802%:%
-%:%1579=803%:%
-%:%1580=803%:%
-%:%1581=804%:%
-%:%1582=804%:%
-%:%1583=804%:%
-%:%1584=805%:%
-%:%1585=806%:%
-%:%1586=806%:%
-%:%1587=807%:%
-%:%1588=807%:%
-%:%1589=807%:%
-%:%1590=808%:%
-%:%1591=809%:%
-%:%1592=809%:%
-%:%1593=810%:%
-%:%1594=811%:%
-%:%1595=812%:%
-%:%1596=813%:%
-%:%1597=814%:%
-%:%1598=814%:%
-%:%1599=815%:%
-%:%1600=815%:%
-%:%1601=815%:%
-%:%1602=816%:%
-%:%1603=817%:%
-%:%1604=817%:%
-%:%1605=818%:%
-%:%1606=819%:%
-%:%1607=820%:%
-%:%1608=820%:%
-%:%1609=821%:%
-%:%1610=821%:%
-%:%1611=821%:%
-%:%1612=822%:%
-%:%1613=823%:%
-%:%1614=823%:%
-%:%1615=824%:%
-%:%1616=825%:%
-%:%1617=825%:%
-%:%1618=826%:%
-%:%1619=826%:%
-%:%1620=826%:%
-%:%1621=827%:%
-%:%1622=828%:%
-%:%1623=828%:%
-%:%1624=829%:%
-%:%1625=830%:%
-%:%1626=831%:%
-%:%1627=832%:%
-%:%1628=833%:%
-%:%1629=834%:%
-%:%1630=835%:%
-%:%1631=835%:%
-%:%1632=836%:%
-%:%1633=836%:%
-%:%1634=836%:%
-%:%1635=837%:%
-%:%1636=838%:%
-%:%1637=838%:%
-%:%1638=839%:%
-%:%1639=839%:%
-%:%1640=839%:%
-%:%1641=840%:%
-%:%1642=841%:%
-%:%1643=841%:%
-%:%1644=842%:%
-%:%1645=843%:%
-%:%1646=844%:%
-%:%1647=845%:%
-%:%1648=845%:%
-%:%1649=846%:%
-%:%1650=846%:%
-%:%1651=847%:%
-%:%1652=847%:%
-%:%1653=848%:%
-%:%1654=849%:%
-%:%1655=850%:%
-%:%1656=851%:%
-%:%1657=851%:%
-%:%1658=852%:%
-%:%1659=852%:%
-%:%1660=853%:%
-%:%1661=853%:%
-%:%1662=854%:%
-%:%1663=855%:%
-%:%1664=855%:%
-%:%1665=856%:%
-%:%1666=856%:%
-%:%1667=857%:%
-%:%1668=857%:%
-%:%1669=858%:%
-%:%1670=858%:%
-%:%1671=858%:%
-%:%1672=859%:%
-%:%1673=859%:%
-%:%1674=860%:%
-%:%1675=860%:%
-%:%1676=861%:%
-%:%1677=861%:%
-%:%1678=861%:%
-%:%1679=862%:%
-%:%1680=863%:%
-%:%1681=863%:%
-%:%1682=864%:%
-%:%1683=864%:%
-%:%1684=865%:%
-%:%1685=865%:%
-%:%1686=866%:%
-%:%1687=866%:%
-%:%1688=867%:%
-%:%1689=867%:%
-%:%1690=868%:%
-%:%1694=872%:%
-%:%1695=873%:%
-%:%1696=873%:%
-%:%1697=874%:%
-%:%1698=875%:%
-%:%1699=876%:%
-%:%1700=876%:%
-%:%1701=877%:%
-%:%1702=877%:%
-%:%1703=878%:%
-%:%1704=878%:%
-%:%1705=879%:%
-%:%1706=879%:%
-%:%1707=880%:%
-%:%1708=881%:%
-%:%1709=882%:%
-%:%1710=882%:%
-%:%1711=883%:%
-%:%1712=884%:%
-%:%1713=885%:%
-%:%1714=885%:%
-%:%1715=886%:%
-%:%1716=886%:%
-%:%1717=887%:%
-%:%1718=887%:%
-%:%1719=888%:%
-%:%1720=889%:%
-%:%1721=890%:%
-%:%1722=890%:%
-%:%1723=891%:%
-%:%1724=891%:%
-%:%1725=891%:%
-%:%1726=892%:%
-%:%1727=892%:%
-%:%1728=893%:%
-%:%1729=894%:%
-%:%1730=894%:%
-%:%1731=895%:%
-%:%1732=895%:%
-%:%1733=896%:%
-%:%1734=896%:%
-%:%1735=897%:%
-%:%1736=897%:%
-%:%1737=898%:%
-%:%1738=898%:%
-%:%1739=899%:%
-%:%1740=899%:%
-%:%1741=900%:%
-%:%1742=900%:%
-%:%1743=901%:%
-%:%1744=901%:%
-%:%1745=902%:%
-%:%1746=903%:%
-%:%1747=904%:%
-%:%1748=904%:%
-%:%1749=905%:%
-%:%1750=906%:%
-%:%1751=907%:%
-%:%1752=907%:%
-%:%1753=908%:%
-%:%1754=909%:%
-%:%1755=910%:%
-%:%1756=911%:%
-%:%1757=911%:%
-%:%1758=912%:%
-%:%1759=912%:%
-%:%1760=913%:%
-%:%1761=914%:%
-%:%1762=915%:%
-%:%1763=915%:%
-%:%1764=916%:%
-%:%1765=916%:%
-%:%1766=917%:%
-%:%1767=917%:%
-%:%1768=918%:%
-%:%1769=919%:%
-%:%1770=920%:%
-%:%1771=921%:%
-%:%1772=922%:%
-%:%1773=922%:%
-%:%1774=923%:%
-%:%1775=923%:%
-%:%1776=924%:%
-%:%1777=925%:%
-%:%1778=926%:%
-%:%1779=926%:%
-%:%1780=927%:%
-%:%1781=928%:%
-%:%1782=929%:%
-%:%1783=929%:%
-%:%1784=930%:%
-%:%1785=930%:%
-%:%1786=931%:%
-%:%1787=932%:%
-%:%1788=932%:%
-%:%1789=933%:%
-%:%1790=934%:%
-%:%1791=934%:%
-%:%1792=935%:%
-%:%1793=935%:%
-%:%1794=936%:%
-%:%1795=937%:%
-%:%1796=938%:%
-%:%1797=938%:%
-%:%1798=939%:%
-%:%1799=940%:%
-%:%1800=941%:%
-%:%1801=942%:%
-%:%1802=943%:%
-%:%1803=943%:%
-%:%1804=944%:%
-%:%1805=944%:%
-%:%1806=945%:%
-%:%1807=945%:%
-%:%1808=946%:%
-%:%1809=946%:%
-%:%1810=947%:%
-%:%1811=947%:%
-%:%1812=948%:%
-%:%1813=949%:%
-%:%1814=950%:%
-%:%1815=950%:%
-%:%1816=951%:%
-%:%1817=952%:%
-%:%1818=952%:%
-%:%1819=953%:%
-%:%1820=953%:%
-%:%1821=954%:%
-%:%1823=956%:%
-%:%1824=957%:%
-%:%1825=957%:%
-%:%1826=958%:%
-%:%1827=959%:%
-%:%1828=959%:%
-%:%1829=960%:%
-%:%1830=960%:%
-%:%1831=961%:%
-%:%1834=964%:%
-%:%1835=965%:%
-%:%1836=965%:%
-%:%1837=966%:%
-%:%1838=966%:%
-%:%1839=967%:%
-%:%1840=967%:%
-%:%1841=968%:%
-%:%1842=968%:%
-%:%1843=969%:%
-%:%1844=970%:%
-%:%1845=971%:%
-%:%1846=972%:%
-%:%1847=973%:%
-%:%1848=973%:%
-%:%1849=974%:%
-%:%1850=974%:%
-%:%1851=975%:%
-%:%1852=975%:%
-%:%1853=976%:%
-%:%1859=976%:%
-%:%1864=977%:%
-%:%1869=978%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Sequential_Majority_Comparison.tex b/virage/src/test/resources/old_theories/output/document/Sequential_Majority_Comparison.tex
deleted file mode 100644
index 61e9bc3..0000000
--- a/virage/src/test/resources/old_theories/output/document/Sequential_Majority_Comparison.tex
+++ /dev/null
@@ -1,1331 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Sequential{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Comparison}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Sequential Majority Comparison%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Sequential{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Comparison\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Plurality{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Drop{\isacharunderscore}{\kern0pt}And{\isacharunderscore}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Compatibility{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Revision{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Maximum{\isacharunderscore}{\kern0pt}Parallel{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-Sequential majority comparison compares two alternatives by plurality voting.
-The loser gets rejected, and the winner is compared to the next alternative.
-This process is repeated until only a single alternative is left, which is
-then elected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ smc\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}smc\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isasymup}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ elect{\isacharunderscore}{\kern0pt}module{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ smc{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}smc\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ elect{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ elect{\isacharunderscore}{\kern0pt}xa\ emptyE\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ reject{\isacharunderscore}{\kern0pt}xa\isanewline
-\ \ \ \ \ \ \ \ \ \ plurality{\isacharunderscore}{\kern0pt}sound\ result{\isacharunderscore}{\kern0pt}disj\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ defer{\isacharunderscore}{\kern0pt}xa\ emptyE\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ reject{\isacharunderscore}{\kern0pt}xa\isanewline
-\ \ \ \ \ \ \ \ \ \ plurality{\isacharunderscore}{\kern0pt}sound\ result{\isacharunderscore}{\kern0pt}disj\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ elect{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}xa\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}mono\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ plurality{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ defer{\isacharunderscore}{\kern0pt}xa\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}mono\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ plurality{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}plurality{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ set\ list\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ par{\isadigit{1}}{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ par{\isadigit{2}}{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ reject{\isacharunderscore}{\kern0pt}xa\ fin{\isacharunderscore}{\kern0pt}A\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ par{\isadigit{1}}{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isadigit{2}}{\isacharunderscore}{\kern0pt}sound\ plurality{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymnotin}\ defer\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymnotin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ order\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ plurality{\isacharunderscore}{\kern0pt}sound\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ fin{\isacharunderscore}{\kern0pt}A\ not{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}xa\isanewline
-\ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}xa\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ smc{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electing\ {\isacharparenleft}{\kern0pt}smc\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}eliminator\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isasymparallel}\isactrlsub {\isasymup}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}loop\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ t\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isadigit{1}}\ in\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}eliminator\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ pass{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\ smc{\isacharunderscore}{\kern0pt}sound\ smc{\isachardot}{\kern0pt}simps\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardot}{\kern0pt}iter{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ order\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{(Weak) Monotonicity Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ smc{\isacharunderscore}{\kern0pt}monotone{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}monotonicity\ {\isacharparenleft}{\kern0pt}smc\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}eliminator\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isasymparallel}\isactrlsub {\isasymup}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}loop\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ t\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isadigit{1}}\ in\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}eliminator\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ pass{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\isanewline
-\ \ \ \ \ \ \ \ \ \ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardot}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ Defer{\isacharunderscore}{\kern0pt}One{\isacharunderscore}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardot}{\kern0pt}iter{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ smc{\isacharunderscore}{\kern0pt}sound\ smc{\isachardot}{\kern0pt}simps\ order\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Sequential_Majority_Comparison.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%35=14%:%
-%:%36=15%:%
-%:%37=16%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%48=21%:%
-%:%49=22%:%
-%:%58=24%:%
-%:%68=26%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%79=31%:%
-%:%89=38%:%
-%:%90=38%:%
-%:%91=39%:%
-%:%92=40%:%
-%:%95=41%:%
-%:%99=41%:%
-%:%100=41%:%
-%:%101=42%:%
-%:%102=42%:%
-%:%103=43%:%
-%:%104=43%:%
-%:%105=44%:%
-%:%106=45%:%
-%:%107=46%:%
-%:%108=47%:%
-%:%109=47%:%
-%:%110=48%:%
-%:%111=48%:%
-%:%112=49%:%
-%:%113=49%:%
-%:%114=50%:%
-%:%116=52%:%
-%:%117=53%:%
-%:%118=53%:%
-%:%119=54%:%
-%:%120=55%:%
-%:%121=56%:%
-%:%122=57%:%
-%:%123=58%:%
-%:%124=59%:%
-%:%125=60%:%
-%:%126=60%:%
-%:%127=61%:%
-%:%128=61%:%
-%:%129=62%:%
-%:%130=63%:%
-%:%131=64%:%
-%:%132=65%:%
-%:%133=66%:%
-%:%134=66%:%
-%:%135=67%:%
-%:%136=67%:%
-%:%137=68%:%
-%:%138=68%:%
-%:%139=69%:%
-%:%140=70%:%
-%:%141=71%:%
-%:%142=72%:%
-%:%143=72%:%
-%:%144=73%:%
-%:%145=73%:%
-%:%146=74%:%
-%:%147=74%:%
-%:%148=75%:%
-%:%150=77%:%
-%:%151=78%:%
-%:%152=78%:%
-%:%153=79%:%
-%:%154=80%:%
-%:%155=81%:%
-%:%156=82%:%
-%:%157=83%:%
-%:%158=84%:%
-%:%159=85%:%
-%:%160=85%:%
-%:%161=86%:%
-%:%162=86%:%
-%:%163=87%:%
-%:%164=88%:%
-%:%165=89%:%
-%:%166=90%:%
-%:%167=91%:%
-%:%168=91%:%
-%:%169=92%:%
-%:%170=92%:%
-%:%171=93%:%
-%:%172=93%:%
-%:%173=94%:%
-%:%174=95%:%
-%:%175=96%:%
-%:%176=97%:%
-%:%177=97%:%
-%:%178=98%:%
-%:%179=98%:%
-%:%180=99%:%
-%:%181=99%:%
-%:%182=100%:%
-%:%184=102%:%
-%:%185=103%:%
-%:%186=103%:%
-%:%187=104%:%
-%:%188=105%:%
-%:%189=106%:%
-%:%190=107%:%
-%:%191=108%:%
-%:%192=108%:%
-%:%193=109%:%
-%:%194=109%:%
-%:%195=110%:%
-%:%196=111%:%
-%:%197=112%:%
-%:%198=113%:%
-%:%199=113%:%
-%:%200=114%:%
-%:%201=114%:%
-%:%202=115%:%
-%:%203=115%:%
-%:%204=116%:%
-%:%205=117%:%
-%:%206=118%:%
-%:%207=119%:%
-%:%208=119%:%
-%:%209=120%:%
-%:%210=120%:%
-%:%211=121%:%
-%:%212=121%:%
-%:%213=122%:%
-%:%215=124%:%
-%:%216=125%:%
-%:%217=125%:%
-%:%218=126%:%
-%:%219=127%:%
-%:%220=128%:%
-%:%221=129%:%
-%:%222=130%:%
-%:%223=130%:%
-%:%224=131%:%
-%:%225=131%:%
-%:%226=132%:%
-%:%227=133%:%
-%:%228=134%:%
-%:%229=135%:%
-%:%230=135%:%
-%:%231=136%:%
-%:%232=136%:%
-%:%233=137%:%
-%:%234=137%:%
-%:%235=138%:%
-%:%236=139%:%
-%:%237=140%:%
-%:%238=141%:%
-%:%239=141%:%
-%:%240=142%:%
-%:%241=142%:%
-%:%242=143%:%
-%:%243=143%:%
-%:%244=144%:%
-%:%246=146%:%
-%:%247=147%:%
-%:%248=147%:%
-%:%249=148%:%
-%:%250=149%:%
-%:%251=150%:%
-%:%252=151%:%
-%:%253=152%:%
-%:%254=152%:%
-%:%255=153%:%
-%:%256=154%:%
-%:%257=155%:%
-%:%258=155%:%
-%:%259=156%:%
-%:%260=156%:%
-%:%261=157%:%
-%:%262=158%:%
-%:%263=158%:%
-%:%264=159%:%
-%:%265=159%:%
-%:%266=160%:%
-%:%267=160%:%
-%:%268=160%:%
-%:%269=161%:%
-%:%270=162%:%
-%:%271=162%:%
-%:%272=163%:%
-%:%273=163%:%
-%:%274=164%:%
-%:%275=164%:%
-%:%276=165%:%
-%:%277=165%:%
-%:%278=166%:%
-%:%279=167%:%
-%:%280=168%:%
-%:%281=169%:%
-%:%282=170%:%
-%:%283=170%:%
-%:%284=171%:%
-%:%285=171%:%
-%:%286=172%:%
-%:%287=172%:%
-%:%288=173%:%
-%:%289=174%:%
-%:%290=175%:%
-%:%291=176%:%
-%:%292=176%:%
-%:%293=177%:%
-%:%294=177%:%
-%:%295=178%:%
-%:%296=178%:%
-%:%297=179%:%
-%:%299=181%:%
-%:%300=182%:%
-%:%301=182%:%
-%:%302=183%:%
-%:%303=184%:%
-%:%304=185%:%
-%:%305=186%:%
-%:%306=187%:%
-%:%307=188%:%
-%:%308=189%:%
-%:%309=190%:%
-%:%310=190%:%
-%:%311=191%:%
-%:%312=191%:%
-%:%313=192%:%
-%:%314=193%:%
-%:%315=194%:%
-%:%316=195%:%
-%:%317=196%:%
-%:%318=196%:%
-%:%319=197%:%
-%:%334=200%:%
-%:%344=208%:%
-%:%345=208%:%
-%:%346=209%:%
-%:%347=210%:%
-%:%354=211%:%
-%:%355=211%:%
-%:%356=212%:%
-%:%357=212%:%
-%:%358=213%:%
-%:%359=213%:%
-%:%360=214%:%
-%:%361=214%:%
-%:%362=215%:%
-%:%363=215%:%
-%:%364=216%:%
-%:%365=216%:%
-%:%366=217%:%
-%:%367=217%:%
-%:%368=218%:%
-%:%369=218%:%
-%:%370=219%:%
-%:%371=220%:%
-%:%372=221%:%
-%:%373=221%:%
-%:%374=222%:%
-%:%375=222%:%
-%:%376=223%:%
-%:%377=223%:%
-%:%378=224%:%
-%:%379=224%:%
-%:%380=225%:%
-%:%381=225%:%
-%:%382=226%:%
-%:%383=226%:%
-%:%384=227%:%
-%:%385=227%:%
-%:%386=228%:%
-%:%387=228%:%
-%:%388=229%:%
-%:%389=229%:%
-%:%390=230%:%
-%:%391=230%:%
-%:%392=231%:%
-%:%393=232%:%
-%:%394=232%:%
-%:%395=233%:%
-%:%396=233%:%
-%:%397=234%:%
-%:%398=234%:%
-%:%399=235%:%
-%:%400=235%:%
-%:%401=236%:%
-%:%402=236%:%
-%:%403=237%:%
-%:%404=237%:%
-%:%405=238%:%
-%:%406=238%:%
-%:%407=239%:%
-%:%408=239%:%
-%:%409=240%:%
-%:%410=240%:%
-%:%411=241%:%
-%:%412=241%:%
-%:%413=242%:%
-%:%414=242%:%
-%:%415=243%:%
-%:%416=243%:%
-%:%417=244%:%
-%:%418=244%:%
-%:%419=245%:%
-%:%420=245%:%
-%:%421=246%:%
-%:%422=246%:%
-%:%423=247%:%
-%:%424=248%:%
-%:%425=248%:%
-%:%426=249%:%
-%:%427=249%:%
-%:%428=250%:%
-%:%429=250%:%
-%:%430=251%:%
-%:%431=251%:%
-%:%432=252%:%
-%:%433=252%:%
-%:%434=253%:%
-%:%435=253%:%
-%:%436=254%:%
-%:%437=254%:%
-%:%438=255%:%
-%:%439=255%:%
-%:%440=256%:%
-%:%441=256%:%
-%:%442=257%:%
-%:%443=257%:%
-%:%444=258%:%
-%:%445=258%:%
-%:%446=259%:%
-%:%447=259%:%
-%:%448=260%:%
-%:%449=260%:%
-%:%450=261%:%
-%:%451=261%:%
-%:%452=262%:%
-%:%453=262%:%
-%:%454=263%:%
-%:%455=263%:%
-%:%456=264%:%
-%:%457=264%:%
-%:%458=265%:%
-%:%459=266%:%
-%:%460=266%:%
-%:%461=267%:%
-%:%462=267%:%
-%:%463=268%:%
-%:%464=268%:%
-%:%465=269%:%
-%:%466=269%:%
-%:%467=270%:%
-%:%468=270%:%
-%:%469=271%:%
-%:%470=271%:%
-%:%471=272%:%
-%:%472=273%:%
-%:%473=273%:%
-%:%474=274%:%
-%:%475=274%:%
-%:%476=275%:%
-%:%477=275%:%
-%:%478=276%:%
-%:%479=276%:%
-%:%480=277%:%
-%:%481=277%:%
-%:%482=278%:%
-%:%483=279%:%
-%:%484=279%:%
-%:%485=280%:%
-%:%486=280%:%
-%:%487=281%:%
-%:%488=282%:%
-%:%489=283%:%
-%:%490=283%:%
-%:%491=284%:%
-%:%506=286%:%
-%:%516=292%:%
-%:%517=292%:%
-%:%518=293%:%
-%:%519=294%:%
-%:%526=295%:%
-%:%527=295%:%
-%:%528=296%:%
-%:%529=297%:%
-%:%530=297%:%
-%:%531=298%:%
-%:%532=298%:%
-%:%533=299%:%
-%:%534=299%:%
-%:%535=300%:%
-%:%536=300%:%
-%:%537=301%:%
-%:%538=301%:%
-%:%539=302%:%
-%:%540=302%:%
-%:%541=303%:%
-%:%542=303%:%
-%:%543=304%:%
-%:%544=305%:%
-%:%545=306%:%
-%:%546=306%:%
-%:%547=307%:%
-%:%548=307%:%
-%:%549=308%:%
-%:%550=308%:%
-%:%551=309%:%
-%:%552=309%:%
-%:%553=310%:%
-%:%554=310%:%
-%:%555=311%:%
-%:%556=311%:%
-%:%557=312%:%
-%:%558=312%:%
-%:%559=313%:%
-%:%560=313%:%
-%:%561=314%:%
-%:%562=314%:%
-%:%563=315%:%
-%:%564=315%:%
-%:%565=316%:%
-%:%566=316%:%
-%:%567=317%:%
-%:%568=317%:%
-%:%569=318%:%
-%:%570=318%:%
-%:%571=319%:%
-%:%572=319%:%
-%:%573=320%:%
-%:%574=320%:%
-%:%575=321%:%
-%:%576=322%:%
-%:%577=322%:%
-%:%578=323%:%
-%:%579=323%:%
-%:%580=324%:%
-%:%581=324%:%
-%:%582=325%:%
-%:%583=325%:%
-%:%584=326%:%
-%:%585=326%:%
-%:%586=327%:%
-%:%587=327%:%
-%:%588=328%:%
-%:%589=328%:%
-%:%590=329%:%
-%:%591=329%:%
-%:%592=330%:%
-%:%593=330%:%
-%:%594=331%:%
-%:%595=331%:%
-%:%596=332%:%
-%:%597=332%:%
-%:%598=333%:%
-%:%599=333%:%
-%:%600=334%:%
-%:%601=334%:%
-%:%602=335%:%
-%:%603=335%:%
-%:%604=336%:%
-%:%605=336%:%
-%:%606=337%:%
-%:%607=337%:%
-%:%608=338%:%
-%:%609=338%:%
-%:%610=339%:%
-%:%611=339%:%
-%:%612=340%:%
-%:%613=340%:%
-%:%614=341%:%
-%:%615=341%:%
-%:%616=342%:%
-%:%617=342%:%
-%:%618=343%:%
-%:%619=344%:%
-%:%620=344%:%
-%:%621=345%:%
-%:%622=345%:%
-%:%623=346%:%
-%:%624=346%:%
-%:%625=347%:%
-%:%626=347%:%
-%:%627=348%:%
-%:%628=348%:%
-%:%629=349%:%
-%:%630=349%:%
-%:%631=350%:%
-%:%632=350%:%
-%:%633=351%:%
-%:%634=351%:%
-%:%635=352%:%
-%:%636=352%:%
-%:%637=353%:%
-%:%637=354%:%
-%:%638=355%:%
-%:%639=355%:%
-%:%640=356%:%
-%:%641=356%:%
-%:%642=357%:%
-%:%643=357%:%
-%:%644=358%:%
-%:%645=358%:%
-%:%646=359%:%
-%:%647=359%:%
-%:%648=360%:%
-%:%649=360%:%
-%:%650=361%:%
-%:%651=361%:%
-%:%652=362%:%
-%:%653=362%:%
-%:%654=363%:%
-%:%655=363%:%
-%:%656=364%:%
-%:%657=364%:%
-%:%658=365%:%
-%:%659=365%:%
-%:%660=366%:%
-%:%661=366%:%
-%:%662=367%:%
-%:%663=367%:%
-%:%664=368%:%
-%:%665=368%:%
-%:%666=369%:%
-%:%667=370%:%
-%:%668=370%:%
-%:%669=371%:%
-%:%670=371%:%
-%:%671=372%:%
-%:%672=372%:%
-%:%673=373%:%
-%:%674=373%:%
-%:%675=374%:%
-%:%676=374%:%
-%:%677=375%:%
-%:%678=375%:%
-%:%679=376%:%
-%:%680=376%:%
-%:%681=377%:%
-%:%682=377%:%
-%:%683=378%:%
-%:%684=378%:%
-%:%685=379%:%
-%:%686=380%:%
-%:%687=380%:%
-%:%688=381%:%
-%:%689=381%:%
-%:%690=382%:%
-%:%691=382%:%
-%:%692=383%:%
-%:%693=383%:%
-%:%694=384%:%
-%:%695=384%:%
-%:%696=385%:%
-%:%697=385%:%
-%:%698=386%:%
-%:%699=386%:%
-%:%700=387%:%
-%:%701=387%:%
-%:%702=388%:%
-%:%703=388%:%
-%:%704=389%:%
-%:%705=389%:%
-%:%706=390%:%
-%:%707=390%:%
-%:%708=391%:%
-%:%709=392%:%
-%:%710=392%:%
-%:%711=393%:%
-%:%712=393%:%
-%:%713=394%:%
-%:%714=395%:%
-%:%715=396%:%
-%:%716=397%:%
-%:%717=397%:%
-%:%718=398%:%
-%:%724=398%:%
-%:%729=399%:%
-%:%734=400%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/Termination_Condition.tex b/virage/src/test/resources/old_theories/output/document/Termination_Condition.tex
deleted file mode 100644
index 7ff53ae..0000000
--- a/virage/src/test/resources/old_theories/output/document/Termination_Condition.tex
+++ /dev/null
@@ -1,99 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Termination{\isacharunderscore}{\kern0pt}Condition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Termination Condition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Termination{\isacharunderscore}{\kern0pt}Condition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Result{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The termination condition is used in loops. It decides whether or not to
-terminate the loop after each iteration, depending on the current state
-of the loop.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/old_theories/Compositional_Structures/Basic_Modules/Component_Types/Termination_Condition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%53=19%:%
-%:%63=21%:%
-%:%64=21%:%
-%:%67=22%:%
-%:%72=23%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/old_theories/output/document/comment.sty b/virage/src/test/resources/old_theories/output/document/comment.sty
deleted file mode 100644
index 658686f..0000000
--- a/virage/src/test/resources/old_theories/output/document/comment.sty
+++ /dev/null
@@ -1,278 +0,0 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Comment.sty   version 3.6, October 1999
-%
-% Purpose:
-% selectively in/exclude pieces of text: the user can define new
-% comment versions, and each is controlled separately.
-% Special comments can be defined where the user specifies the
-% action that is to be taken with each comment line.
-%
-% Author
-%    Victor Eijkhout
-%    Department of Computer Science
-%    University of Tennessee
-%    107 Ayres Hall
-%    Knoxville TN 37996
-%    USA
-%
-%    victor@eijkhout.net
-%
-% This program is free software; you can redistribute it and/or
-% modify it under the terms of the GNU General Public License
-% as published by the Free Software Foundation; either version 2
-% of the License, or (at your option) any later version.
-% 
-% This program is distributed in the hope that it will be useful,
-% but WITHOUT ANY WARRANTY; without even the implied warranty of
-% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
-% GNU General Public License for more details.
-%
-% For a copy of the GNU General Public License, write to the 
-% Free Software Foundation, Inc.,
-% 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA,
-% or find it on the net, for instance at
-% http://www.gnu.org/copyleft/gpl.html
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% This style can be used with plain TeX or LaTeX, and probably
-% most other packages too.
-%
-% Usage: all text included between
-%    \comment ... \endcomment
-% or \begin{comment} ... \end{comment}
-% is discarded. 
-%
-% The opening and closing commands should appear on a line
-% of their own. No starting spaces, nothing after it.
-% This environment should work with arbitrary amounts
-% of comment, and the comment can be arbitrary text.
-%
-% Other `comment' environments are defined by
-% and are selected/deselected with
-% \includecomment{versiona}
-% \excludecoment{versionb}
-%
-% These environments are used as
-% \versiona ... \endversiona
-% or \begin{versiona} ... \end{versiona}
-% with the opening and closing commands again on a line of 
-% their own.
-%
-% LaTeX users note: for an included comment, the
-% \begin and \end lines act as if they don't exist.
-% In particular, they don't imply grouping, so assignments 
-% &c are not local.
-%
-% Special comments are defined as
-% \specialcomment{name}{before commands}{after commands}
-% where the second and third arguments are executed before
-% and after each comment block. You can use this for global
-% formatting commands.
-% To keep definitions &c local, you can include \begingroup
-% in the `before commands' and \endgroup in the `after commands'.
-% ex:
-% \specialcomment{smalltt}
-%     {\begingroup\ttfamily\footnotesize}{\endgroup}
-% You do *not* have to do an additional
-% \includecomment{smalltt}
-% To remove 'smalltt' blocks, give \excludecomment{smalltt}
-% after the definition.
-%
-% Processing comments can apply processing to each line.
-% \processcomment{name}{each-line commands}%
-%    {before commands}{after commands}
-% By defining a control sequence 
-% \def\Thiscomment##1{...} in the before commands the user can
-% specify what is to be done with each comment line.
-% BUG this does not work quite yet BUG
-%
-% Trick for short in/exclude macros (such as \maybe{this snippet}):
-%\includecomment{cond}
-%\newcommand{\maybe}[1]{}
-%\begin{cond}
-%\renewcommand{\maybe}[1]{#1}
-%\end{cond}
-%
-% Basic approach of the implementation:
-% to comment something out, scoop up  every line in verbatim mode
-% as macro argument, then throw it away.
-% For inclusions, in LaTeX the block is written out to
-% a file \CommentCutFile (default "comment.cut"), which is
-% then included.
-% In plain TeX (and other formats) both the opening and
-% closing comands are defined as noop.
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Changes in version 3.1
-% - updated author's address
-% - cleaned up some code
-% - trailing contents on \begin{env} line is always discarded
-%  even if you've done \includecomment{env}
-% - comments no longer define grouping!! you can even
-%   \includecomment{env}
-%   \begin{env}
-%   \begin{itemize}
-%   \end{env}
-%  Isn't that something ...
-% - included comments are written to file and input again.
-% Changes in 3.2
-% - \specialcomment brought up to date (thanks to Ivo Welch).
-% Changes in 3.3
-% - updated author's address again
-% - parametrised \CommentCutFile
-% Changes in 3.4
-% - added GNU public license
-% - added \processcomment, because Ivo's fix (above) brought an
-%   inconsistency to light.
-% Changes in 3.5
-% - corrected typo in header.
-% - changed author email
-% - corrected \specialcomment yet again.
-% - fixed excludecomment of an earlier defined environment.
-% Changes in 3.6
-% - The 'cut' file is now written more verbatim, using \meaning;
-%   some people reported having trouble with ISO latin 1, or umlaute.sty.
-% - removed some \newif statements.
-%   Has this suddenly become \outer again?
-%
-% Known bugs:
-% - excludecomment leads to one superfluous space
-% - processcomment leads to a superfluous line break
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\def\makeinnocent#1{\catcode`#1=12 }
-\def\csarg#1#2{\expandafter#1\csname#2\endcsname}
-\def\latexname{lplain}\def\latexename{LaTeX2e}
-\newwrite\CommentStream
-\def\CommentCutFile{comment.cut}
-
-\def\ProcessComment#1% start it all of
-   {\begingroup
-    \def\CurrentComment{#1}%
-    \let\do\makeinnocent \dospecials 
-    \makeinnocent\^^L% and whatever other special cases
-    \endlinechar`\^^M \catcode`\^^M=12 \xComment}
-%\def\ProcessCommentWithArg#1#2% to be used in \leveledcomment
-%   {\begingroup
-%    \def\CurrentComment{#1}%
-%    \let\do\makeinnocent \dospecials 
-%    \makeinnocent\^^L% and whatever other special cases
-%    \endlinechar`\^^M \catcode`\^^M=12 \xComment}
-{\catcode`\^^M=12 \endlinechar=-1 %
- \gdef\xComment#1^^M{%
-    \expandafter\ProcessCommentLine}
- \gdef\ProcessCommentLine#1^^M{\def\test{#1}
-      \csarg\ifx{End\CurrentComment Test}\test
-          \edef\next{\noexpand\EndOfComment{\CurrentComment}}%
-      \else \ThisComment{#1}\let\next\ProcessCommentLine
-      \fi \next}
-}
-
-\def\CSstringmeaning#1{\expandafter\CSgobblearrow\meaning#1}
-\def\CSstringcsnoescape#1{\expandafter\CSgobbleescape\string#1}
-{\escapechar-1
-\expandafter\expandafter\expandafter\gdef
-  \expandafter\expandafter\expandafter\CSgobblearrow
-    \expandafter\string\csname macro:->\endcsname{}
-}
-\def\CSgobbleescape#1{\ifnum`\\=`#1 \else #1\fi}
-\def\WriteCommentLine#1{\def\CStmp{#1}%
-    \immediate\write\CommentStream{\CSstringmeaning\CStmp}}
-
-% 3.1 change: in LaTeX and LaTeX2e prevent grouping
-\if 0%
-\ifx\fmtname\latexename 
-    0%
-\else \ifx\fmtname\latexname 
-          0%
-      \else 
-          1%
-\fi   \fi
-%%%%
-%%%% definitions for LaTeX
-%%%%
-\def\AfterIncludedComment
-   {\immediate\closeout\CommentStream
-    \input{\CommentCutFile}\relax
-    }%
-\def\TossComment{\immediate\closeout\CommentStream}
-\def\BeforeIncludedComment
-   {\immediate\openout\CommentStream=\CommentCutFile
-    \let\ThisComment\WriteCommentLine}
-\def\includecomment
- #1{\message{Include comment '#1'}%
-    \csarg\let{After#1Comment}\AfterIncludedComment
-    \csarg\def{#1}{\BeforeIncludedComment
-        \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\long\def\specialcomment
- #1#2#3{\message{Special comment '#1'}%
-    % note: \AfterIncludedComment does \input, so #2 goes here!
-    \csarg\def{After#1Comment}{#2\AfterIncludedComment#3}%
-    \csarg\def{#1}{\BeforeIncludedComment\relax
-          \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\long\def\processcomment
- #1#2#3#4{\message{Lines-Processing comment '#1'}%
-    \csarg\def{After#1Comment}{#3\AfterIncludedComment#4}%
-    \csarg\def{#1}{\BeforeIncludedComment#2\relax
-          \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\def\leveledcomment
- #1#2{\message{Include comment '#1' up to level '#2'}%
-    %\csname #1IsLeveledCommenttrue\endcsname
-    \csarg\let{After#1Comment}\AfterIncludedComment
-    \csarg\def{#1}{\BeforeIncludedComment
-        \ProcessCommentWithArg{#1}}%
-    \CommentEndDef{#1}}
-\else 
-%%%%
-%%%%plain TeX and other formats
-%%%%
-\def\includecomment
- #1{\message{Including comment '#1'}%
-    \csarg\def{#1}{}%
-    \csarg\def{end#1}{}}
-\long\def\specialcomment
- #1#2#3{\message{Special comment '#1'}%
-    \csarg\def{#1}{\def\ThisComment{}\def\AfterComment{#3}#2%
-           \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\fi
-
-%%%%
-%%%% general definition of skipped comment
-%%%%
-\def\excludecomment
- #1{\message{Excluding comment '#1'}%
-    \csarg\def{#1}{\let\AfterComment\relax
-           \def\ThisComment####1{}\ProcessComment{#1}}%
-    \csarg\let{After#1Comment}\TossComment
-    \CommentEndDef{#1}}
-
-\if 0%
-\ifx\fmtname\latexename 
-    0%
-\else \ifx\fmtname\latexname 
-          0%
-      \else 
-          1%
-\fi   \fi
-% latex & latex2e:
-\def\EndOfComment#1{\endgroup\end{#1}%
-    \csname After#1Comment\endcsname}
-\def\CommentEndDef#1{{\escapechar=-1\relax
-    \csarg\xdef{End#1Test}{\string\\end\string\{#1\string\}}%
-    }}
-\else
-% plain & other
-\def\EndOfComment#1{\endgroup\AfterComment}
-\def\CommentEndDef#1{{\escapechar=-1\relax
-    \csarg\xdef{End#1Test}{\string\\end#1}%
-    }}
-\fi
-
-\excludecomment{comment}
-
-\endinput
diff --git a/virage/src/test/resources/old_theories/output/document/isabelle.sty b/virage/src/test/resources/old_theories/output/document/isabelle.sty
deleted file mode 100644
index b6dcb3e..0000000
--- a/virage/src/test/resources/old_theories/output/document/isabelle.sty
+++ /dev/null
@@ -1,267 +0,0 @@
-%%
-%% macros for Isabelle generated LaTeX output
-%%
-
-%%% Simple document preparation (based on theory token language and symbols)
-
-% isabelle environments
-
-\newcommand{\isabellecontext}{UNKNOWN}
-\newcommand{\setisabellecontext}[1]{\def\isabellecontext{#1}}
-
-\newcommand{\isastyle}{\UNDEF}
-\newcommand{\isastylett}{\UNDEF}
-\newcommand{\isastyleminor}{\UNDEF}
-\newcommand{\isastyleminortt}{\UNDEF}
-\newcommand{\isastylescript}{\UNDEF}
-\newcommand{\isastyletext}{\normalsize\normalfont\rmfamily}
-\newcommand{\isastyletxt}{\normalfont\rmfamily}
-\newcommand{\isastylecmt}{\normalfont\rmfamily}
-
-\newcommand{\isaspacing}{%
-  \sfcode 42 1000 % .
-  \sfcode 63 1000 % ?
-  \sfcode 33 1000 % !
-  \sfcode 58 1000 % :
-  \sfcode 59 1000 % ;
-  \sfcode 44 1000 % ,
-}
-
-%symbol markup -- \emph achieves decent spacing via italic corrections
-\newcommand{\isamath}[1]{\emph{$#1$}}
-\newcommand{\isatext}[1]{\emph{#1}}
-\DeclareRobustCommand{\isascriptstyle}{\def\isamath##1{##1}\def\isatext##1{\mbox{\isaspacing\isastylescript##1}}}
-\newcommand{\isactrlsub}[1]{\emph{\isascriptstyle${}\sb{#1}$}}
-\newcommand{\isactrlsup}[1]{\emph{\isascriptstyle${}\sp{#1}$}}
-\DeclareRobustCommand{\isactrlbsub}{\emph\bgroup\math{}\sb\bgroup\mbox\bgroup\isaspacing\isastylescript}
-\DeclareRobustCommand{\isactrlesub}{\egroup\egroup\endmath\egroup}
-\DeclareRobustCommand{\isactrlbsup}{\emph\bgroup\math{}\sp\bgroup\mbox\bgroup\isaspacing\isastylescript}
-\DeclareRobustCommand{\isactrlesup}{\egroup\egroup\endmath\egroup}
-\newcommand{\isactrlbold}[1]{{\bfseries\upshape\boldmath#1}}
-
-\newenvironment{isaantiq}{{\isacharat\isacharbraceleft}}{{\isacharbraceright}}
-
-\newdimen\isa@parindent\newdimen\isa@parskip
-
-\newenvironment{isabellebody}{%
-\isamarkuptrue\par%
-\isa@parindent\parindent\parindent0pt%
-\isa@parskip\parskip\parskip0pt%
-\isaspacing\isastyle}{\par}
-
-\newenvironment{isabellebodytt}{%
-\isamarkuptrue\par%
-\isa@parindent\parindent\parindent0pt%
-\isa@parskip\parskip\parskip0pt%
-\isaspacing\isastylett}{\par}
-
-\newenvironment{isabelle}
-{\begin{trivlist}\begin{isabellebody}\item\relax}
-{\end{isabellebody}\end{trivlist}}
-
-\newenvironment{isabellett}
-{\begin{trivlist}\begin{isabellebodytt}\item\relax}
-{\end{isabellebodytt}\end{trivlist}}
-
-\newcommand{\isa}[1]{\emph{\isaspacing\isastyleminor #1}}
-\newcommand{\isatt}[1]{\emph{\isaspacing\isastyleminortt #1}}
-
-\newcommand{\isaindent}[1]{\hphantom{#1}}
-\newcommand{\isanewline}{\mbox{}\par\mbox{}}
-\newcommand{\isasep}{}
-\newcommand{\isadigit}[1]{#1}
-
-\newcommand{\isachardefaults}{%
-\def\isacharbell{\isamath{\bigbox}}%requires stmaryrd
-\chardef\isacharbang=`\!%
-\chardef\isachardoublequote=`\"%
-\chardef\isachardoublequoteopen=`\"%
-\chardef\isachardoublequoteclose=`\"%
-\chardef\isacharhash=`\#%
-\chardef\isachardollar=`\$%
-\chardef\isacharpercent=`\%%
-\chardef\isacharampersand=`\&%
-\chardef\isacharprime=`\'%
-\chardef\isacharparenleft=`\(%
-\chardef\isacharparenright=`\)%
-\chardef\isacharasterisk=`\*%
-\chardef\isacharplus=`\+%
-\chardef\isacharcomma=`\,%
-\chardef\isacharminus=`\-%
-\chardef\isachardot=`\.%
-\chardef\isacharslash=`\/%
-\chardef\isacharcolon=`\:%
-\chardef\isacharsemicolon=`\;%
-\chardef\isacharless=`\<%
-\chardef\isacharequal=`\=%
-\chardef\isachargreater=`\>%
-\chardef\isacharquery=`\?%
-\chardef\isacharat=`\@%
-\chardef\isacharbrackleft=`\[%
-\chardef\isacharbackslash=`\\%
-\chardef\isacharbrackright=`\]%
-\chardef\isacharcircum=`\^%
-\chardef\isacharunderscore=`\_%
-\def\isacharunderscorekeyword{\_}%
-\chardef\isacharbackquote=`\`%
-\chardef\isacharbackquoteopen=`\`%
-\chardef\isacharbackquoteclose=`\`%
-\chardef\isacharbraceleft=`\{%
-\chardef\isacharbar=`\|%
-\chardef\isacharbraceright=`\}%
-\chardef\isachartilde=`\~%
-\def\isacharverbatimopen{\isacharbraceleft\isacharasterisk}%
-\def\isacharverbatimclose{\isacharasterisk\isacharbraceright}%
-\def\isacartoucheopen{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\langle$}}}%
-\def\isacartoucheclose{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\rangle$}}}%
-}
-
-
-% keyword and section markup
-
-\newcommand{\isakeyword}[1]
-{\emph{\normalfont\bfseries\def\isachardot{.}\def\isacharunderscore{\isacharunderscorekeyword}%
-\def\isacharbraceleft{\{}\def\isacharbraceright{\}}#1}}
-\newcommand{\isacommand}[1]{\isakeyword{#1}}
-
-\newcommand{\isakeywordcontrol}[1]
-{\emph{\normalfont\bfseries\itshape\def\isacharunderscore{\isacharunderscorekeyword}#1\,}}
-
-\newcommand{\isamarkupchapter}[1]{\chapter{#1}}
-\newcommand{\isamarkupsection}[1]{\section{#1}}
-\newcommand{\isamarkupsubsection}[1]{\subsection{#1}}
-\newcommand{\isamarkupsubsubsection}[1]{\subsubsection{#1}}
-\newcommand{\isamarkupparagraph}[1]{\paragraph{#1}}
-\newcommand{\isamarkupsubparagraph}[1]{\subparagraph{#1}}
-
-\newif\ifisamarkup
-\newcommand{\isabeginpar}{\par\ifisamarkup\relax\else\medskip\fi}
-\newcommand{\isaendpar}{\par\medskip}
-\newenvironment{isapar}{\parindent\isa@parindent\parskip\isa@parskip\isabeginpar}{\isaendpar}
-\newenvironment{isamarkuptext}{\par\isastyletext\begin{isapar}}{\end{isapar}}
-\newenvironment{isamarkuptxt}{\par\isastyletxt\begin{isapar}}{\end{isapar}}
-\newcommand{\isamarkupcmt}[1]{{\isastylecmt--- #1}}
-
-
-% styles
-
-\def\isabellestyle#1{\csname isabellestyle#1\endcsname}
-
-\newcommand{\isabellestyledefault}{%
-\def\isastyle{\small\normalfont\ttfamily\slshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\small\normalfont\ttfamily\slshape}%
-\def\isastyleminortt{\small\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\ttfamily\slshape}%
-\isachardefaults%
-}
-\isabellestyledefault
-
-\newcommand{\isabellestylett}{%
-\def\isastyle{\small\normalfont\ttfamily}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\small\normalfont\ttfamily}%
-\def\isastyleminortt{\small\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\ttfamily}%
-\isachardefaults%
-}
-
-\newcommand{\isabellestyleit}{%
-\def\isastyle{\small\normalfont\itshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\normalfont\itshape}%
-\def\isastyleminortt{\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\itshape}%
-\isachardefaults%
-\def\isacharunderscorekeyword{\mbox{-}}%
-\def\isacharbang{\isamath{!}}%
-\def\isachardoublequote{}%
-\def\isachardoublequoteopen{}%
-\def\isachardoublequoteclose{}%
-\def\isacharhash{\isamath{\#}}%
-\def\isachardollar{\isamath{\$}}%
-\def\isacharpercent{\isamath{\%}}%
-\def\isacharampersand{\isamath{\&}}%
-\def\isacharprime{\isamath{\mskip2mu{'}\mskip-2mu}}%
-\def\isacharparenleft{\isamath{(}}%
-\def\isacharparenright{\isamath{)}}%
-\def\isacharasterisk{\isamath{*}}%
-\def\isacharplus{\isamath{+}}%
-\def\isacharcomma{\isamath{\mathord,}}%
-\def\isacharminus{\isamath{-}}%
-\def\isachardot{\isamath{\mathord.}}%
-\def\isacharslash{\isamath{/}}%
-\def\isacharcolon{\isamath{\mathord:}}%
-\def\isacharsemicolon{\isamath{\mathord;}}%
-\def\isacharless{\isamath{<}}%
-\def\isacharequal{\isamath{=}}%
-\def\isachargreater{\isamath{>}}%
-\def\isacharat{\isamath{@}}%
-\def\isacharbrackleft{\isamath{[}}%
-\def\isacharbackslash{\isamath{\backslash}}%
-\def\isacharbrackright{\isamath{]}}%
-\def\isacharunderscore{\mbox{-}}%
-\def\isacharbraceleft{\isamath{\{}}%
-\def\isacharbar{\isamath{\mid}}%
-\def\isacharbraceright{\isamath{\}}}%
-\def\isachartilde{\isamath{{}\sp{\sim}}}%
-\def\isacharbackquoteopen{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\langle$}}}%
-\def\isacharbackquoteclose{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\rangle$}}}%
-}
-
-\newcommand{\isabellestyleliteral}{%
-\isabellestyleit%
-\def\isacharunderscore{\_}%
-\def\isacharunderscorekeyword{\_}%
-\chardef\isacharbackquoteopen=`\`%
-\chardef\isacharbackquoteclose=`\`%
-}
-
-\newcommand{\isabellestyleliteralunderscore}{%
-\isabellestyleliteral%
-\def\isacharunderscore{\textunderscore}%
-\def\isacharunderscorekeyword{\textunderscore}%
-}
-
-\newcommand{\isabellestylesl}{%
-\isabellestyleit%
-\def\isastyle{\small\normalfont\slshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\normalfont\slshape}%
-\def\isastyleminortt{\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\slshape}%
-}
-
-
-% cancel text
-
-\usepackage[normalem]{ulem}
-\newcommand{\isamarkupcancel}[1]{\isa{\xout{#1}}}
-
-
-% tagged regions
-
-%plain TeX version of comment package -- much faster!
-\let\isafmtname\fmtname\def\fmtname{plain}
-\usepackage{comment}
-\let\fmtname\isafmtname
-
-\newcommand{\isafold}[1]{\emph{$\langle\mathord{\mathit{#1}}\rangle$}}
-
-\newcommand{\isakeeptag}[1]%
-{\includecomment{isadelim#1}\includecomment{isatag#1}\csarg\def{isafold#1}{}}
-\newcommand{\isadroptag}[1]%
-{\excludecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{}}
-\newcommand{\isafoldtag}[1]%
-{\includecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{\isafold{#1}}}
-
-\isakeeptag{document}
-\isakeeptag{theory}
-\isakeeptag{proof}
-\isakeeptag{ML}
-\isakeeptag{visible}
-\isadroptag{invisible}
-\isakeeptag{important}
-\isakeeptag{unimportant}
-
-\IfFileExists{isabelletags.sty}{\usepackage{isabelletags}}{}
diff --git a/virage/src/test/resources/old_theories/output/document/isabellesym.sty b/virage/src/test/resources/old_theories/output/document/isabellesym.sty
deleted file mode 100644
index b0d076c..0000000
--- a/virage/src/test/resources/old_theories/output/document/isabellesym.sty
+++ /dev/null
@@ -1,427 +0,0 @@
-%%
-%% definitions of standard Isabelle symbols
-%%
-
-\newcommand{\isasymzero}{\isamath{\mathbf{0}}}  %requires amssymb
-\newcommand{\isasymone}{\isamath{\mathbf{1}}}  %requires amssymb
-\newcommand{\isasymtwo}{\isamath{\mathbf{2}}}  %requires amssymb
-\newcommand{\isasymthree}{\isamath{\mathbf{3}}}  %requires amssymb
-\newcommand{\isasymfour}{\isamath{\mathbf{4}}}  %requires amssymb
-\newcommand{\isasymfive}{\isamath{\mathbf{5}}}  %requires amssymb
-\newcommand{\isasymsix}{\isamath{\mathbf{6}}}  %requires amssymb
-\newcommand{\isasymseven}{\isamath{\mathbf{7}}}  %requires amssymb
-\newcommand{\isasymeight}{\isamath{\mathbf{8}}}  %requires amssymb
-\newcommand{\isasymnine}{\isamath{\mathbf{9}}}  %requires amssymb
-\newcommand{\isasymA}{\isamath{\mathcal{A}}}
-\newcommand{\isasymB}{\isamath{\mathcal{B}}}
-\newcommand{\isasymC}{\isamath{\mathcal{C}}}
-\newcommand{\isasymD}{\isamath{\mathcal{D}}}
-\newcommand{\isasymE}{\isamath{\mathcal{E}}}
-\newcommand{\isasymF}{\isamath{\mathcal{F}}}
-\newcommand{\isasymG}{\isamath{\mathcal{G}}}
-\newcommand{\isasymH}{\isamath{\mathcal{H}}}
-\newcommand{\isasymI}{\isamath{\mathcal{I}}}
-\newcommand{\isasymJ}{\isamath{\mathcal{J}}}
-\newcommand{\isasymK}{\isamath{\mathcal{K}}}
-\newcommand{\isasymL}{\isamath{\mathcal{L}}}
-\newcommand{\isasymM}{\isamath{\mathcal{M}}}
-\newcommand{\isasymN}{\isamath{\mathcal{N}}}
-\newcommand{\isasymO}{\isamath{\mathcal{O}}}
-\newcommand{\isasymP}{\isamath{\mathcal{P}}}
-\newcommand{\isasymQ}{\isamath{\mathcal{Q}}}
-\newcommand{\isasymR}{\isamath{\mathcal{R}}}
-\newcommand{\isasymS}{\isamath{\mathcal{S}}}
-\newcommand{\isasymT}{\isamath{\mathcal{T}}}
-\newcommand{\isasymU}{\isamath{\mathcal{U}}}
-\newcommand{\isasymV}{\isamath{\mathcal{V}}}
-\newcommand{\isasymW}{\isamath{\mathcal{W}}}
-\newcommand{\isasymX}{\isamath{\mathcal{X}}}
-\newcommand{\isasymY}{\isamath{\mathcal{Y}}}
-\newcommand{\isasymZ}{\isamath{\mathcal{Z}}}
-\newcommand{\isasyma}{\isamath{\mathrm{a}}}
-\newcommand{\isasymb}{\isamath{\mathrm{b}}}
-\newcommand{\isasymc}{\isamath{\mathrm{c}}}
-\newcommand{\isasymd}{\isamath{\mathrm{d}}}
-\newcommand{\isasyme}{\isamath{\mathrm{e}}}
-\newcommand{\isasymf}{\isamath{\mathrm{f}}}
-\newcommand{\isasymg}{\isamath{\mathrm{g}}}
-\newcommand{\isasymh}{\isamath{\mathrm{h}}}
-\newcommand{\isasymi}{\isamath{\mathrm{i}}}
-\newcommand{\isasymj}{\isamath{\mathrm{j}}}
-\newcommand{\isasymk}{\isamath{\mathrm{k}}}
-\newcommand{\isasyml}{\isamath{\mathrm{l}}}
-\newcommand{\isasymm}{\isamath{\mathrm{m}}}
-\newcommand{\isasymn}{\isamath{\mathrm{n}}}
-\newcommand{\isasymo}{\isamath{\mathrm{o}}}
-\newcommand{\isasymp}{\isamath{\mathrm{p}}}
-\newcommand{\isasymq}{\isamath{\mathrm{q}}}
-\newcommand{\isasymr}{\isamath{\mathrm{r}}}
-\newcommand{\isasyms}{\isamath{\mathrm{s}}}
-\newcommand{\isasymt}{\isamath{\mathrm{t}}}
-\newcommand{\isasymu}{\isamath{\mathrm{u}}}
-\newcommand{\isasymv}{\isamath{\mathrm{v}}}
-\newcommand{\isasymw}{\isamath{\mathrm{w}}}
-\newcommand{\isasymx}{\isamath{\mathrm{x}}}
-\newcommand{\isasymy}{\isamath{\mathrm{y}}}
-\newcommand{\isasymz}{\isamath{\mathrm{z}}}
-\newcommand{\isasymAA}{\isamath{\mathfrak{A}}}  %requires eufrak
-\newcommand{\isasymBB}{\isamath{\mathfrak{B}}}  %requires eufrak
-\newcommand{\isasymCC}{\isamath{\mathfrak{C}}}  %requires eufrak
-\newcommand{\isasymDD}{\isamath{\mathfrak{D}}}  %requires eufrak
-\newcommand{\isasymEE}{\isamath{\mathfrak{E}}}  %requires eufrak
-\newcommand{\isasymFF}{\isamath{\mathfrak{F}}}  %requires eufrak
-\newcommand{\isasymGG}{\isamath{\mathfrak{G}}}  %requires eufrak
-\newcommand{\isasymHH}{\isamath{\mathfrak{H}}}  %requires eufrak
-\newcommand{\isasymII}{\isamath{\mathfrak{I}}}  %requires eufrak
-\newcommand{\isasymJJ}{\isamath{\mathfrak{J}}}  %requires eufrak
-\newcommand{\isasymKK}{\isamath{\mathfrak{K}}}  %requires eufrak
-\newcommand{\isasymLL}{\isamath{\mathfrak{L}}}  %requires eufrak
-\newcommand{\isasymMM}{\isamath{\mathfrak{M}}}  %requires eufrak
-\newcommand{\isasymNN}{\isamath{\mathfrak{N}}}  %requires eufrak
-\newcommand{\isasymOO}{\isamath{\mathfrak{O}}}  %requires eufrak
-\newcommand{\isasymPP}{\isamath{\mathfrak{P}}}  %requires eufrak
-\newcommand{\isasymQQ}{\isamath{\mathfrak{Q}}}  %requires eufrak
-\newcommand{\isasymRR}{\isamath{\mathfrak{R}}}  %requires eufrak
-\newcommand{\isasymSS}{\isamath{\mathfrak{S}}}  %requires eufrak
-\newcommand{\isasymTT}{\isamath{\mathfrak{T}}}  %requires eufrak
-\newcommand{\isasymUU}{\isamath{\mathfrak{U}}}  %requires eufrak
-\newcommand{\isasymVV}{\isamath{\mathfrak{V}}}  %requires eufrak
-\newcommand{\isasymWW}{\isamath{\mathfrak{W}}}  %requires eufrak
-\newcommand{\isasymXX}{\isamath{\mathfrak{X}}}  %requires eufrak
-\newcommand{\isasymYY}{\isamath{\mathfrak{Y}}}  %requires eufrak
-\newcommand{\isasymZZ}{\isamath{\mathfrak{Z}}}  %requires eufrak
-\newcommand{\isasymaa}{\isamath{\mathfrak{a}}}  %requires eufrak
-\newcommand{\isasymbb}{\isamath{\mathfrak{b}}}  %requires eufrak
-\newcommand{\isasymcc}{\isamath{\mathfrak{c}}}  %requires eufrak
-\newcommand{\isasymdd}{\isamath{\mathfrak{d}}}  %requires eufrak
-\newcommand{\isasymee}{\isamath{\mathfrak{e}}}  %requires eufrak
-\newcommand{\isasymff}{\isamath{\mathfrak{f}}}  %requires eufrak
-\newcommand{\isasymgg}{\isamath{\mathfrak{g}}}  %requires eufrak
-\newcommand{\isasymhh}{\isamath{\mathfrak{h}}}  %requires eufrak
-\newcommand{\isasymii}{\isamath{\mathfrak{i}}}  %requires eufrak
-\newcommand{\isasymjj}{\isamath{\mathfrak{j}}}  %requires eufrak
-\newcommand{\isasymkk}{\isamath{\mathfrak{k}}}  %requires eufrak
-\newcommand{\isasymll}{\isamath{\mathfrak{l}}}  %requires eufrak
-\newcommand{\isasymmm}{\isamath{\mathfrak{m}}}  %requires eufrak
-\newcommand{\isasymnn}{\isamath{\mathfrak{n}}}  %requires eufrak
-\newcommand{\isasymoo}{\isamath{\mathfrak{o}}}  %requires eufrak
-\newcommand{\isasympp}{\isamath{\mathfrak{p}}}  %requires eufrak
-\newcommand{\isasymqq}{\isamath{\mathfrak{q}}}  %requires eufrak
-\newcommand{\isasymrr}{\isamath{\mathfrak{r}}}  %requires eufrak
-\newcommand{\isasymss}{\isamath{\mathfrak{s}}}  %requires eufrak
-\newcommand{\isasymtt}{\isamath{\mathfrak{t}}}  %requires eufrak
-\newcommand{\isasymuu}{\isamath{\mathfrak{u}}}  %requires eufrak
-\newcommand{\isasymvv}{\isamath{\mathfrak{v}}}  %requires eufrak
-\newcommand{\isasymww}{\isamath{\mathfrak{w}}}  %requires eufrak
-\newcommand{\isasymxx}{\isamath{\mathfrak{x}}}  %requires eufrak
-\newcommand{\isasymyy}{\isamath{\mathfrak{y}}}  %requires eufrak
-\newcommand{\isasymzz}{\isamath{\mathfrak{z}}}  %requires eufrak
-\newcommand{\isasymalpha}{\isamath{\alpha}}
-\newcommand{\isasymbeta}{\isamath{\beta}}
-\newcommand{\isasymgamma}{\isamath{\gamma}}
-\newcommand{\isasymdelta}{\isamath{\delta}}
-\newcommand{\isasymepsilon}{\isamath{\varepsilon}}
-\newcommand{\isasymzeta}{\isamath{\zeta}}
-\newcommand{\isasymeta}{\isamath{\eta}}
-\newcommand{\isasymtheta}{\isamath{\vartheta}}
-\newcommand{\isasymiota}{\isamath{\iota}}
-\newcommand{\isasymkappa}{\isamath{\kappa}}
-\newcommand{\isasymlambda}{\isamath{\lambda}}
-\newcommand{\isasymmu}{\isamath{\mu}}
-\newcommand{\isasymnu}{\isamath{\nu}}
-\newcommand{\isasymxi}{\isamath{\xi}}
-\newcommand{\isasympi}{\isamath{\pi}}
-\newcommand{\isasymrho}{\isamath{\varrho}}
-\newcommand{\isasymsigma}{\isamath{\sigma}}
-\newcommand{\isasymtau}{\isamath{\tau}}
-\newcommand{\isasymupsilon}{\isamath{\upsilon}}
-\newcommand{\isasymphi}{\isamath{\varphi}}
-\newcommand{\isasymchi}{\isamath{\chi}}
-\newcommand{\isasympsi}{\isamath{\psi}}
-\newcommand{\isasymomega}{\isamath{\omega}}
-\newcommand{\isasymGamma}{\isamath{\Gamma}}
-\newcommand{\isasymDelta}{\isamath{\Delta}}
-\newcommand{\isasymTheta}{\isamath{\Theta}}
-\newcommand{\isasymLambda}{\isamath{\Lambda}}
-\newcommand{\isasymXi}{\isamath{\Xi}}
-\newcommand{\isasymPi}{\isamath{\Pi}}
-\newcommand{\isasymSigma}{\isamath{\Sigma}}
-\newcommand{\isasymUpsilon}{\isamath{\Upsilon}}
-\newcommand{\isasymPhi}{\isamath{\Phi}}
-\newcommand{\isasymPsi}{\isamath{\Psi}}
-\newcommand{\isasymOmega}{\isamath{\Omega}}
-\newcommand{\isasymbool}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{B}}}
-\newcommand{\isasymcomplex}{\isamath{\mathrm{C}\mkern-15mu{\phantom{\mathrm{t}}\vrule}\mkern9mu}}
-\newcommand{\isasymnat}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{N}}}
-\newcommand{\isasymrat}{\isamath{\mathrm{Q}\mkern-16mu{\phantom{\mathrm{t}}\vrule}\mkern10mu}}
-\newcommand{\isasymreal}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{R}}}
-\newcommand{\isasymint}{\isamath{\mathsf{Z}\mkern-7.5mu\mathsf{Z}}}
-\newcommand{\isasymleftarrow}{\isamath{\leftarrow}}
-\newcommand{\isasymrightarrow}{\isamath{\rightarrow}}
-\newcommand{\isasymlongleftarrow}{\isamath{\longleftarrow}}
-\newcommand{\isasymlongrightarrow}{\isamath{\longrightarrow}}
-\newcommand{\isasymlonglongleftarrow}{\isamath{\xleftarrow{\hphantom{AAA}}}}  %requires amsmath
-\newcommand{\isasymlonglongrightarrow}{\isamath{\xrightarrow{\hphantom{AAA}}}}  %requires amsmath
-\newcommand{\isasymlonglonglongleftarrow}{\isamath{\xleftarrow{\hphantom{AAAA}}}}  %requires amsmath
-\newcommand{\isasymlonglonglongrightarrow}{\isamath{\xrightarrow{\hphantom{AAAA}}}}  %requires amsmath
-\newcommand{\isasymLeftarrow}{\isamath{\Leftarrow}}
-\newcommand{\isasymRightarrow}{\isamath{\Rightarrow}}
-\newcommand{\isasymLongleftarrow}{\isamath{\Longleftarrow}}
-\newcommand{\isasymLongrightarrow}{\isamath{\Longrightarrow}}
-\newcommand{\isasymLleftarrow}{\isamath{\Lleftarrow}}  %requires amssymb
-\newcommand{\isasymRrightarrow}{\isamath{\Rrightarrow}}  %requires amssymb
-\newcommand{\isasymleftrightarrow}{\isamath{\leftrightarrow}}
-\newcommand{\isasymLeftrightarrow}{\isamath{\Leftrightarrow}}
-\newcommand{\isasymlongleftrightarrow}{\isamath{\longleftrightarrow}}
-\newcommand{\isasymLongleftrightarrow}{\isamath{\Longleftrightarrow}}
-\newcommand{\isasymmapsto}{\isamath{\mapsto}}
-\newcommand{\isasymlongmapsto}{\isamath{\longmapsto}}
-\newcommand{\isasymmidarrow}{\isamath{\relbar}}
-\newcommand{\isasymMidarrow}{\isamath{\Relbar}}
-\newcommand{\isasymhookleftarrow}{\isamath{\hookleftarrow}}
-\newcommand{\isasymhookrightarrow}{\isamath{\hookrightarrow}}
-\newcommand{\isasymleftharpoondown}{\isamath{\leftharpoondown}}
-\newcommand{\isasymrightharpoondown}{\isamath{\rightharpoondown}}
-\newcommand{\isasymleftharpoonup}{\isamath{\leftharpoonup}}
-\newcommand{\isasymrightharpoonup}{\isamath{\rightharpoonup}}
-\newcommand{\isasymrightleftharpoons}{\isamath{\rightleftharpoons}}
-\newcommand{\isasymleadsto}{\isamath{\leadsto}}  %requires amssymb
-\newcommand{\isasymdownharpoonleft}{\isamath{\downharpoonleft}}  %requires amssymb
-\newcommand{\isasymdownharpoonright}{\isamath{\downharpoonright}}  %requires amssymb
-\newcommand{\isasymupharpoonleft}{\isamath{\upharpoonleft}}  %requires amssymb
-\newcommand{\isasymupharpoonright}{\isamath{\upharpoonright}}  %requires amssymb
-\newcommand{\isasymrestriction}{\isamath{\restriction}}  %requires amssymb
-\newcommand{\isasymColon}{\isamath{\mathrel{::}}}
-\newcommand{\isasymup}{\isamath{\uparrow}}
-\newcommand{\isasymUp}{\isamath{\Uparrow}}
-\newcommand{\isasymdown}{\isamath{\downarrow}}
-\newcommand{\isasymDown}{\isamath{\Downarrow}}
-\newcommand{\isasymupdown}{\isamath{\updownarrow}}
-\newcommand{\isasymUpdown}{\isamath{\Updownarrow}}
-\newcommand{\isasymlangle}{\isamath{\langle}}
-\newcommand{\isasymrangle}{\isamath{\rangle}}
-\newcommand{\isasymllangle}{\isamath{\langle\mskip-5mu\langle}}
-\newcommand{\isasymrrangle}{\isamath{\rangle\mskip-5mu\rangle}}
-\newcommand{\isasymlceil}{\isamath{\lceil}}
-\newcommand{\isasymrceil}{\isamath{\rceil}}
-\newcommand{\isasymlfloor}{\isamath{\lfloor}}
-\newcommand{\isasymrfloor}{\isamath{\rfloor}}
-\newcommand{\isasymlparr}{\isamath{\mathopen{(\mkern-3mu\mid}}}
-\newcommand{\isasymrparr}{\isamath{\mathclose{\mid\mkern-3mu)}}}
-\newcommand{\isasymlbrakk}{\isamath{\mathopen{\lbrack\mkern-3mu\lbrack}}}
-\newcommand{\isasymrbrakk}{\isamath{\mathclose{\rbrack\mkern-3mu\rbrack}}}
-\newcommand{\isasymlbrace}{\isamath{\mathopen{\lbrace\mkern-4.5mu\mid}}}
-\newcommand{\isasymrbrace}{\isamath{\mathclose{\mid\mkern-4.5mu\rbrace}}}
-\newcommand{\isasymguillemotleft}{\isatext{\flqq}}  %requires babel
-\newcommand{\isasymguillemotright}{\isatext{\frqq}}  %requires babel
-\newcommand{\isasymbottom}{\isamath{\bot}}
-\newcommand{\isasymtop}{\isamath{\top}}
-\newcommand{\isasymand}{\isamath{\wedge}}
-\newcommand{\isasymAnd}{\isamath{\bigwedge}}
-\newcommand{\isasymor}{\isamath{\vee}}
-\newcommand{\isasymOr}{\isamath{\bigvee}}
-\newcommand{\isasymforall}{\isamath{\forall\,}}
-\newcommand{\isasymexists}{\isamath{\exists\,}}
-\newcommand{\isasymnot}{\isamath{\neg}}
-\newcommand{\isasymnexists}{\isamath{\nexists\,}}  %requires amssymb
-\newcommand{\isasymcircle}{\isamath{\ocircle}}  %requires wasysym
-\newcommand{\isasymbox}{\isamath{\Box}}  %requires amssymb
-\newcommand{\isasymdiamond}{\isamath{\Diamond}}  %requires amssymb
-\newcommand{\isasymdiamondop}{\isamath{\diamond}}
-\newcommand{\isasymsurd}{\isamath{\surd}}
-\newcommand{\isasymturnstile}{\isamath{\vdash}}
-\newcommand{\isasymTurnstile}{\isamath{\models}}
-\newcommand{\isasymtturnstile}{\isamath{\vdash\!\!\!\vdash}}
-\newcommand{\isasymTTurnstile}{\isamath{\mid\!\models}}
-\newcommand{\isasymstileturn}{\isamath{\dashv}}
-\newcommand{\isasymle}{\isamath{\le}}
-\newcommand{\isasymge}{\isamath{\ge}}
-\newcommand{\isasymlless}{\isamath{\ll}}
-\newcommand{\isasymggreater}{\isamath{\gg}}
-\newcommand{\isasymlesssim}{\isamath{\lesssim}}  %requires amssymb
-\newcommand{\isasymgreatersim}{\isamath{\gtrsim}}  %requires amssymb
-\newcommand{\isasymlessapprox}{\isamath{\lessapprox}}  %requires amssymb
-\newcommand{\isasymgreaterapprox}{\isamath{\gtrapprox}}  %requires amssymb
-\newcommand{\isasymin}{\isamath{\in}}
-\newcommand{\isasymnotin}{\isamath{\notin}}
-\newcommand{\isasymsubset}{\isamath{\subset}}
-\newcommand{\isasymsupset}{\isamath{\supset}}
-\newcommand{\isasymsubseteq}{\isamath{\subseteq}}
-\newcommand{\isasymsupseteq}{\isamath{\supseteq}}
-\newcommand{\isasymsqsubset}{\isamath{\sqsubset}}  %requires amssymb
-\newcommand{\isasymsqsupset}{\isamath{\sqsupset}}  %requires amssymb
-\newcommand{\isasymsqsubseteq}{\isamath{\sqsubseteq}}
-\newcommand{\isasymsqsupseteq}{\isamath{\sqsupseteq}}
-\newcommand{\isasyminter}{\isamath{\cap}}
-\newcommand{\isasymInter}{\isamath{\bigcap\,}}
-\newcommand{\isasymunion}{\isamath{\cup}}
-\newcommand{\isasymUnion}{\isamath{\bigcup\,}}
-\newcommand{\isasymsqunion}{\isamath{\sqcup}}
-\newcommand{\isasymSqunion}{\isamath{\bigsqcup\,}}
-\newcommand{\isasymsqinter}{\isamath{\sqcap}}
-\newcommand{\isasymSqinter}{\isamath{\bigsqcap\,}}  %requires stmaryrd
-\newcommand{\isasymsetminus}{\isamath{\setminus}}
-\newcommand{\isasympropto}{\isamath{\propto}}
-\newcommand{\isasymuplus}{\isamath{\uplus}}
-\newcommand{\isasymUplus}{\isamath{\biguplus\,}}
-\newcommand{\isasymnoteq}{\isamath{\not=}}
-\newcommand{\isasymsim}{\isamath{\sim}}
-\newcommand{\isasymdoteq}{\isamath{\doteq}}
-\newcommand{\isasymsimeq}{\isamath{\simeq}}
-\newcommand{\isasymapprox}{\isamath{\approx}}
-\newcommand{\isasymasymp}{\isamath{\asymp}}
-\newcommand{\isasymcong}{\isamath{\cong}}
-\newcommand{\isasymsmile}{\isamath{\smile}}
-\newcommand{\isasymequiv}{\isamath{\equiv}}
-\newcommand{\isasymfrown}{\isamath{\frown}}
-\newcommand{\isasymJoin}{\isamath{\Join}}  %requires amssymb
-\newcommand{\isasymbowtie}{\isamath{\bowtie}}
-\newcommand{\isasymprec}{\isamath{\prec}}
-\newcommand{\isasymsucc}{\isamath{\succ}}
-\newcommand{\isasympreceq}{\isamath{\preceq}}
-\newcommand{\isasymsucceq}{\isamath{\succeq}}
-\newcommand{\isasymparallel}{\isamath{\parallel}}
-\newcommand{\isasymbar}{\isamath{\mid}}
-\newcommand{\isasymbbar}{\isamath{[\mskip-1.5mu]}}
-\newcommand{\isasymplusminus}{\isamath{\pm}}
-\newcommand{\isasymminusplus}{\isamath{\mp}}
-\newcommand{\isasymtimes}{\isamath{\times}}
-\newcommand{\isasymdiv}{\isamath{\div}}
-\newcommand{\isasymcdot}{\isamath{\cdot}}
-\newcommand{\isasymsqdot}{\isamath{\sbox\z@{$\centerdot$}\ht\z@=.33333\ht\z@\vcenter{\box\z@}}}  %requires amssymb
-\newcommand{\isasymstar}{\isamath{\star}}
-\newcommand{\isasymbullet}{\boldmath\isamath{\mathchoice{\displaystyle{\cdot}}{\textstyle{\cdot}}{\scriptstyle{\bullet}}{\scriptscriptstyle{\bullet}}}}
-\newcommand{\isasymcirc}{\isamath{\circ}}
-\newcommand{\isasymdagger}{\isamath{\dagger}}
-\newcommand{\isasymddagger}{\isamath{\ddagger}}
-\newcommand{\isasymlhd}{\isamath{\lhd}}  %requires amssymb
-\newcommand{\isasymrhd}{\isamath{\rhd}}  %requires amssymb
-\newcommand{\isasymunlhd}{\isamath{\unlhd}}  %requires amssymb
-\newcommand{\isasymunrhd}{\isamath{\unrhd}}  %requires amssymb
-\newcommand{\isasymtriangleleft}{\isamath{\triangleleft}}
-\newcommand{\isasymtriangleright}{\isamath{\triangleright}}
-\newcommand{\isasymtriangle}{\isamath{\triangle}}
-\newcommand{\isasymtriangleq}{\isamath{\triangleq}}  %requires amssymb
-\newcommand{\isasymoplus}{\isamath{\oplus}}
-\newcommand{\isasymOplus}{\isamath{\bigoplus\,}}
-\newcommand{\isasymotimes}{\isamath{\otimes}}
-\newcommand{\isasymOtimes}{\isamath{\bigotimes\,}}
-\newcommand{\isasymodot}{\isamath{\odot}}
-\newcommand{\isasymOdot}{\isamath{\bigodot\,}}
-\newcommand{\isasymominus}{\isamath{\ominus}}
-\newcommand{\isasymoslash}{\isamath{\oslash}}
-\newcommand{\isasymdots}{\isamath{\dots}}
-\newcommand{\isasymcdots}{\isamath{\cdots}}
-\newcommand{\isasymSum}{\isamath{\sum\,}}
-\newcommand{\isasymProd}{\isamath{\prod\,}}
-\newcommand{\isasymCoprod}{\isamath{\coprod\,}}
-\newcommand{\isasyminfinity}{\isamath{\infty}}
-\newcommand{\isasymintegral}{\isamath{\int\,}}
-\newcommand{\isasymointegral}{\isamath{\oint\,}}
-\newcommand{\isasymclubsuit}{\isamath{\clubsuit}}
-\newcommand{\isasymdiamondsuit}{\isamath{\diamondsuit}}
-\newcommand{\isasymheartsuit}{\isamath{\heartsuit}}
-\newcommand{\isasymspadesuit}{\isamath{\spadesuit}}
-\newcommand{\isasymaleph}{\isamath{\aleph}}
-\newcommand{\isasymemptyset}{\isamath{\emptyset}}
-\newcommand{\isasymnabla}{\isamath{\nabla}}
-\newcommand{\isasympartial}{\isamath{\partial}}
-\newcommand{\isasymRe}{\isamath{\Re}}
-\newcommand{\isasymIm}{\isamath{\Im}}
-\newcommand{\isasymflat}{\isamath{\flat}}
-\newcommand{\isasymnatural}{\isamath{\natural}}
-\newcommand{\isasymsharp}{\isamath{\sharp}}
-\newcommand{\isasymangle}{\isamath{\angle}}
-\newcommand{\isasymcopyright}{\isatext{\normalfont\rmfamily\copyright}}
-\newcommand{\isasymregistered}{\isatext{\normalfont\rmfamily\textregistered}}
-\newcommand{\isasyminverse}{\isamath{{}^{-1}}}
-\newcommand{\isasymonequarter}{\isatext{\normalfont\rmfamily\textonequarter}}  %requires textcomp
-\newcommand{\isasymonehalf}{\isatext{\normalfont\rmfamily\textonehalf}}  %requires textcomp
-\newcommand{\isasymthreequarters}{\isatext{\normalfont\rmfamily\textthreequarters}}  %requires textcomp
-\newcommand{\isasymordfeminine}{\isatext{\normalfont\rmfamily\textordfeminine}}
-\newcommand{\isasymordmasculine}{\isatext{\normalfont\rmfamily\textordmasculine}}
-\newcommand{\isasymsection}{\isatext{\normalfont\rmfamily\S}}
-\newcommand{\isasymparagraph}{\isatext{\normalfont\rmfamily\P}}
-\newcommand{\isasymexclamdown}{\isatext{\normalfont\rmfamily\textexclamdown}}
-\newcommand{\isasymquestiondown}{\isatext{\normalfont\rmfamily\textquestiondown}}
-\newcommand{\isasymeuro}{\isatext{\euro}}  %requires eurosym
-\newcommand{\isasympounds}{\isamath{\pounds}}
-\newcommand{\isasymyen}{\isatext{\yen}}  %requires amssymb
-\newcommand{\isasymcent}{\isatext{\textcent}}  %requires textcomp
-\newcommand{\isasymcurrency}{\isatext{\textcurrency}} %requires textcomp
-\newcommand{\isasymdegree}{\isatext{\normalfont\rmfamily\textdegree}}  %requires textcomp
-\newcommand{\isasymhyphen}{\isatext{\normalfont\rmfamily-}}
-\newcommand{\isasymamalg}{\isamath{\amalg}}
-\newcommand{\isasymmho}{\isamath{\mho}}  %requires amssymb
-\newcommand{\isasymlozenge}{\isamath{\lozenge}}  %requires amssymb
-\newcommand{\isasymwp}{\isamath{\wp}}
-\newcommand{\isasymwrong}{\isamath{\wr}}
-\newcommand{\isasymacute}{\isatext{\'\relax}}
-\newcommand{\isasymindex}{\isatext{\i}}
-\newcommand{\isasymdieresis}{\isatext{\"\relax}}
-\newcommand{\isasymcedilla}{\isatext{\c\relax}}
-\newcommand{\isasymhungarumlaut}{\isatext{\H\relax}}
-\newcommand{\isasymsome}{\isamath{\epsilon\,}}
-\newcommand{\isasymbind}{\isamath{\mathbin{>\!\!\!>\mkern-6.7mu=}}}
-\newcommand{\isasymthen}{\isamath{\mathbin{>\!\!\!>}}}
-\newcommand{\isasymopen}{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\langle$}}}
-\newcommand{\isasymclose}{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\rangle$}}}
-\newcommand{\isasymhole}{\isatext{\normalfont\rmfamily\wasylozenge}}  %requires wasysym
-\newcommand{\isasymnewline}{\isatext{\fbox{$\hookleftarrow$}}}
-\newcommand{\isasymcomment}{\isatext{\isastylecmt---}}
-\newcommand{\isasymproof}{\isamath{\,\langle\mathit{proof}\rangle}}
-
-\newcommand{\isactrlmarker}{\isatext{\ding{48}}}  %requires pifont
-\newcommand{\isactrlassert}{\isakeywordcontrol{assert}}
-\newcommand{\isactrlcancel}{\isakeywordcontrol{cancel}}
-\newcommand{\isactrlbinding}{\isakeywordcontrol{binding}}
-\newcommand{\isactrlclass}{\isakeywordcontrol{class}}
-\newcommand{\isactrlclassUNDERSCOREsyntax}{\isakeywordcontrol{class{\isacharunderscore}syntax}}
-\newcommand{\isactrlcommandUNDERSCOREkeyword}{\isakeywordcontrol{command{\isacharunderscore}keyword}}
-\newcommand{\isactrlconst}{\isakeywordcontrol{const}}
-\newcommand{\isactrlconstUNDERSCOREabbrev}{\isakeywordcontrol{const{\isacharunderscore}abbrev}}
-\newcommand{\isactrlconstUNDERSCOREname}{\isakeywordcontrol{const{\isacharunderscore}name}}
-\newcommand{\isactrlconstUNDERSCOREsyntax}{\isakeywordcontrol{const{\isacharunderscore}syntax}}
-\newcommand{\isactrlcontext}{\isakeywordcontrol{context}}
-\newcommand{\isactrlcprop}{\isakeywordcontrol{cprop}}
-\newcommand{\isactrlcterm}{\isakeywordcontrol{cterm}}
-\newcommand{\isactrlctyp}{\isakeywordcontrol{ctyp}}
-\newcommand{\isactrldir}{\isakeywordcontrol{dir}}
-\newcommand{\isactrlfile}{\isakeywordcontrol{file}}
-\newcommand{\isactrlhere}{\isakeywordcontrol{here}}
-\newcommand{\isactrlkeyword}{\isakeywordcontrol{keyword}}
-\newcommand{\isactrllatex}{\isakeywordcontrol{latex}}
-\newcommand{\isactrllocale}{\isakeywordcontrol{locale}}
-\newcommand{\isactrlmakeUNDERSCOREstring}{\isakeywordcontrol{make{\isacharunderscore}string}}
-\newcommand{\isactrlmasterUNDERSCOREdir}{\isakeywordcontrol{master{\isacharunderscore}dir}}
-\newcommand{\isactrlmethod}{\isakeywordcontrol{method}}
-\newcommand{\isactrlnamedUNDERSCOREtheorems}{\isakeywordcontrol{named{\isacharunderscore}theorems}}
-\newcommand{\isactrlnonterminal}{\isakeywordcontrol{nonterminal}}
-\newcommand{\isactrloracleUNDERSCOREname}{\isakeywordcontrol{oracle{\isacharunderscore}name}}
-\newcommand{\isactrlpath}{\isakeywordcontrol{path}}
-\newcommand{\isactrlpathUNDERSCOREbinding}{\isakeywordcontrol{path{\isacharunderscore}binding}}
-\newcommand{\isactrlplugin}{\isakeywordcontrol{plugin}}
-\newcommand{\isactrlprint}{\isakeywordcontrol{print}}
-\newcommand{\isactrlprop}{\isakeywordcontrol{prop}}
-\newcommand{\isactrlscala}{\isakeywordcontrol{scala}}
-\newcommand{\isactrlscalaUNDERSCOREfunction}{\isakeywordcontrol{scala{\isacharunderscore}function}}
-\newcommand{\isactrlscalaUNDERSCOREmethod}{\isakeywordcontrol{scala{\isacharunderscore}method}}
-\newcommand{\isactrlscalaUNDERSCOREobject}{\isakeywordcontrol{scala{\isacharunderscore}object}}
-\newcommand{\isactrlscalaUNDERSCOREtype}{\isakeywordcontrol{scala{\isacharunderscore}type}}
-\newcommand{\isactrlsimproc}{\isakeywordcontrol{simproc}}
-\newcommand{\isactrlsort}{\isakeywordcontrol{sort}}
-\newcommand{\isactrlsyntaxUNDERSCOREconst}{\isakeywordcontrol{syntax{\isacharunderscore}const}}
-\newcommand{\isactrlsystemUNDERSCOREoption}{\isakeywordcontrol{system{\isacharunderscore}option}}
-\newcommand{\isactrlterm}{\isakeywordcontrol{term}}
-\newcommand{\isactrltheory}{\isakeywordcontrol{theory}}
-\newcommand{\isactrltheoryUNDERSCOREcontext}{\isakeywordcontrol{theory{\isacharunderscore}context}}
-\newcommand{\isactrltyp}{\isakeywordcontrol{typ}}
-\newcommand{\isactrltypeUNDERSCOREabbrev}{\isakeywordcontrol{type{\isacharunderscore}abbrev}}
-\newcommand{\isactrltypeUNDERSCOREname}{\isakeywordcontrol{type{\isacharunderscore}name}}
-\newcommand{\isactrltypeUNDERSCOREsyntax}{\isakeywordcontrol{type{\isacharunderscore}syntax}}
-\newcommand{\isactrlundefined}{\isakeywordcontrol{undefined}}
-
-\newcommand{\isactrlcode}{\isakeywordcontrol{code}}
-\newcommand{\isactrlcomputation}{\isakeywordcontrol{computation}}
-\newcommand{\isactrlcomputationUNDERSCOREconv}{\isakeywordcontrol{computation{\isacharunderscore}conv}}
-\newcommand{\isactrlcomputationUNDERSCOREcheck}{\isakeywordcontrol{computation{\isacharunderscore}check}}
diff --git a/virage/src/test/resources/old_theories/output/document/isabelletags.sty b/virage/src/test/resources/old_theories/output/document/isabelletags.sty
deleted file mode 100644
index 8b13789..0000000
--- a/virage/src/test/resources/old_theories/output/document/isabelletags.sty
+++ /dev/null
@@ -1 +0,0 @@
-
diff --git a/virage/src/test/resources/old_theories/output/document/pdfsetup.sty b/virage/src/test/resources/old_theories/output/document/pdfsetup.sty
deleted file mode 100644
index 2bf5ccf..0000000
--- a/virage/src/test/resources/old_theories/output/document/pdfsetup.sty
+++ /dev/null
@@ -1,7 +0,0 @@
-%%
-%% default hyperref setup (both for pdf and dvi output)
-%%
-
-\usepackage{color}
-\definecolor{linkcolor}{rgb}{0,0,0.5}
-\usepackage[colorlinks=true,linkcolor=linkcolor,citecolor=linkcolor,filecolor=linkcolor,urlcolor=linkcolor,pdfpagelabels]{hyperref}
diff --git a/virage/src/test/resources/old_theories/output/document/railsetup.sty b/virage/src/test/resources/old_theories/output/document/railsetup.sty
deleted file mode 100644
index 93cae05..0000000
--- a/virage/src/test/resources/old_theories/output/document/railsetup.sty
+++ /dev/null
@@ -1,1202 +0,0 @@
-% rail.sty - style file to support railroad diagrams
-%
-% 09-Jul-90 L. Rooijakkers
-% 08-Oct-90 L. Rooijakkers	fixed centering bug when \rail@tmpc<0.
-% 07-Feb-91 L. Rooijakkers	added \railoptions command, indexing
-% 08-Feb-91 L. Rooijakkers	minor fixes
-% 28-Jun-94 K. Barthelmann	turned into LaTeX2e package
-% 08-Dec-96 K. Barthelmann	replaced \@writefile
-% 13-Dec-96 K. Barthelmann	cleanup
-% 22-Feb-98 K. Barthelmann	fixed catcodes of special characters
-% 18-Apr-98 K. Barthelmann	fixed \par handling
-% 19-May-98 J. Olsson		Added new macros to support arrow heads.
-% 26-Jul-98 K. Barthelmann	changed \par to output newlines
-% 02-May-11 M. Wenzel           default setup for Isabelle
-%
-% This style file needs to be used in conjunction with the 'rail'
-% program. Running LaTeX as 'latex file' produces file.rai, which should be
-% processed by Rail with 'rail file'. This produces file.rao, which will
-% be picked up by LaTeX on the next 'latex file' run.
-%
-% LaTeX will warn if there is no file.rao or it's out of date.
-%
-% The macros in this file thus consist of two parts: those that read and
-% write the .rai and .rao files, and those that do the actual formatting
-% of the railroad diagrams.
-
-\NeedsTeXFormat{LaTeX2e}
-\ProvidesPackage{rail}[1998/05/19]
-
-% railroad diagram formatting parameters (user level)
-% all of these are copied into their internal versions by \railinit
-%
-% \railunit : \unitlength within railroad diagrams
-% \railextra : extra length at outside of diagram
-% \railboxheight : height of ovals and frames
-% \railboxskip : vertical space between lines
-% \railboxleft : space to the left of a box
-% \railboxright : space to the right of a box
-% \railovalspace : extra space around contents of oval
-% \railframespace : extra space around contents of frame
-% \railtextleft : space to the left of text
-% \railtextright : space to the right of text
-% \railtextup : space to lift text up
-% \railjoinsize : circle size of join/split arcs
-% \railjoinadjust : space to adjust join
-%
-% \railnamesep : separator between name and rule body
-
-\newlength\railunit
-\newlength\railextra
-\newlength\railboxheight
-\newlength\railboxskip
-\newlength\railboxleft
-\newlength\railboxright
-\newlength\railovalspace
-\newlength\railframespace
-\newlength\railtextleft
-\newlength\railtextright
-\newlength\railtextup
-\newlength\railjoinsize
-\newlength\railjoinadjust
-\newlength\railnamesep
-
-% initialize the parameters
-
-\setlength\railunit{1sp}
-\setlength\railextra{4ex}
-\setlength\railboxleft{1ex}
-\setlength\railboxright{1ex}
-\setlength\railovalspace{2ex}
-\setlength\railframespace{2ex}
-\setlength\railtextleft{1ex}
-\setlength\railtextright{1ex}
-\setlength\railjoinadjust{0pt}
-\setlength\railnamesep{1ex}
-
-\DeclareOption{10pt}{
-  \setlength\railboxheight{16pt}
-  \setlength\railboxskip{24pt}
-  \setlength\railtextup{5pt}
-  \setlength\railjoinsize{16pt}
-}
-\DeclareOption{11pt}{
-  \setlength\railboxheight{16pt}
-  \setlength\railboxskip{24pt}
-  \setlength\railtextup{5pt}
-  \setlength\railjoinsize{16pt}
-}
-\DeclareOption{12pt}{
-  \setlength\railboxheight{20pt}
-  \setlength\railboxskip{28pt}
-  \setlength\railtextup{6pt}
-  \setlength\railjoinsize{20pt}
-}
-
-\ExecuteOptions{10pt}
-\ProcessOptions
-
-% internal versions of the formatting parameters
-%
-% \rail@extra   : \railextra
-% \rail@boxht   : \railboxheight
-% \rail@boxsp   : \railboxskip
-% \rail@boxlf   : \railboxleft
-% \rail@boxrt   : \railboxright
-% \rail@boxhht  : \railboxheight / 2
-% \rail@ovalsp  : \railovalspace
-% \rail@framesp : \railframespace
-% \rail@textlf  : \railtextleft
-% \rail@textrt  : \railtextright
-% \rail@textup  : \railtextup
-% \rail@joinsz  : \railjoinsize
-% \rail@joinhsz : \railjoinsize / 2
-% \rail@joinadj : \railjoinadjust
-%
-% \railinit : internalize all of the parameters.
-
-\newcount\rail@extra
-\newcount\rail@boxht
-\newcount\rail@boxsp
-\newcount\rail@boxlf
-\newcount\rail@boxrt
-\newcount\rail@boxhht
-\newcount\rail@ovalsp
-\newcount\rail@framesp
-\newcount\rail@textlf
-\newcount\rail@textrt
-\newcount\rail@textup
-\newcount\rail@joinsz
-\newcount\rail@joinhsz
-\newcount\rail@joinadj
-
-\newcommand\railinit{
-\rail@extra=\railextra
-\divide\rail@extra by \railunit
-\rail@boxht=\railboxheight
-\divide\rail@boxht by \railunit
-\rail@boxsp=\railboxskip
-\divide\rail@boxsp by \railunit
-\rail@boxlf=\railboxleft
-\divide\rail@boxlf by \railunit
-\rail@boxrt=\railboxright
-\divide\rail@boxrt by \railunit
-\rail@boxhht=\railboxheight
-\divide\rail@boxhht by \railunit
-\divide\rail@boxhht by 2
-\rail@ovalsp=\railovalspace
-\divide\rail@ovalsp by \railunit
-\rail@framesp=\railframespace
-\divide\rail@framesp by \railunit
-\rail@textlf=\railtextleft
-\divide\rail@textlf by \railunit
-\rail@textrt=\railtextright
-\divide\rail@textrt by \railunit
-\rail@textup=\railtextup
-\divide\rail@textup by \railunit
-\rail@joinsz=\railjoinsize
-\divide\rail@joinsz by \railunit
-\rail@joinhsz=\railjoinsize
-\divide\rail@joinhsz by \railunit
-\divide\rail@joinhsz by 2
-\rail@joinadj=\railjoinadjust
-\divide\rail@joinadj by \railunit
-}
-
-\AtBeginDocument{\railinit}
-
-% \rail@param : declarations for list environment
-%
-% \railparam{TEXT} : sets \rail@param to TEXT
-%
-% \rail@reserved : characters reserved for grammar
-
-\newcommand\railparam[1]{
-\def\rail@param{
-  \setlength\leftmargin{0pt}\setlength\rightmargin{0pt}
-  \setlength\labelwidth{0pt}\setlength\labelsep{0pt}
-  \setlength\itemindent{0pt}\setlength\listparindent{0pt}
-  #1
-}
-}
-\railparam{}
-
-\newtoks\rail@reserved
-\rail@reserved={:;|*+?[]()'"}
-
-% \rail@termfont : format setup for terminals
-%
-% \rail@nontfont : format setup for nonterminals
-%
-% \rail@annofont : format setup for annotations
-%
-% \rail@rulefont : format setup for rule names
-%
-% \rail@indexfont : format setup for index entry
-%
-% \railtermfont{TEXT} : set terminal format setup to TEXT
-%
-% \railnontermfont{TEXT} : set nonterminal format setup to TEXT
-%
-% \railannotatefont{TEXT} : set annotation format setup to TEXT
-%
-% \railnamefont{TEXT} : set rule name format setup to TEXT
-%
-% \railindexfont{TEXT} : set index entry format setup to TEXT
-
-\def\rail@termfont{\ttfamily\upshape}
-\def\rail@nontfont{\rmfamily\upshape}
-\def\rail@annofont{\rmfamily\itshape}
-\def\rail@namefont{\rmfamily\itshape}
-\def\rail@indexfont{\rmfamily\itshape}
-
-\newcommand\railtermfont[1]{
-\def\rail@termfont{#1}
-}
-
-\newcommand\railnontermfont[1]{
-\def\rail@nontfont{#1}
-}
-
-\newcommand\railannotatefont[1]{
-\def\rail@annofont{#1}
-}
-
-\newcommand\railnamefont[1]{
-\def\rail@namefont{#1}
-}
-
-\newcommand\railindexfont[1]{
-\def\rail@indexfont{#1}
-}
-
-% railroad read/write macros
-%
-% \begin{rail} TEXT \end{rail} : TEXT is written out to the .rai file,
-%                                as \rail@i{NR}{TEXT}. Then the matching
-%                                \rail@o{NR}{FMT} from the .rao file is
-%                                executed (if defined).
-%
-% \railoptions{OPTIONS} : OPTIONS are written out to the .rai file,
-%                         as \rail@p{OPTIONS}.
-%
-% \railterm{IDENT,IDENT,...} : format IDENT as terminals. writes out
-%                              \rail@t{IDENT} to the .rai file
-%
-% \railalias{IDENT}{TEXT} : format IDENT as TEXT. defines \rail@t@IDENT as
-%                           TEXT.
-%
-% \railtoken{IDENT}{TEXT} : abbreviates \railalias{IDENT}{TEXT}\railterm{IDENT}
-%                           (for backward compatibility)
-%
-% \rail@setcodes : guards special characters
-%
-% \rail@makeother{CHARACTER} : sets \catcode of CHARACTER to "other"
-%                              used inside a loop for \rail@setcodes
-%
-% \rail@nr : railroad diagram counter
-%
-% \ifrail@match : current \rail@i{NR}{TEXT} matches
-%
-% \rail@first : actions to be done first. read in .rao file,
-%               open .rai file if \@filesw true, undefine \rail@first.
-%               executed from \begin{rail}, \railoptions and \railterm.
-%
-% \rail@i{NR}{TEXT} : defines \rail@i@NR as TEXT. written to the .rai
-%                     file by \rail, read from the .rao file by
-%                     \rail@first
-%
-% \rail@t{IDENT} : tells Rail that IDENT is to be custom formatted,
-%                  written to the .rai file by \railterm.
-%
-% \rail@o{NR}{TEXT} : defines \rail@o@NR as TEXT, read from the .rao
-%                     file by \rail@first.
-%
-% \rail@p{OPTIONS} : pass options to rail, written to the .rai file by
-%                    \railoptions
-%
-% \rail@write{TEXT} : write TEXT to the .rai file
-%
-% \rail@warn : warn user for mismatching diagrams
-%
-% \rail@endwarn : either \relax or \rail@warn
-%
-% \ifrail@all : checked at the end of the document
-
-\def\rail@makeother#1{
-  \expandafter\catcode\expandafter`\csname\string #1\endcsname=12
-}
-
-\def\rail@setcodes{
-\let\par=\relax
-\let\\=\relax
-\expandafter\@tfor\expandafter\rail@symbol\expandafter:\expandafter=%
-  \the\rail@reserved
-\do{\expandafter\rail@makeother\rail@symbol}
-}
-
-\newcount\rail@nr
-
-\newif\ifrail@all
-\rail@alltrue
-
-\newif\ifrail@match
-
-\def\rail@first{
-\begingroup
-\makeatletter
-\rail@setcodes
-\InputIfFileExists{\jobname.rao}{}{\PackageInfo{rail}{No file \jobname.rao}}
-\makeatother
-\endgroup
-\if@filesw
-\newwrite\tf@rai
-\immediate\openout\tf@rai=\jobname.rai
-\fi
-\global\let\rail@first=\relax
-}
-
-\long\def\rail@body#1\end{
-{
-\newlinechar=`^^J
-\def\par{\string\par^^J}
-\rail@write{\string\rail@i{\number\rail@nr}{#1}}
-}
-\xdef\rail@i@{#1}
-\end
-}
-
-\newenvironment{rail}{
-\global\advance\rail@nr by 1
-\rail@first
-\begingroup
-\rail@setcodes
-\rail@body
-}{
-\endgroup
-\rail@matchtrue
-\@ifundefined{rail@o@\number\rail@nr}{\rail@matchfalse}{}
-\expandafter\ifx\csname rail@i@\number\rail@nr\endcsname\rail@i@
-\else
-\rail@matchfalse
-\fi
-\ifrail@match
-\csname rail@o@\number\rail@nr\endcsname
-\else
-\PackageWarning{rail}{Railroad diagram {\number\rail@nr} doesn't match}
-\global\let\rail@endwarn=\rail@warn
-\begin{list}{}{\rail@param}
-\rail@begin{1}{}
-\rail@setbox{\bfseries ???}
-\rail@oval
-\rail@end
-\end{list}
-\fi
-}
-
-\newcommand\railoptions[1]{
-\rail@first
-\rail@write{\string\rail@p{#1}}
-}
-
-\newcommand\railterm[1]{
-\rail@first
-\@for\rail@@:=#1\do{
-\rail@write{\string\rail@t{\rail@@}}
-}
-}
-
-\newcommand\railalias[2]{
-\expandafter\def\csname rail@t@#1\endcsname{#2}
-}
-
-\newcommand\railtoken[2]{\railalias{#1}{#2}\railterm{#1}}
-
-\long\def\rail@i#1#2{
-\expandafter\gdef\csname rail@i@#1\endcsname{#2}
-}
-
-\def\rail@o#1#2{
-\expandafter\gdef\csname rail@o@#1\endcsname{
-\begin{list}{}{\rail@param}
-#2
-\end{list}
-}
-}
-
-\def\rail@t#1{}
-
-\def\rail@p#1{}
-
-\long\def\rail@write#1{\@ifundefined{tf@rai}{}{\immediate\write\tf@rai{#1}}}
-
-\def\rail@warn{
-\PackageWarningNoLine{rail}{Railroad diagram(s) may have changed.
-                            Use 'rail' and rerun}
-}
-
-\let\rail@endwarn=\relax
-
-\AtEndDocument{\rail@endwarn}
-
-% index entry macro
-%
-% \rail@index{IDENT} : add index entry for IDENT
-
-\def\rail@index#1{
-\index{\rail@indexfont#1}
-}
-
-% railroad formatting primitives
-%
-% \rail@x : current x
-% \rail@y : current y
-% \rail@ex : current end x
-% \rail@sx : starting x for \rail@cr
-% \rail@rx : rightmost previous x for \rail@cr
-%
-% \rail@tmpa : temporary count
-% \rail@tmpb : temporary count
-% \rail@tmpc : temporary count
-%
-% \rail@put : put at (\rail@x,\rail@y)
-% \rail@vput : put vector at (\rail@x,\rail@y)
-%
-% \rail@eline : end line by drawing from \rail@ex to \rail@x
-%
-% \rail@vreline : end line by drawing a vector from \rail@x to \rail@ex
-%
-% \rail@vleline : end line by drawing a vector from \rail@ex to \rail@x
-%
-% \rail@sety{LEVEL} : set \rail@y to level LEVEL
-
-\newcount\rail@x
-\newcount\rail@y
-\newcount\rail@ex
-\newcount\rail@sx
-\newcount\rail@rx
-
-\newcount\rail@tmpa
-\newcount\rail@tmpb
-\newcount\rail@tmpc
-
-\def\rail@put{\put(\number\rail@x,\number\rail@y)}
-
-\def\rail@vput{\put(\number\rail@ex,\number\rail@y)}
-
-\def\rail@eline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@put{\line(-1,0){\number\rail@tmpb}}
-}
-
-\def\rail@vreline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@vput{\vector(1,0){\number\rail@tmpb}}
-}
-
-\def\rail@vleline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@put{\vector(-1,0){\number\rail@tmpb}}
-}
-
-\def\rail@sety#1{
-\rail@y=#1
-\multiply\rail@y by -\rail@boxsp
-\advance\rail@y by -\rail@boxht
-}
-
-% \rail@begin{HEIGHT}{NAME} : begin a railroad diagram of height HEIGHT
-%
-% \rail@end : end a railroad diagram
-%
-% \rail@expand{IDENT} : expand IDENT
-
-\def\rail@begin#1#2{
-\item
-\begin{minipage}[t]{\linewidth}
-\ifx\@empty#2\else
-{\rail@namefont \rail@expand{#2}}\\*[\railnamesep]
-\fi
-\unitlength=\railunit
-\rail@tmpa=#1
-\multiply\rail@tmpa by \rail@boxsp
-\begin{picture}(0,\number\rail@tmpa)(0,-\number\rail@tmpa)
-\rail@ex=0
-\rail@rx=0
-\rail@x=\rail@extra
-\rail@sx=\rail@x
-\rail@sety{0}
-}
-
-\def\rail@end{
-\advance\rail@x by \rail@extra
-\rail@eline
-\end{picture}
-\end{minipage}
-}
-
-\def\rail@vend{
-\advance\rail@x by \rail@extra
-\rail@vreline
-\end{picture}
-\end{minipage}
-}
-
-\def\rail@expand#1{\@ifundefined{rail@t@#1}{#1}{\csname rail@t@#1\endcsname}}
-
-% \rail@token{TEXT}[ANNOT] : format token TEXT with annotation
-% \rail@ltoken{TEXT}[ANNOT] : format token TEXT with annotation, arrow left
-% \rail@rtoken{TEXT}[ANNOT] : format token TEXT with annotation, arrow right
-%
-% \rail@ctoken{TEXT}[ANNOT] : format token TEXT centered with annotation
-% \rail@lctoken{TEXT}[ANNOT] : format token TEXT centered with annotation, arrow left
-% \rail@rctoken{TEXT}[ANNOT] : format token TEXT centered with annotation, arrow right
-%
-% \rail@nont{TEXT}[ANNOT] : format nonterminal TEXT with annotation
-% \rail@lnont{TEXT}[ANNOT] : format nonterminal TEXT with annotation, arrow left
-% \rail@rnont{TEXT}[ANNOT] : format nonterminal TEXT with annotation. arrow right
-%
-% \rail@cnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation
-% \rail@lcnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation,
-%                             arrow left
-% \rail@rcnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation,
-%                             arrow right
-%
-% \rail@term{TEXT}[ANNOT] : format terminal TEXT with annotation
-% \rail@lterm{TEXT}[ANNOT] : format terminal TEXT with annotation, arrow left
-% \rail@rterm{TEXT}[ANNOT] : format terminal TEXT with annotation, arrow right
-%
-% \rail@cterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation
-% \rail@lcterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation, arrow left
-% \rail@rcterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation,
-%                             arrow right
-%
-% \rail@annote[TEXT] : format TEXT as annotation
-
-\def\rail@token#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@oval
-}
-
-\def\rail@ltoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vloval
-}
-
-\def\rail@rtoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vroval
-}
-
-\def\rail@ctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@coval
-}
-
-\def\rail@lctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcoval
-}
-
-\def\rail@rctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcoval
-}
-
-\def\rail@nont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@frame
-}
-
-\def\rail@lnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlframe
-}
-
-\def\rail@rnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrframe
-}
-
-\def\rail@cnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@cframe
-}
-
-\def\rail@lcnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcframe
-}
-
-\def\rail@rcnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcframe
-}
-
-\def\rail@term#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@oval
-}
-
-\def\rail@lterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vloval
-}
-
-\def\rail@rterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vroval
-}
-
-\def\rail@cterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@coval
-}
-
-\def\rail@lcterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcoval
-}
-
-\def\rail@rcterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcoval
-}
-
-\def\rail@annote[#1]{
-\rail@setbox{\rail@annofont #1}
-\rail@text
-}
-
-% \rail@box : temporary box for \rail@oval and \rail@frame
-%
-% \rail@setbox{TEXT} : set \rail@box to TEXT, set \rail@tmpa to width
-%
-% \rail@oval : format \rail@box of width \rail@tmpa inside an oval
-% \rail@vloval : format \rail@box of width \rail@tmpa inside an oval, vector left
-% \rail@vroval : format \rail@box of width \rail@tmpa inside an oval, vector right
-%
-% \rail@coval : same as \rail@oval, but centered between \rail@x and
-%               \rail@mx
-% \rail@vlcoval : same as \rail@oval, but centered between \rail@x and
-%                 \rail@mx, vector left
-% \rail@vrcoval : same as \rail@oval, but centered between \rail@x and
-%                 \rail@mx, vector right
-%
-% \rail@frame : format \rail@box of width \rail@tmpa inside a frame
-% \rail@vlframe : format \rail@box of width \rail@tmpa inside a frame, vector left
-% \rail@vrframe : format \rail@box of width \rail@tmpa inside a frame, vector right
-%
-% \rail@cframe : same as \rail@frame, but centered between \rail@x and
-%                \rail@mx
-% \rail@vlcframe : same as \rail@frame, but centered between \rail@x and
-%                  \rail@mx, vector left
-% \rail@vrcframe : same as \rail@frame, but centered between \rail@x and
-%                  \rail@mx, vector right
-%
-% \rail@text : format \rail@box of width \rail@tmpa above the line
-
-\newbox\rail@box
-
-\def\rail@setbox#1{
-\setbox\rail@box\hbox{\strut#1}
-\rail@tmpa=\wd\rail@box
-\divide\rail@tmpa by \railunit
-}
-
-\def\rail@oval{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@vloval{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-\rail@vleline
-}
-
-\def\rail@vroval{
-\advance\rail@x by \rail@boxlf
-\rail@vreline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@coval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@oval
-}
-
-\def\rail@vlcoval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vloval
-}
-
-\def\rail@vrcoval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vroval
-}
-
-\def\rail@frame{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@vlframe{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-\rail@vleline
-}
-
-\def\rail@vrframe{
-\advance\rail@x by \rail@boxlf
-\rail@vreline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@cframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@frame
-}
-
-\def\rail@vlcframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vlframe
-}
-
-\def\rail@vrcframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vrframe
-}
-
-\def\rail@text{
-\advance\rail@x by \rail@textlf
-\advance\rail@y by \rail@textup
-\rail@put{\box\rail@box}
-\advance\rail@y by -\rail@textup
-\advance\rail@x by \rail@tmpa
-\advance\rail@x by \rail@textrt
-}
-
-% alternatives
-%
-% \rail@jx \rail@jy : current join point
-%
-% \rail@gx \rail@gy \rail@gex \rail@grx : global versions of \rail@x etc,
-%                                         to pass values over group closings
-%
-% \rail@mx : maximum x so far
-%
-% \rail@sy : starting \rail@y for alternatives
-%
-% \rail@jput : put at (\rail@jx,\rail@jy)
-%
-% \rail@joval[PART] : put \oval[PART] with adjust
-
-\newcount\rail@jx
-\newcount\rail@jy
-
-\newcount\rail@gx
-\newcount\rail@gy
-\newcount\rail@gex
-\newcount\rail@grx
-
-\newcount\rail@sy
-\newcount\rail@mx
-
-\def\rail@jput{
-\put(\number\rail@jx,\number\rail@jy)
-}
-
-\def\rail@joval[#1]{
-\advance\rail@jx by \rail@joinadj
-\rail@jput{\oval(\number\rail@joinsz,\number\rail@joinsz)[#1]}
-\advance\rail@jx by -\rail@joinadj
-}
-
-% \rail@barsplit : incoming split for '|'
-%
-% \rail@plussplit : incoming split for '+'
-%
-
-\def\rail@barsplit{
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-}
-
-\def\rail@plussplit{
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by \rail@joinsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-}
-
-% \rail@alt{SPLIT} : start alternatives with incoming split SPLIT
-%
-
-\def\rail@alt#1{
-\rail@sy=\rail@y
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@x by \rail@joinsz
-\rail@mx=0
-\let\rail@list=\@empty
-\let\rail@comma=\@empty
-\let\rail@split=#1
-\begingroup
-\rail@sx=\rail@x
-\rail@rx=0
-}
-
-% \rail@nextalt{FIX}{Y} : start next alternative at vertical position Y
-%                         and fix-up FIX
-%
-
-\def\rail@nextalt#1#2{
-\global\rail@gx=\rail@x
-\global\rail@gy=\rail@y
-\global\rail@gex=\rail@ex
-\global\rail@grx=\rail@rx
-\endgroup
-#1
-\ifnum\rail@gx>\rail@mx\rail@mx=\rail@gx\fi
-\ifnum\rail@grx>\rail@mx\rail@mx=\rail@grx\fi
-\edef\rail@list{\rail@list\rail@comma\number\rail@gex:\number\rail@gy}
-\def\rail@comma{,}
-\rail@split
-\let\rail@split=\@empty
-\rail@sety{#2}
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[bl]
-\advance\rail@jx by -\rail@joinhsz
-\rail@ex=\rail@x
-\begingroup
-\rail@sx=\rail@x
-\rail@rx=0
-}
-
-% \rail@barjoin : outgoing join for first '|' alternative
-%
-% \rail@plusjoin : outgoing join for first '+' alternative
-%
-% \rail@altjoin : join for subsequent alternative
-%
-
-\def\rail@barjoin{
-\ifnum\rail@y<\rail@sy
-\global\rail@gex=\rail@jx
-\else
-\global\rail@gex=\rail@ex
-\fi
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-\ifnum\rail@y<\rail@sy
-\rail@altjoin
-\fi
-}
-
-\def\rail@plusjoin{
-\global\rail@gex=\rail@ex
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by -\rail@joinsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-}
-
-\def\rail@altjoin{
-\rail@eline
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by -\rail@joinhsz
-\rail@joval[br]
-\advance\rail@jx by \rail@joinhsz
-}
-
-% \rail@eltsplit EX:Y; : split EX:Y into \rail@ex \rail@y
-%
-% \rail@endalt{JOIN} : end alternatives with outgoing join JOIN
-
-\def\rail@eltsplit#1:#2;{\rail@ex=#1\rail@y=#2}
-
-\def\rail@endalt#1{
-\global\rail@gx=\rail@x
-\global\rail@gy=\rail@y
-\global\rail@gex=\rail@ex
-\global\rail@grx=\rail@rx
-\endgroup
-\ifnum\rail@gx>\rail@mx\rail@mx=\rail@gx\fi
-\ifnum\rail@grx>\rail@mx\rail@mx=\rail@grx\fi
-\edef\rail@list{\rail@list\rail@comma\number\rail@gex:\number\rail@gy}
-\rail@x=\rail@mx
-\rail@jx=\rail@x
-\rail@jy=\rail@sy
-\advance\rail@jx by \rail@joinsz
-\let\rail@join=#1
-\@for\rail@elt:=\rail@list\do{
-\expandafter\rail@eltsplit\rail@elt;
-\rail@join
-\let\rail@join=\rail@altjoin
-}
-\rail@x=\rail@mx
-\rail@y=\rail@sy
-\rail@ex=\rail@gex
-\advance\rail@x by \rail@joinsz
-}
-
-% \rail@bar : start '|' alternatives
-%
-% \rail@nextbar : next '|' alternative
-%
-% \rail@endbar : end '|' alternatives
-%
-
-\def\rail@bar{
-\rail@alt\rail@barsplit
-}
-
-\def\rail@nextbar{
-\rail@nextalt\relax
-}
-
-\def\rail@endbar{
-\rail@endalt\rail@barjoin
-}
-
-% \rail@plus : start '+' alternatives
-%
-% \rail@nextplus: next '+' alternative
-%
-% \rail@endplus : end '+' alternatives
-%
-
-\def\rail@plus{
-\rail@alt\rail@plussplit
-}
-
-\def\rail@nextplus{
-\rail@nextalt\rail@fixplus
-}
-
-\def\rail@fixplus{
-\ifnum\rail@gy<\rail@sy
-\begingroup
-\rail@x=\rail@gx
-\rail@y=\rail@gy
-\rail@ex=\rail@gex
-\rail@rx=\rail@grx
-\ifnum\rail@x<\rail@rx
-\rail@x=\rail@rx
-\fi
-\rail@eline
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\rail@joval[br]
-\advance\rail@jx by \rail@joinhsz
-\rail@tmpa=\rail@sy
-\advance\rail@tmpa by -\rail@joinhsz
-\advance\rail@tmpa by -\rail@jy
-\rail@jput{\line(0,1){\number\rail@tmpa}}
-\rail@jy=\rail@sy
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jy by \rail@joinhsz
-\global\rail@gx=\rail@jx
-\global\rail@gy=\rail@jy
-\global\rail@gex=\rail@gx
-\global\rail@grx=\rail@rx
-\endgroup
-\fi
-}
-
-\def\rail@endplus{
-\rail@endalt\rail@plusjoin
-}
-
-% \rail@cr{Y} : carriage return to vertical position Y
-
-\def\rail@cr#1{
-\rail@tmpa=\rail@sx
-\advance\rail@tmpa by \rail@joinsz
-\ifnum\rail@x<\rail@tmpa\rail@x=\rail@tmpa\fi
-\rail@eline
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@x by \rail@joinsz
-\ifnum\rail@x>\rail@rx\rail@rx=\rail@x\fi
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-\rail@sety{#1}
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@boxsp
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@boxsp
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by -\rail@joinhsz
-\rail@joval[br]
-\advance\rail@jy by -\rail@joinhsz
-\rail@tmpa=\rail@jx
-\advance\rail@tmpa by -\rail@sx
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(-1,0){\number\rail@tmpa}}
-\rail@jx=\rail@sx
-\advance\rail@jx by \rail@joinhsz
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-\rail@tmpa=\rail@boxsp
-\advance\rail@tmpa by -\rail@joinsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\advance\rail@jy by -\rail@tmpa
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[bl]
-\rail@x=\rail@jx
-\rail@ex=\rail@x
-}
-
-% default setup for Isabelle
-\newenvironment{railoutput}%
-{\begin{list}{}{\rail@param}\def\rail@expand{\relax}\makeatletter}{\makeatother\end{list}}
-
-\def\rail@termfont{\isabellestyle{tt}}
-\def\rail@nontfont{\isabellestyle{it}}
-\def\rail@namefont{\isabellestyle{it}}
diff --git a/virage/src/test/resources/old_theories/output/document/root.bib b/virage/src/test/resources/old_theories/output/document/root.bib
deleted file mode 100644
index 6f429f0..0000000
--- a/virage/src/test/resources/old_theories/output/document/root.bib
+++ /dev/null
@@ -1,71 +0,0 @@
-@InProceedings{adt2019,
-    author    = {Karsten Diekhoff and
-                 Michael Kirsten and
-                 Jonas Kr{\"{a}}mer},
-    editor    = {Sa{\v{s}}a Peke{\v{c}} and
-                 {Kristen Brent} Venable},
-    title     = {Formal Property-Oriented Design of
-                 Voting Rules Using Composable Modules},
-    booktitle = {6th International Conference on Algorithmic
-                 Decision Theory ({ADT} 2019)},
-    series    = {Lecture Notes in Artificial Intelligence},
-    part      = {Short Papers},
-    year      = {2019},
-    abstract  = {Voting rules aggregate multiple individual preferences in order to
-                 make a collective decision. Commonly, these mechanisms are expected
-                 to respect a multitude of different notions of fairness and reliability,
-                 which must be carefully balanced to avoid inconsistencies.
-
-                 We present an approach for the sound and flexible design of voting rules
-                 from composable modules. Formal composition rules guarantee social
-                 choice properties from properties of the individual components. The
-                 approach can be applied to many voting rules from the literature.},
-    publisher = {Springer},
-    volume    = {11834},
-    pages     = {164--166},
-    doi       = {10.1007/978-3-030-31489-7}
-}
-
-@InProceedings{lopstr2019,
-  author        = {Karsten Diekhoff and
-                   Michael Kirsten and
-                   Jonas Kr{\"{a}}mer},
-  title         = {Verified Construction of Fair Voting Rules},
-  booktitle     = {29th International Symposium on Logic-Based Program
-                   Synthesis and Transformation ({LOPSTR} 2019),
-                   Revised Selected Papers},
-  series        = {Lecture Notes in Computer Science},
-  editor        = {Maurizio Gabbrielli},
-  year          = {2020},
-  abstract      = {Voting rules aggregate multiple individual preferences in order to
-                   make collective decisions. Commonly, these mechanisms are expected
-                   to respect a multitude of different fairness and reliability
-                   properties, e.g., to ensure that each voter's ballot accounts for
-                   the same proportion of the elected alternatives, or that a voter
-                   cannot change the election outcome in her favor by insincerely
-                   filling out her ballot. However, no voting rule is fair in all
-                   respects, and trade-off attempts between such properties often
-                   bring out inconsistencies, which makes the construction of arguably
-                   practical and fair voting rules non-trivial and error-prone.
- 
-                   In this paper, we present a formal and systematic approach for the
-                   flexible and verified construction of voting rules from composable
-                   core modules to respect such properties by construction. Formal
-                   composition rules guarantee resulting properties from properties of
-                   the individual components, which are of generic nature to be reused
-                   for various voting rules. We provide a prototypical logic-based
-                   implementation with proofs for a selected set of structures and
-                   composition rules within the theorem prover Isabelle/HOL. The
-                   approach can be readily extended in order to support many voting
-                   rules from the literature by extending the set of basic modules and
-                   composition rules. We exemplarily construct the well-known voting
-                   rule sequential majority comparison (SMC) from simple generic
-                   modules, and automatically produce a formal proof that SMC
-                   satisfies the fairness property monotonicity. Monotonicity is a
-                   well-known social-choice property that is easily violated by voting
-                   rules in practice.},
-  publisher     = {Springer},
-  doi           = {10.1007/978-3-030-45260-5_6},
-  volume        = {12042},
-  pages         = {90--104}
-}
diff --git a/virage/src/test/resources/old_theories/output/document/root.tex b/virage/src/test/resources/old_theories/output/document/root.tex
deleted file mode 100644
index ec9759f..0000000
--- a/virage/src/test/resources/old_theories/output/document/root.tex
+++ /dev/null
@@ -1,56 +0,0 @@
-\documentclass[11pt,a4paper,notitlepage]{report}
-\input{settings}
-\usepackage{isabelle,isabellesym}
-\usepackage{amssymb}
-\usepackage{pdfsetup}
-
-\urlstyle{rm}
-\isabellestyle{it}
-
-% for uniform font size
-%\renewcommand{\isastyle}{\isastyleminor}
-
-% This is required due to some unusual behaviour by Isabelle.
-% It should be included in isabellesym.sty automatically, but it is not.
-\newcommand{\isasymcirclearrowleft}{\isamath{\circlearrowleft}}
-
-\begin{document}
-
-\title{Verified Construction of Fair Voting Rules}
-\author{Michael Kirsten}
-\institute{Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany\\
-\email{kirsten@kit.edu}}
-\maketitle
-
-\begin{abstract}
-Voting rules aggregate multiple individual preferences in order to make a
-collective decision.
-Commonly, these mechanisms are expected to respect a multitude of different
-notions of fairness and reliability, which must be carefully balanced to avoid
-inconsistencies.
-
-This article contains a formalisation of a framework for the construction of
-such fair voting rules using composable modules~\cite{adt2019,lopstr2019}.
-The framework is a formal and systematic approach for the flexible and verified
-construction of voting rules from individual composable modules to respect such
-social-choice properties by construction.
-Formal composition rules guarantee resulting social-choice properties from
-properties of the individual components which are of generic nature to be
-reused for various voting rules.
-We provide proofs for a selected set of structures and composition rules.
-The approach can be readily extended in order to support more voting rules,
-e.g., from the literature by extending the sets of modules and composition
-rules.
-\end{abstract}
-
-\tableofcontents
-
-\parindent 0pt\parskip 0.5ex
-
-\input{session}
-
-% optional bibliography
-\bibliographystyle{abbrv}
-\bibliography{root}
-
-\end{document}
diff --git a/virage/src/test/resources/old_theories/output/document/session.tex b/virage/src/test/resources/old_theories/output/document/session.tex
deleted file mode 100644
index 43d554b..0000000
--- a/virage/src/test/resources/old_theories/output/document/session.tex
+++ /dev/null
@@ -1,36 +0,0 @@
-\input{Preference_Relation.tex}
-\input{Result.tex}
-\input{Profile.tex}
-\input{Electoral_Module.tex}
-\input{Evaluation_Function.tex}
-\input{Elimination_Module.tex}
-\input{Aggregator.tex}
-\input{Maximum_Aggregator.tex}
-\input{Termination_Condition.tex}
-\input{Defer_Equal_Condition.tex}
-\input{Defer_Module.tex}
-\input{Drop_Module.tex}
-\input{Pass_Module.tex}
-\input{Elect_Module.tex}
-\input{Plurality_Module.tex}
-\input{Borda_Module.tex}
-\input{Condorcet_Module.tex}
-\input{Copeland_Module.tex}
-\input{Minimax_Module.tex}
-\input{Drop_And_Pass_Compatibility.tex}
-\input{Revision_Composition.tex}
-\input{Sequential_Composition.tex}
-\input{Parallel_Composition.tex}
-\input{Loop_Composition.tex}
-\input{Maximum_Parallel_Composition.tex}
-\input{Elect_Composition.tex}
-\input{Defer_One_Loop_Composition.tex}
-\input{Borda_Rule.tex}
-\input{Pairwise_Majority_Rule.tex}
-\input{Copeland_Rule.tex}
-\input{Minimax_Rule.tex}
-\input{Blacks_Rule.tex}
-\input{Nanson_Baldwin_Rule.tex}
-\input{Classic_Nanson_Rule.tex}
-\input{Schwartz_Rule.tex}
-\input{Sequential_Majority_Comparison.tex}
diff --git a/virage/src/test/resources/old_theories/output/document/settings.tex b/virage/src/test/resources/old_theories/output/document/settings.tex
deleted file mode 100644
index 9a1f927..0000000
--- a/virage/src/test/resources/old_theories/output/document/settings.tex
+++ /dev/null
@@ -1,30 +0,0 @@
-\usepackage[T1]{fontenc}
-\usepackage{lmodern}
-\usepackage{sectsty}
-\usepackage{xpatch}
-\usepackage{upquote}
-
-\makeatletter
-\xpatchcmd{\@maketitle}
-  {\@title}
-{%\vspace{-2.45\baselineskip}
-\bfseries\@title\vspace{.25\baselineskip}}
-  {}{}
-\makeatother
-
-\makeatletter
-\providecommand{\institute}[1]{%
-    \apptocmd{\@author}{\end{tabular}%
-    \vspace{.5\baselineskip}\par
-    \begin{tabular}[t]{c}
-    #1}{}{}
-}
-\makeatother
-
-\newcommand{\email}[1]{%
-    \texttt{\href{mailto:#1}{\color{black}{#1}}}%
-}
-
-\pretocmd{\tableofcontents}{\sectionfont{\Huge}}{}{}
-%\let\clearpage\relax
-\apptocmd{\tableofcontents}{\sectionfont{\Large}}{}{}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Plurality_Module.thy b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Plurality_Module.thy
index 65d7b0a..8cff2e6 100644
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Plurality_Module.thy
+++ b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Plurality_Module.thy
@@ -59,23 +59,4 @@ proof -
     by (simp add: electoral_modI)
 qed
 
-ML \<open>
-    (* val w = (Defs.all_specifications_of (Theory.defs_of @{theory}))
-    val x = filter (fn x => String.isPrefix "Seq" x) (map (fn x => snd (fst x))
-        (Defs.all_specifications_of (Theory.defs_of @{theory}))) *)
-    val y = filter (fn x => (String.isPrefix ("Plurality_Module.plurality") (snd (fst (x))))) 
-        (Defs.all_specifications_of (Theory.defs_of @{theory}))
-    val z = filter (fn x => String.isPrefix ("Plurality_Module.plurality_def") (#description x)) (map (fn x => (hd (snd x))) y)
-
-    val def = hd z
-    val def_rhs = #rhs def
-
-    val def_funs = filter (fn x => String.isPrefix ("fun") (snd (fst x))) (def_rhs)
-    val def_consts = filter (fn x => not (String.isPrefix ("fun") (snd (fst x)))) (def_rhs)
-
-    val fun_type = Defs.pretty_entry (Defs.global_context @{theory}) (List.last def_funs)
-
-   (* val a = Defs.pretty_entry (Defs.global_context @{theory}) (nth (snd (#lhs (hd z), #rhs (hd z)) )2) *)
- \<close>
-
 end
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/#Preference_Relation.thy# b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/#Preference_Relation.thy#
deleted file mode 100644
index aae5f7a..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/#Preference_Relation.thy#
+++ /dev/null
@@ -1,1115 +0,0 @@
-(*  File:       Preference_Relation.thy
-    Copyright   2021  Karlsruhe Institute of Technology (KIT)
-*)
-\<^marker>\<open>creator "Jonas Kraemer, Karlsruhe Institute of Technology (KIT)"\<close>
-\<^marker>\<open>contributor "Karsten Diekhoff, Karlsruhe Institute of Technology (KIT)"\<close>
-\<^marker>\<open>contributor "Michael Kirsten, Karlsruhe Institute of Technology (KIT)"\<close>
-
-chapter \<open>Social-Choice Types\<close>
-
-section \<open>Preference Relation\<close>
-
-theory Preference_Relation
-  imports Main
-begin
-
-text
-\<open>The very core of the composable modules voting framework: types and
-functions, derivations, lemmata, operations on preference relations, etc.\<close>
-
-subsection \<open>Definition\<close>
-
-type_synonym Alternative = Nat.nat
-
-(*
-  Each voter expresses pairwise relations between all alternatives,
-  thereby inducing a linear order.
-*)
-type_synonym Preference_Relation = "Alternative rel"
-
-fun is_less_preferred_than ::
-  "Alternative \<Rightarrow> Preference_Relation \<Rightarrow> Alternative \<Rightarrow> bool" ("_ \<preceq>\<^sub>_ _" [50, 1000, 51] 50) where
-    "x \<preceq>\<^sub>r y = ((x, y) \<in> r)"
-
-lemma lin_imp_antisym:
-  assumes "linear_order_on A r"
-  shows "antisym r"
-  using assms linear_order_on_def partial_order_on_def
-  by auto
-
-lemma lin_imp_trans:
-  assumes "linear_order_on A r"
-  shows "trans r"
-  using assms order_on_defs
-  by blast
-
-subsection \<open>Ranking\<close>
-
-fun rank :: "'a Preference_Relation \<Rightarrow> 'a \<Rightarrow> nat" where
-  "rank r x = card (above r x)"
-
-lemma rank_gt_zero:
-  assumes
-    refl: "x \<preceq>\<^sub>r x" and
-    fin:  "finite r"
-  shows "rank r x \<ge> 1"
-proof -
-  have "x \<in> {y \<in> Field r. (x, y) \<in> r}"
-    using FieldI2 refl
-    by fastforce
-  hence "{y \<in> Field r. (x, y) \<in> r} \<noteq> {}"
-    by blast
-  hence "card {y \<in> Field r. (x, y) \<in> r} \<noteq> 0"
-    by (simp add: fin finite_Field)
-  moreover have "card{y \<in> Field r. (x, y) \<in> r} \<ge> 0"
-    using fin
-    by auto
-  ultimately show ?thesis
-    using Collect_cong FieldI2 above_def
-          less_one not_le_imp_less rank.elims
-    by (metis (no_types, lifting))
-qed
-
-subsection \<open>Limited Preference\<close>
-
-definition limited :: "'a set \<Rightarrow> 'a Preference_Relation \<Rightarrow> bool" where
-  "limited A r \<equiv> r \<subseteq> A \<times> A"
-
-lemma limitedI:
-  "(\<And>x y. \<lbrakk> x \<preceq>\<^sub>r y \<rbrakk> \<Longrightarrow>  x \<in> A \<and> y \<in> A) \<Longrightarrow> limited A r"
-  unfolding limited_def
-  by auto
-
-lemma limited_dest:
-  "(\<And>x y. \<lbrakk> x \<preceq>\<^sub>r y; limited A r \<rbrakk> \<Longrightarrow>  x \<in> A \<and> y \<in> A)"
-  unfolding limited_def
-  by auto
-
-fun limit :: "'a set \<Rightarrow> 'a Preference_Relation \<Rightarrow> 'a Preference_Relation" where
-  "limit A r = {(a, b) \<in> r. a \<in> A \<and> b \<in> A}"
-
-definition connex :: "'a set \<Rightarrow> 'a Preference_Relation \<Rightarrow> bool" where
-  "connex A r \<equiv> limited A r \<and> (\<forall>x \<in> A. \<forall>y \<in> A. x \<preceq>\<^sub>r y \<or> y \<preceq>\<^sub>r x)"
-
-lemma connex_imp_refl:
-  assumes "connex A r"
-  shows "refl_on A r"
-proof
-  show "r \<subseteq> A \<times> A"
-    using assms connex_def limited_def
-    by metis
-next
-  fix
-    x :: "'a"
-  assume
-    x_in_A: "x \<in> A"
-  have "x \<preceq>\<^sub>r x"
-    using assms connex_def x_in_A
-    by metis
-  thus "(x, x) \<in> r"
-    by simp
-qed
-
-lemma lin_ord_imp_connex:
-  assumes "linear_order_on A r"
-  shows "connex A r"
-  unfolding connex_def limited_def
-proof (safe)
-  fix
-    a :: "'a" and
-    b :: "'a"
-  assume
-    asm1: "(a, b) \<in> r"
-  show "a \<in> A"
-    using asm1 assms partial_order_onD(1)
-          order_on_defs(3) refl_on_domain
-    by metis
-next
-  fix
-    a :: "'a" and
-    b :: "'a"
-  assume
-    asm1: "(a, b) \<in> r"
-  show "b \<in> A"
-    using asm1 assms partial_order_onD(1)
-          order_on_defs(3) refl_on_domain
-    by metis
-next
-  fix
-    x :: "'a" and
-    y :: "'a"
-  assume
-    asm1: "x \<in> A" and
-    asm2: "y \<in> A" and
-    asm3: "\<not> y \<preceq>\<^sub>r x"
-  have "(y, x) \<notin> r"
-    using asm3
-    by simp
-  hence "(x, y) \<in> r"
-    using asm1 asm2 assms partial_order_onD(1)
-          linear_order_on_def refl_onD total_on_def
-    by metis
-  thus "x \<preceq>\<^sub>r y"
-    by simp
-qed
-
-lemma connex_antsym_and_trans_imp_lin_ord:
-  assumes
-    connex_r: "connex A r" and
-    antisym_r: "antisym r" and
-    trans_r: "trans r"
-  shows "linear_order_on A r"
-  unfolding connex_def linear_order_on_def partial_order_on_def
-            preorder_on_def refl_on_def total_on_def
-proof (safe)
-  fix
-    a :: "'a" and
-    b :: "'a"
-  assume
-    asm1: "(a, b) \<in> r"
-  show "a \<in> A"
-    using asm1 connex_r refl_on_domain connex_imp_refl
-    by metis
-next
-  fix
-    a :: "'a" and
-    b :: "'a"
-  assume
-    asm1: "(a, b) \<in> r"
-  show "b \<in> A"
-    using asm1 connex_r refl_on_domain connex_imp_refl
-    by metis
-next
-  fix
-    x :: "'a"
-  assume
-    asm1: "x \<in> A"
-  show "(x, x) \<in> r"
-    using asm1 connex_r connex_imp_refl refl_onD
-  by metis
-next
-  show "trans r"
-    using trans_r
-    by simp
-next
-  show "antisym r"
-    using antisym_r
-    by simp
-next
-  fix
-    x :: "'a" and
-    y :: "'a"
-  assume
-    asm1: "x \<in> A" and
-    asm2: "y \<in> A" and
-    asm3: "x \<noteq> y" and
-    asm4: "(y, x) \<notin> r"
-  have "x \<preceq>\<^sub>r y \<or> y \<preceq>\<^sub>r x"
-    using asm1 asm2 connex_r connex_def
-    by metis
-  hence "(x, y) \<in> r \<or> (y, x) \<in> r"
-    by simp
-  thus "(x, y) \<in> r"
-    using asm4
-    by metis
-qed
-
-lemma limit_to_limits: "limited A (limit A r)"
-  unfolding limited_def
-  by auto
-
-lemma limit_presv_connex:
-  assumes
-    connex: "connex S r" and
-    subset: "A \<subseteq> S"
-  shows "connex A (limit A r)"
-  unfolding connex_def limited_def
-proof (simp, safe)
-  let ?s = "{(a, b). (a, b) \<in> r \<and> a \<in> A \<and> b \<in> A}"
-  fix
-    x :: "'a" and
-    y :: "'a" and
-    a :: "'a" and
-    b :: "'a"
-  assume
-    asm1: "x \<in> A" and
-    asm2: "y \<in> A" and
-    asm3: "(y, x) \<notin> r"
-  have "y \<preceq>\<^sub>r x \<or> x \<preceq>\<^sub>r y"
-    using asm1 asm2 connex connex_def in_mono subset
-    by metis
-  hence
-    "x \<preceq>\<^sub>?s y \<or> y \<preceq>\<^sub>?s x"
-    using asm1 asm2
-    by auto
-  hence "x \<preceq>\<^sub>?s y"
-    using asm3
-    by simp
-  thus "(x, y) \<in> r"
-    by simp
-qed
-
-lemma limit_presv_antisym:
-  assumes
-    antisymmetric: "antisym r" and
-    subset: "A \<subseteq> S"
-  shows "antisym (limit A r)"
-  using antisym_def antisymmetric
-  by auto
-
-lemma limit_presv_trans:
-  assumes
-    transitive: "trans r" and
-    subset:     "A \<subseteq> S"
-  shows "trans (limit A r)"
-  unfolding trans_def
-proof (simp, safe)
-  fix
-    x :: "'a" and
-    y :: "'a" and
-    z :: "'a"
-  assume
-    asm1: "(x, y) \<in> r" and
-    asm2: "x \<in> A" and
-    asm3: "y \<in> A" and
-    asm4: "(y, z) \<in> r" and
-    asm5: "z \<in> A"
-  show  "(x, z) \<in> r"
-    using asm1 asm4 transE transitive
-    by metis
-qed
-
-lemma limit_presv_lin_ord:
-  assumes
-    "linear_order_on S r" and
-      "A \<subseteq> S"
-    shows "linear_order_on A (limit A r)"
-  using assms connex_antsym_and_trans_imp_lin_ord
-            limit_presv_antisym limit_presv_connex
-            limit_presv_trans lin_ord_imp_connex
-            order_on_defs(1) order_on_defs(2)
-            order_on_defs(3)
-  by metis
-
-lemma limit_presv_prefs1:
-  assumes
-    x_less_y: "x \<preceq>\<^sub>r y" and
-    x_in_A: "x \<in> A" and
-    y_in_A: "y \<in> A"
-  shows "let s = limit A r in x \<preceq>\<^sub>s y"
-  using x_in_A x_less_y y_in_A
-  by simp
-
-lemma limit_presv_prefs2:
-  assumes x_less_y: "(x, y) \<in> limit A r"
-  shows "x \<preceq>\<^sub>r y"
-  using mem_Collect_eq x_less_y
-  by auto
-
-lemma limit_trans:
-  assumes
-    "B \<subseteq> A" and
-    "C \<subseteq> B" and
-    "linear_order_on A r"
-  shows "limit C r = limit C (limit B r)"
-  using assms
-  by auto
-
-lemma lin_ord_not_empty:
-  assumes "r \<noteq> {}"
-  shows "\<not> linear_order_on {} r"
-  using assms connex_imp_refl lin_ord_imp_connex
-        refl_on_domain subrelI
-  by fastforce
-
-lemma lin_ord_singleton:
-  "\<forall>r. linear_order_on {a} r \<longrightarrow> r = {(a, a)}"
-proof
-  fix r :: "'a Preference_Relation"
-  show "linear_order_on {a} r \<longrightarrow> r = {(a, a)}"
-  proof
-    assume asm: "linear_order_on {a} r"
-    hence "a \<preceq>\<^sub>r a"
-      using connex_def lin_ord_imp_connex singletonI
-      by metis
-    moreover have "\<forall>(x, y) \<in> r. x = a \<and> y = a"
-      using asm connex_imp_refl lin_ord_imp_connex
-            refl_on_domain split_beta
-      by fastforce
-    ultimately show "r = {(a, a)}"
-      by auto
-  qed
-qed
-
-subsection \<open>Auxiliary Lemmata\<close>
-
-lemma above_trans:
-  assumes
-    "trans r" and
-    "(a, b) \<in> r"
-  shows "above r b \<subseteq> above r a"
-  using Collect_mono above_def assms transE
-  by metis
-
-lemma above_refl:
-  assumes
-    "refl_on A r" and
-     "a \<in> A"
-  shows "a \<in> above r a"
-  using above_def assms refl_onD
-  by fastforce
-
-lemma above_subset_geq_one:
-  assumes
-    "linear_order_on A r \<and> linear_order_on A s" and
-    "above r a \<subseteq> above s a" and
-    "above s a = {a}"
-  shows "above r a = {a}"
-  using above_def assms connex_imp_refl above_refl insert_absorb
-        lin_ord_imp_connex mem_Collect_eq refl_on_domain
-        singletonI subset_singletonD
-  by metis
-
-lemma above_connex:
-  assumes
-    "connex A r" and
-    "a \<in> A"
-  shows "a \<in> above r a"
-  using assms connex_imp_refl above_refl
-  by metis
-
-lemma pref_imp_in_above: "a \<preceq>\<^sub>r b \<longleftrightarrow> b \<in> above r a"
-  by (simp add: above_def)
-
-lemma limit_presv_above:
-  assumes
-    "b \<in> above r a" and
-    (*"linear_order_on A r" and
-    "B \<subseteq> A" and ??? *)
-    "a \<in> B \<and> b \<in> B"
-  shows "b \<in> above (limit B r) a"
-  using pref_imp_in_above assms limit_presv_prefs1
-  by metis
-
-lemma limit_presv_above2:
-  assumes
-    "b \<in> above (limit B r) a" and
-    "linear_order_on A r" and
-    "B \<subseteq> A" and
-    "a \<in> B" and
-    "b \<in> B"
-  shows "b \<in> above r a"
-  unfolding above_def
-  using above_def assms(1) limit_presv_prefs2
-        mem_Collect_eq pref_imp_in_above
-  by metis
-
-lemma above_one:
-  assumes
-    "linear_order_on A r" and
-    "finite A \<and> A \<noteq> {}"
-  shows "\<exists>a\<in>A. above r a = {a} \<and> (\<forall>x\<in>A. above r x = {x} \<longrightarrow> x = a)"
-proof -
-  obtain n::nat where n: "n+1 = card A"
-    using Suc_eq_plus1 antisym_conv2 assms(2) card_eq_0_iff
-          gr0_implies_Suc le0
-    by metis
-  have
-    "(linear_order_on A r \<and> finite A \<and> A \<noteq> {} \<and> n+1 = card A)
-          \<longrightarrow> (\<exists>a. a\<in>A \<and> above r a = {a})"
-  proof (induction n arbitrary: A r)
-    case 0
-    show ?case
-    proof
-      assume asm: "linear_order_on A r \<and> finite A \<and> A \<noteq> {} \<and> 0+1 = card A"
-      then obtain a where "{a} = A"
-        using card_1_singletonE add.left_neutral
-        by metis
-      hence "a \<in> A \<and> above r a = {a}"
-        using above_def asm connex_imp_refl above_refl
-              lin_ord_imp_connex refl_on_domain
-        by fastforce
-      thus "\<exists>a. a\<in>A \<and> above r a = {a}"
-        by auto
-    qed
-  next
-    case (Suc n)
-    show ?case
-    proof
-      assume asm:
-        "linear_order_on A r \<and> finite A \<and> A \<noteq> {} \<and> Suc n+1 = card A"
-      then obtain B where B: "card B = n+1 \<and> B \<subseteq> A"
-        using Suc_inject add_Suc card.insert_remove finite.cases
-              insert_Diff_single subset_insertI
-        by (metis (mono_tags, lifting))
-      then obtain a where a: "{a} = A - B"
-        using Suc_eq_plus1 add_diff_cancel_left' asm card_1_singletonE
-              card_Diff_subset finite_subset
-        by metis
-      have "\<exists>b\<in>B. above (limit B r) b = {b}"
-        using B One_nat_def Suc.IH add_diff_cancel_left' asm
-              card_eq_0_iff diff_le_self finite_subset leD lessI
-              limit_presv_lin_ord
-        by metis
-      then obtain b where b: "above (limit B r) b = {b}"
-        by blast
-      show "\<exists>a. a \<in> A \<and> above r a = {a}"
-      proof cases
-        assume
-          asm1: "a \<preceq>\<^sub>r b"
-        have f1:
-          "\<forall>A r a aa.
-            \<not> refl_on A r \<or> (a::'a, aa) \<notin> r \<or> a \<in> A \<and> aa \<in> A"
-          using refl_on_domain
-          by metis
-        have f2:
-          "\<forall>A r. \<not> connex (A::'a set) r \<or> refl_on A r"
-          using connex_imp_refl
-          by metis
-        have f3:
-          "\<forall>A r. \<not> linear_order_on (A::'a set) r \<or> connex A r"
-          by (simp add: lin_ord_imp_connex)
-        hence "refl_on A r"
-          using f2 asm
-          by metis
-        hence "a \<in> A \<and> b \<in> A"
-          using f1 asm1
-          by simp
-        hence f4:
-          "\<forall>a. a \<notin> A \<or> b = a \<or> (b, a) \<in> r \<or> (a, b) \<in> r"
-          using asm order_on_defs(3) total_on_def
-          by metis
-        have f5:
-          "(b, b) \<in> limit B r"
-          using above_def b mem_Collect_eq singletonI
-          by metis
-        have f6:
-          "\<forall>a A Aa. (a::'a) \<notin> A - Aa \<or> a \<in> A \<and> a \<notin> Aa"
-          by simp
-        have ff1:
-          "{a. (b, a) \<in> limit B r} = {b}"
-          using above_def b
-          by (metis (no_types))
-        have ff2:
-          "(b, b) \<in> {(aa, a). (aa, a) \<in> r \<and> aa \<in> B \<and> a \<in> B}"
-          using f5
-          by simp
-        moreover have b_wins_B:
-          "\<forall>x \<in> B. b \<in> above r x"
-          using B above_def f4 ff1 ff2 CollectI
-                Product_Type.Collect_case_prodD
-          by fastforce
-        moreover have "b \<in> above r a"
-          using asm1 pref_imp_in_above
-          by metis
-        ultimately have b_wins:
-          "\<forall>x \<in> A. b \<in> above r x"
-          using Diff_iff a empty_iff insert_iff
-          by (metis (no_types))
-        hence "\<forall>x \<in> A. x \<in> above r b \<longrightarrow> x = b"
-          using CollectD above_def antisym_def asm lin_imp_antisym
-          by metis
-        hence "\<forall>x \<in> A. x \<in> above r b \<longleftrightarrow> x = b"
-          using b_wins
-          by blast
-        moreover have above_b_in_A: "above r b \<subseteq> A"
-          using above_def asm connex_imp_refl lin_ord_imp_connex
-                mem_Collect_eq refl_on_domain subsetI
-          by metis
-        ultimately have "above r b = {b}"
-          using above_def b
-          by fastforce
-        thus ?thesis
-          using above_b_in_A
-          by blast
-      next
-        assume "\<not>a \<preceq>\<^sub>r b"
-        hence b_smaller_a: "b \<preceq>\<^sub>r a"
-          using B DiffE a asm b limit_to_limits connex_def
-                limited_dest singletonI subset_iff
-                lin_ord_imp_connex pref_imp_in_above
-          by metis
-        hence b_smaller_a_0: "(b, a) \<in> r"
-          by simp
-        have g1:
-          "\<forall>A r Aa.
-            \<not> linear_order_on (A::'a set) r \<or>
-              \<not> Aa \<subseteq> A \<or>
-              linear_order_on Aa (limit Aa r)"
-          using limit_presv_lin_ord
-          by metis
-        have
-          "{a. (b, a) \<in> limit B r} = {b}"
-          using above_def b
-          by metis
-        hence g2: "b \<in> B"
-          by auto
-        have g3:
-          "partial_order_on B (limit B r) \<and> total_on B (limit B r)"
-          using g1 B asm order_on_defs(3)
-          by metis
-        have
-          "\<forall>A r.
-            total_on A r = (\<forall>a. (a::'a) \<notin> A \<or>
-              (\<forall>aa. (aa \<notin> A \<or> a = aa) \<or> (a, aa) \<in> r \<or> (aa, a) \<in> r))"
-          using total_on_def
-          by metis
-        hence
-          "\<forall>a. a \<notin> B \<or>
-            (\<forall>aa. aa \<notin> B \<or> a = aa \<or>
-                (a, aa) \<in> limit B r \<or> (aa, a) \<in> limit B r)"
-          using g3
-          by simp
-        have "\<forall>x \<in> B. b \<in> above r x"
-          using limit_presv_above2 B pref_imp_in_above asm b above_def
-                limit_presv_lin_ord order_on_defs(3) singletonD
-                singletonI total_on_def mem_Collect_eq g2
-          by (smt (verit, ccfv_threshold))
-        hence b_wins2:
-          "\<forall>x \<in> B. x \<preceq>\<^sub>r b"
-          by (simp add: above_def)
-        hence b_wins2_0:
-          "\<forall>x \<in> B. (x, b) \<in> r"
-          by simp
-        have "trans r"
-          using asm lin_imp_trans
-          by metis
-        hence "\<forall>x \<in> B. (x, a) \<in> r"
-          using transE b_smaller_a_0 b_wins2_0
-          by metis
-        hence "\<forall>x \<in> B. x \<preceq>\<^sub>r a"
-          by simp
-        hence nothing_above_a: "\<forall>x \<in> A. x \<preceq>\<^sub>r a"
-          using a asm lin_ord_imp_connex above_connex Diff_iff
-                empty_iff insert_iff pref_imp_in_above
-          by metis
-        have "\<forall>x \<in> A. x \<in> above r a \<longleftrightarrow> x = a"
-          using antisym_def asm lin_imp_antisym
-                nothing_above_a pref_imp_in_above
-                CollectD above_def
-          by metis
-        moreover have above_a_in_A: "above r a \<subseteq> A"
-          using above_def asm connex_imp_refl lin_ord_imp_connex
-                mem_Collect_eq refl_on_domain
-          by fastforce
-        ultimately have "above r a = {a}"
-          using above_def a
-          by auto
-        thus ?thesis
-          using above_a_in_A
-          by blast
-      qed
-    qed
-  qed
-  hence "\<exists>a. a\<in>A \<and> above r a = {a}"
-    using assms n
-    by blast
-  thus ?thesis
-    using Diff_eq_empty_iff above_trans assms(1) empty_Diff insertE
-          insert_Diff_if insert_absorb insert_not_empty order_on_defs(1)
-          order_on_defs(2) order_on_defs(3) total_on_def
-    by (smt (verit, ccfv_SIG))
-qed
-
-lemma above_one2:
-  assumes
-    lin_ord: "linear_order_on A r" and
-    fin_not_emp: "finite A \<and> A \<noteq> {}" and
-    above1: "above r a = {a} \<and> above r b = {b}"
-  shows "a = b"
-proof -
-  have "a \<preceq>\<^sub>r a \<and> b \<preceq>\<^sub>r b"
-    using above1 singletonI pref_imp_in_above
-    by metis
-  also have
-    "\<exists>a\<in>A. above r a = {a} \<and>
-      (\<forall>x\<in>A. above r x = {x} \<longrightarrow> x = a)"
-    using lin_ord fin_not_emp
-    by (simp add: above_one)
-  moreover have "connex A r"
-    using lin_ord
-    by (simp add: lin_ord_imp_connex)
-  ultimately show "a = b"
-    using above1 connex_def limited_dest
-    by metis
-qed
-
-lemma above_presv_limit:
-  assumes "linear_order r"
-  shows "above (limit A r) x \<subseteq> A"
-  unfolding above_def
-  by auto
-
-subsection \<open>Lifting Property\<close>
-
-definition equiv_rel_except_a :: "'a set \<Rightarrow> 'a Preference_Relation \<Rightarrow>
-                                    'a Preference_Relation \<Rightarrow> 'a \<Rightarrow> bool" where
-  "equiv_rel_except_a A r s a \<equiv>
-    linear_order_on A r \<and> linear_order_on A s \<and> a \<in> A \<and>
-    (\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y)"
-
-definition lifted :: "'a set \<Rightarrow> 'a Preference_Relation \<Rightarrow>
-                        'a Preference_Relation \<Rightarrow> 'a \<Rightarrow> bool" where
-  "lifted A r s a \<equiv>
-    equiv_rel_except_a A r s a \<and> (\<exists>x \<in> A - {a}. a \<preceq>\<^sub>r x \<and> x \<preceq>\<^sub>s a)"
-
-lemma trivial_equiv_rel:
-  assumes order: "linear_order_on A p"
-  shows "\<forall>a \<in> A. equiv_rel_except_a A p p a"
-  by (simp add: equiv_rel_except_a_def order)
-
-lemma lifted_imp_equiv_rel_except_a:
-  assumes lifted: "lifted A r s a"
-  shows "equiv_rel_except_a A r s a"
-proof -
-  from lifted have
-    "linear_order_on A r \<and> linear_order_on A s \<and> a \<in> A \<and>
-      (\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y)"
-    by (simp add: lifted_def equiv_rel_except_a_def)
-  thus ?thesis
-    by (simp add: equiv_rel_except_a_def)
-qed
-
-lemma lifted_mono:
-  assumes lifted: "lifted A r s a"
-  shows "\<forall>x \<in> A - {a}. \<not>(x \<preceq>\<^sub>r a \<and> a \<preceq>\<^sub>s x)"
-proof (safe)
-  fix
-    x :: "'a"
-  assume
-    x_in_A:   "x \<in> A" and
-    x_exist:  "x \<notin> {}" and
-    x_neq_a:  "x \<noteq> a" and
-    x_pref_a: "x \<preceq>\<^sub>r a" and
-    a_pref_x: "a \<preceq>\<^sub>s x"
-  from x_pref_a
-  have x_pref_a_0: "(x, a) \<in> r"
-    by simp
-  from a_pref_x
-  have a_pref_x_0: "(a, x) \<in> s"
-    by simp
-  have "antisym r"
-    using equiv_rel_except_a_def lifted
-          lifted_imp_equiv_rel_except_a
-          lin_imp_antisym
-    by metis
-  hence antisym_r:
-    "(\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
-    using antisym_def
-    by metis
-  hence imp_x_eq_a_0:
-    "\<lbrakk>(x, a) \<in> r; (a, x) \<in> r\<rbrakk> \<Longrightarrow> x = a"
-    by simp
-  have lift_ex: "\<exists>x \<in> A - {a}. a \<preceq>\<^sub>r x \<and> x \<preceq>\<^sub>s a"
-    using lifted lifted_def
-    by metis
-  from lift_ex obtain y :: 'a where
-    f1: "y \<in> A - {a} \<and> a \<preceq>\<^sub>r y \<and> y \<preceq>\<^sub>s a"
-    by metis
-  hence f1_0:
-    "y \<in> A - {a} \<and> (a, y) \<in> r \<and> (y, a) \<in> s"
-    by simp
-  have f2:
-    "equiv_rel_except_a A r s a"
-    using lifted lifted_def
-    by metis
-  hence f2_0:
-    "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y"
-    using equiv_rel_except_a_def
-    by metis
-  hence f2_1:
-    "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. (x, y) \<in> r \<longleftrightarrow> (x, y) \<in> s"
-    by simp
-  have trans: "\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r"
-    using f2 equiv_rel_except_a_def linear_order_on_def
-          partial_order_on_def preorder_on_def trans_def
-    by metis
-  have x_pref_y_0: "(x, y) \<in> s"
-    using equiv_rel_except_a_def f1_0 f2 f2_1 insertE
-          insert_Diff x_in_A x_neq_a x_pref_a_0 trans
-    by metis
-  have a_pref_y_0: "(a, y) \<in> s"
-    using a_pref_x_0 imp_x_eq_a_0 x_neq_a x_pref_a_0
-          equiv_rel_except_a_def f2 lin_imp_trans
-          transE x_pref_y_0
-    by metis
-  show "False"
-    using a_pref_y_0 antisymD equiv_rel_except_a_def
-          DiffD2 f1_0 f2 lin_imp_antisym singletonI
-    by metis
-qed
-
-lemma lifted_mono2:
-  assumes
-    lifted: "lifted A r s a" and
-    x_pref_a: "x \<preceq>\<^sub>r a"
-  shows "x \<preceq>\<^sub>s a"
-proof (simp)
-  have x_pref_a_0: "(x, a) \<in> r"
-    using x_pref_a
-    by simp
-  have x_in_A: "x \<in> A"
-    using connex_imp_refl equiv_rel_except_a_def
-          lifted lifted_def lin_ord_imp_connex
-          refl_on_domain x_pref_a_0
-    by metis
-  have "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y"
-    using lifted lifted_def equiv_rel_except_a_def
-    by metis
-  hence rest_eq:
-    "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. (x, y) \<in> r \<longleftrightarrow> (x, y) \<in> s"
-    by simp
-  have "\<exists>x \<in> A - {a}. a \<preceq>\<^sub>r x \<and> x \<preceq>\<^sub>s a"
-    using lifted lifted_def
-    by metis
-  hence ex_lifted:
-    "\<exists>x \<in> A - {a}. (a, x) \<in> r \<and> (x, a) \<in> s"
-    by simp
-  show "(x, a) \<in> s"
-  proof (cases "x = a")
-    case True
-    thus ?thesis
-      using connex_imp_refl equiv_rel_except_a_def refl_onD
-            lifted lifted_def lin_ord_imp_connex
-      by metis
-  next
-    case False
-    thus ?thesis
-      using equiv_rel_except_a_def insertE insert_Diff
-            lifted lifted_imp_equiv_rel_except_a x_in_A
-            x_pref_a_0 ex_lifted lin_imp_trans rest_eq
-            trans_def
-      by metis
-  qed
-qed
-
-lemma lifted_above:
-  assumes "lifted A r s a"
-  shows "above s a \<subseteq> above r a"
-  unfolding above_def
-proof (safe)
-  fix
-    x :: "'a"
-  assume
-    a_pref_x: "(a, x) \<in> s"
-  have "\<exists>x \<in> A - {a}. a \<preceq>\<^sub>r x \<and> x \<preceq>\<^sub>s a"
-    using assms lifted_def
-    by metis
-  hence lifted_r:
-    "\<exists>x \<in> A - {a}. (a, x) \<in> r \<and> (x, a) \<in> s"
-    by simp
-  have "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y"
-    using assms lifted_def equiv_rel_except_a_def
-    by metis
-  hence rest_eq:
-    "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. (x, y) \<in> r \<longleftrightarrow> (x, y) \<in> s"
-    by simp
-  have trans_r:
-    "\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r"
-    using trans_def lifted_def lin_imp_trans
-          equiv_rel_except_a_def assms
-    by metis
-  have trans_s:
-    "\<forall>x y z. (x, y) \<in> s \<longrightarrow> (y, z) \<in> s \<longrightarrow> (x, z) \<in> s"
-    using trans_def lifted_def lin_imp_trans
-          equiv_rel_except_a_def assms
-    by metis
-  have refl_r:
-    "(a, a) \<in> r"
-    using assms connex_imp_refl equiv_rel_except_a_def
-          lifted_def lin_ord_imp_connex refl_onD
-    by metis
-  have x_in_A: "x \<in> A"
-    using a_pref_x assms connex_imp_refl equiv_rel_except_a_def
-          lifted_def lin_ord_imp_connex refl_onD2
-    by metis
-  show "(a, x) \<in> r"
-    using Diff_iff a_pref_x lifted_r rest_eq singletonD
-          trans_r trans_s x_in_A refl_r
-    by (metis (full_types))
-qed
-
-lemma lifted_above2:
-  assumes
-    "lifted A r s a" and
-    "x \<in> A-{a}"
-  shows "above r x \<subseteq> above s x \<union> {a}"
-proof (safe, simp)
-  fix y :: "'a"
-  assume
-    y_in_above_r: "y \<in> above r x" and
-    y_not_in_above_s: "y \<notin> above s x"
-  have "\<forall>z \<in> A-{a}. x \<preceq>\<^sub>r z \<longleftrightarrow> x \<preceq>\<^sub>s z"
-    using assms lifted_def equiv_rel_except_a_def
-    by metis
-  hence "\<forall>z \<in> A-{a}. (x, z) \<in> r \<longleftrightarrow> (x, z) \<in> s"
-    by simp
-  hence "\<forall>z \<in> A-{a}. z \<in> above r x \<longleftrightarrow> z \<in> above s x"
-    by (simp add: above_def)
-  hence "y \<in> above r x \<longleftrightarrow> y \<in> above s x"
-    using y_not_in_above_s assms(1) connex_def
-          equiv_rel_except_a_def lifted_def lifted_mono2
-          limited_dest lin_ord_imp_connex member_remove
-          pref_imp_in_above remove_def
-    by metis
-  thus "y = a"
-    using y_in_above_r y_not_in_above_s
-    by simp
-qed
-
-lemma limit_lifted_imp_eq_or_lifted:
-  assumes
-    lifted: "lifted S r s a" and
-    subset: "A \<subseteq> S"
-  shows
-    "limit A r = limit A s \<or>
-      lifted A (limit A r) (limit A s) a"
-proof -
-  from lifted have
-    "\<forall>x \<in> S - {a}. \<forall>y \<in> S - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y"
-    by (simp add: lifted_def equiv_rel_except_a_def)
-  with subset have temp:
-    "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y"
-    by auto
-  hence eql_rs:
-      "\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}.
-      (x, y) \<in> (limit A r) \<longleftrightarrow> (x, y) \<in> (limit A s)"
-    using DiffD1 limit_presv_prefs1 limit_presv_prefs2
-    by auto
-  show ?thesis
-  proof cases
-    assume a1: "a \<in> A"
-    thus ?thesis
-    proof cases
-      (* (\<forall>x \<in> A - {a}. \<forall>y \<in> A - {a}. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y) *)
-      assume a1_1: "\<exists>x \<in> A - {a}. a \<preceq>\<^sub>r x \<and> x \<preceq>\<^sub>s a" (* \<and> a1: "a \<in> A" *)
-      from lifted subset have
-        "linear_order_on A (limit A r) \<and> linear_order_on A (limit A s)"
-        using lifted_def equiv_rel_except_a_def limit_presv_lin_ord
-        by metis
-      moreover from a1 a1_1 have keep_lift:
-        "\<exists>x \<in> A - {a}. (let q = limit A r in a \<preceq>\<^sub>q x) \<and>
-            (let u = limit A s in x \<preceq>\<^sub>u a)"
-        using DiffD1 limit_presv_prefs1
-        by simp
-      ultimately show ?thesis
-        using a1 temp
-        by (simp add: lifted_def equiv_rel_except_a_def)
-    next
-      assume
-        "\<not>(\<exists>x \<in> A - {a}. a \<preceq>\<^sub>r x \<and> x \<preceq>\<^sub>s a)" (* \<and> a1: "a \<in> A" *)
-      hence a1_2:
-        "\<forall>x \<in> A - {a}. \<not>(a \<preceq>\<^sub>r x \<and> x \<preceq>\<^sub>s a)"
-        by auto
-      moreover have not_worse:
-        "\<forall>x \<in> A - {a}. \<not>(x \<preceq>\<^sub>r a \<and> a \<preceq>\<^sub>s x)"
-        using lifted subset lifted_mono
-        by fastforce
-      moreover have connex:
-        "connex A (limit A r) \<and> connex A (limit A s)"
-        using lifted subset lifted_def equiv_rel_except_a_def
-              limit_presv_lin_ord lin_ord_imp_connex
-        by metis
-      moreover have connex1:
-        "\<forall>A r. connex A r =
-          (limited A r \<and> (\<forall>a. (a::'a) \<in> A \<longrightarrow>
-            (\<forall>aa. aa \<in> A \<longrightarrow> a \<preceq>\<^sub>r aa \<or> aa \<preceq>\<^sub>r a)))"
-        by (simp add: Ball_def_raw connex_def)
-      hence limit1:
-        "limited A (limit A r) \<and>
-          (\<forall>a. a \<notin> A \<or>
-            (\<forall>aa.
-              aa \<notin> A \<or> (a, aa) \<in> limit A r \<or>
-                (aa, a) \<in> limit A r ))"
-        using connex connex1
-        by simp
-      have limit2:
-        "\<forall>a aa A r. (a::'a, aa) \<notin> limit A r \<or> a \<preceq>\<^sub>r aa"
-        using limit_presv_prefs2
-        by metis
-      have
-        "limited A (limit A s) \<and>
-          (\<forall>a. a \<notin> A \<or>
-            (\<forall>aa. aa \<notin> A \<or>
-              (let q = limit A s in a \<preceq>\<^sub>q aa \<or> aa \<preceq>\<^sub>q a)))"
-        using connex connex_def
-        by metis
-      hence connex2:
-        "limited A (limit A s) \<and>
-          (\<forall>a. a \<notin> A \<or>
-            (\<forall>aa. aa \<notin> A \<or>
-              ((a, aa) \<in> limit A s \<or> (aa, a) \<in> limit A s)))"
-        by simp
-      ultimately have
-          "\<forall>x \<in> A - {a}. (a \<preceq>\<^sub>r x \<and> a \<preceq>\<^sub>s x) \<or> (x \<preceq>\<^sub>r a \<and> x \<preceq>\<^sub>s a)"
-        using DiffD1 limit1 limit_presv_prefs2 a1
-        by metis
-      hence r_eq_s_on_A_0:
-        "\<forall>x \<in> A - {a}. ((a, x) \<in> r \<and> (a, x) \<in> s) \<or> ((x, a) \<in> r \<and> (x, a) \<in> s)"
-        by simp
-      have
-        "\<forall>x \<in> A - {a}. (a, x) \<in> (limit A r) \<longleftrightarrow> (a, x) \<in> (limit A s)"
-        using DiffD1 limit2 limit1 connex2 a1 a1_2 not_worse
-        by metis
-      hence
-        "\<forall>x \<in> A - {a}.
-          (let q = limit A r in a \<preceq>\<^sub>q x) \<longleftrightarrow> (let q = limit A s in a \<preceq>\<^sub>q x)"
-        by simp
-      moreover have
-        "\<forall>x \<in> A - {a}. (x, a) \<in> (limit A r) \<longleftrightarrow> (x, a) \<in> (limit A s)"
-        using a1 a1_2 not_worse DiffD1 limit_presv_prefs2 connex2 limit1
-        by metis
-      moreover have
-        "(a, a) \<in> (limit A r) \<and> (a, a) \<in> (limit A s)"
-        using a1 connex connex_imp_refl refl_onD
-        by metis
-      moreover have
-        "limited A (limit A r) \<and> limited A (limit A s)"
-        using limit_to_limits
-        by metis
-      ultimately have
-        "\<forall>x y. (x, y) \<in> limit A r \<longleftrightarrow> (x, y) \<in> limit A s"
-        using eql_rs
-        by auto
-      thus ?thesis
-        by simp
-    qed
-  next
-    assume a2: "a \<notin> A"
-    with eql_rs have
-      "\<forall>x \<in> A. \<forall>y \<in> A. (x, y) \<in> (limit A r) \<longleftrightarrow> (x, y) \<in> (limit A s)"
-      by simp
-    thus ?thesis
-      using limit_to_limits limited_dest subrelI subset_antisym
-      by auto
-  qed
-qed
-
-lemma negl_diff_imp_eq_limit:
-  assumes
-    change: "equiv_rel_except_a S r s a" and
-    subset: "A \<subseteq> S" and
-    notInA: "a \<notin> A"
-  shows "limit A r = limit A s"
-proof -
-  have "A \<subseteq> S-{a}"
-    by (simp add: notInA subset subset_Diff_insert)
-  hence "\<forall>x \<in> A. \<forall>y \<in> A. x \<preceq>\<^sub>r y \<longleftrightarrow> x \<preceq>\<^sub>s y"
-    by (meson change equiv_rel_except_a_def in_mono)
-  thus ?thesis
-    by auto
-qed
-
-theorem lifted_above_winner:
-  assumes
-    lifted_a: "lifted A r s a" and
-    above_x: "above r x = {x}" and
-    fin_A: "finite A"
-  shows "above s x = {x} \<or> above s a = {a}"
-proof cases
-  assume "x = a"
-  thus ?thesis
-    using above_subset_geq_one lifted_a above_x
-          lifted_above lifted_def equiv_rel_except_a_def
-    by metis
-next
-  assume asm1: "x \<noteq> a"
-  thus ?thesis
-  proof cases
-    assume "above s x = {x}"
-    thus ?thesis
-      by simp
-  next
-    assume asm2: "above s x \<noteq> {x}" (* \<and> asm1: x \<noteq> a *)
-    have "\<forall>y \<in> A. y \<preceq>\<^sub>r x"
-    proof -
-      fix aa :: 'a
-      have imp_a: "x \<preceq>\<^sub>r aa \<longrightarrow> aa \<notin> A \<or> aa \<preceq>\<^sub>r x"
-        using singletonD pref_imp_in_above above_x
-        by metis
-      also have f1:
-        "\<forall>A r.
-          (connex A r \<or>
-            (\<exists>a. (\<exists>aa. \<not> (aa::'a) \<preceq>\<^sub>r a \<and> \<not> a \<preceq>\<^sub>r aa \<and> aa \<in> A) \<and> a \<in> A) \<or>
-              \<not> limited A r) \<and>
-            ((\<forall>a. (\<forall>aa. aa \<preceq>\<^sub>r a \<or> a \<preceq>\<^sub>r aa \<or> aa \<notin> A) \<or> a \<notin> A) \<and> limited A r \<or>
-              \<not> connex A r)"
-        using connex_def
-        by metis
-      moreover have eq_exc_a:
-        "equiv_rel_except_a A r s a"
-        using lifted_def lifted_a
-        by metis
-      ultimately have "aa \<notin> A \<or> aa \<preceq>\<^sub>r x"
-        using pref_imp_in_above above_x equiv_rel_except_a_def
-              lin_ord_imp_connex limited_dest insertCI
-        by metis
-      thus ?thesis
-        using f1 eq_exc_a above_one above_one2 above_x fin_A
-              equiv_rel_except_a_def insert_not_empty pref_imp_in_above
-              lin_ord_imp_connex mk_disjoint_insert insertE
-        by metis
-    qed
-    moreover have "equiv_rel_except_a A r s a"
-      using lifted_a lifted_def
-      by metis
-    moreover have "x \<in> A-{a}"
-      using above_one above_one2 asm1 assms calculation
-            equiv_rel_except_a_def insert_not_empty
-            member_remove remove_def insert_absorb
-      by metis
-    ultimately have "\<forall>y \<in> A-{a}. y \<preceq>\<^sub>s x"
-      using DiffD1 lifted_a equiv_rel_except_a_def
-      by metis
-    hence not_others: "\<forall>y \<in> A-{a}. above s y \<noteq> {y}"
-      using asm2 empty_iff insert_iff pref_imp_in_above
-      by metis
-    hence "above s a = {a}"
-      using Diff_iff all_not_in_conv lifted_a fin_A lifted_def
-            equiv_rel_except_a_def above_one singleton_iff
-      by metis
-    thus ?thesis
-      by simp
-  qed
-qed
-
-theorem lifted_above_winner2:
-  assumes
-    "lifted A r s a" and
-    "above r a = {a}" and
-    "finite A"
-  shows "above s a = {a}"
-  using assms lifted_above_winner
-  by metis
-
-theorem lifted_above_winner3:
-  assumes
-    lifted_a: "lifted A r s a" and
-    above_x: "above s x = {x}" and
-    fin_A: "finite A" and
-    x_not_a: "x \<noteq> a"
-  shows "above r x = {x}"
-proof (rule ccontr)
-  assume asm: "above r x \<noteq> {x}"
-  then obtain y where y: "above r y = {y}"
-    using lifted_a fin_A insert_Diff insert_not_empty
-          lifted_def equiv_rel_except_a_def above_one
-    by metis
-  hence "above s y = {y} \<or> above s a = {a}"
-    using lifted_a fin_A lifted_above_winner
-    by metis
-  moreover have "\<forall>b. above s b = {b} \<longrightarrow> b = x"
-    using all_not_in_conv lifted_a above_x lifted_def
-          fin_A equiv_rel_except_a_def above_one2
-    by metis
-  ultimately have "y = x"
-    using x_not_a
-    by presburger
-  moreover have "y \<noteq> x"
-    using asm y
-    by blast
-  ultimately show "False"
-    by simp
-qed
-
-end
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/#Profile.thy# b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/#Profile.thy#
deleted file mode 100644
index 507abac..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/#Profile.thy#
+++ /dev/null
@@ -1,853 +0,0 @@
-(*  File:       Profile.thy
-    Copyright   2021  Karlsruhe Institute of Technology (KIT)
-*)
-\<^marker>\<open>creator "Jonas Kraemer, Karlsruhe Institute of Technology (KIT)"\<close>
-\<^marker>\<open>contributor "Karsten Diekhoff, Karlsruhe Institute of Technology (KIT)"\<close>
-\<^marker>\<open>contributor "Michael Kirsten, Karlsruhe Institute of Technology (KIT)"\<close>
-\<^marker>\<open>contributor "Stephan Bohr, Karlsruhe Institute of Technology (KIT)"\<close>
-
-section \<open>Preference Profile\<close>
-
-theory Profile
-  imports Preference_Relation
-begin
-
-text
-\<open>Preference profiles denote the decisions made by the individual voters on
-the eligible alternatives. They are represented in the form of one preference
-relation (e.g., selected on a ballot) per voter, collectively captured in a
-list of such preference relations.
-Unlike a the common preference profiles in the social-choice sense, the
-profiles described here considers only the (sub-)set of alternatives that are
-received.\<close>
-
-subsection \<open>Definition\<close>
-
-type_synonym Alternative = Nat.nat
-
-(*A profile contains one ballot for each voter.*)
-type_synonym Profile = "(Alternative Preference_Relation) list"
-
-(*
-   A profile on a finite set of alternatives A contains only ballots that are
-   linear orders on A.
-*)
-definition profile :: "'a set \<Rightarrow> Profile \<Rightarrow> bool" where
-  "profile A p \<equiv> \<forall>i::nat. i < length p \<longrightarrow> linear_order_on A (p!i)"
-
-lemma profile_set : "profile A p \<equiv> (\<forall>b \<in> (set p). linear_order_on A b)"
-  by (simp add: all_set_conv_all_nth profile_def)
-
-abbreviation finite_profile :: "'a set \<Rightarrow> 'a Profile \<Rightarrow> bool" where
-  "finite_profile A p \<equiv> finite A \<and> profile A p"
-
-subsection \<open>Preference Counts and Comparisons\<close>
-
-(*
-   The win count for an alternative a in a profile p is
-   the amount of ballots in p that rank alternative a in first position.
-*)
-fun win_count :: "'a Profile \<Rightarrow> 'a \<Rightarrow> nat" where
-  "win_count p a =
-    card {i::nat. i < length p \<and> above (p!i) a = {a}}"
-
-fun win_count_code :: "'a Profile \<Rightarrow> 'a \<Rightarrow> nat" where
-  "win_count_code Nil a = 0" |
-  "win_count_code (p#ps) a =
-      (if (above p a = {a}) then 1 else 0) + win_count_code ps a"
-
-lemma win_count_equiv[code]: "win_count p x = win_count_code p x"
-proof (induction p rule: rev_induct, simp)
-  case (snoc a p)
-  fix
-    a :: "'a Preference_Relation" and
-    p :: "'a Profile"
-  assume
-    base_case:
-    "win_count p x = win_count_code p x"
-  have size_one: "length [a] = 1"
-    by simp
-  have p_pos_in_ps:
-    "\<forall>i<length p. p!i = (p@[a])!i"
-    by (simp add: nth_append)
-  have
-    "win_count [a] x =
-      (let q = [a] in
-        card {i::nat. i < length q \<and>
-              (let r = (q!i) in (above r x = {x}))})"
-    by simp
-  hence one_ballot_equiv:
-    "win_count [a] x = win_count_code [a] x"
-    using size_one
-    by (simp add: nth_Cons')
-  have pref_count_induct:
-    "win_count (p@[a]) x =
-      win_count p x + win_count [a] x"
-  proof (simp)
-    have
-      "{i. i = 0 \<and> (above ([a]!i) x = {x})} =
-        (if (above a x = {x}) then {0} else {})"
-      by (simp add: Collect_conv_if)
-    hence shift_idx_a:
-      "card {i. i = length p \<and> (above ([a]!0) x = {x})} =
-        card {i. i = 0 \<and> (above ([a]!i) x = {x})}"
-      by simp
-    have set_prof_eq:
-      "{i. i < Suc (length p) \<and> (above ((p@[a])!i) x = {x})} =
-        {i. i < length p \<and> (above (p!i) x = {x})} \<union>
-          {i. i = length p \<and> (above ([a]!0) x = {x})}"
-    proof (safe, simp_all)
-      fix
-        xa :: nat and
-        xaa :: "'a"
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "xa \<noteq> length p" and
-        "xaa \<in> above (p!xa) x"
-      thus "xaa = x"
-        using less_antisym p_pos_in_ps singletonD
-        by metis
-    next
-      fix
-        xa :: nat
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "xa \<noteq> length p"
-      thus "x \<in> above (p!xa) x"
-        using less_antisym insertI1 p_pos_in_ps
-        by metis
-    next
-      fix
-        xa :: nat and
-        xaa :: "'a"
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "xaa \<in> above a x" and
-        "xaa \<noteq> x"
-      thus "xa < length p"
-        using less_antisym nth_append_length
-              p_pos_in_ps singletonD
-        by metis
-    next
-      fix
-        xa :: nat and
-        xaa :: "'a" and
-        xb :: "'a"
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "xaa \<in> above a x" and
-        "xaa \<noteq> x" and
-        "xb \<in> above (p!xa) x"
-      thus "xb = x"
-        using less_antisym p_pos_in_ps
-              nth_append_length singletonD
-        by metis
-    next
-      fix
-        xa :: nat and
-        xaa :: "'a"
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "xaa \<in> above a x" and
-        "xaa \<noteq> x"
-      thus "x \<in> above (p!xa) x"
-        using insertI1 less_antisym nth_append
-              nth_append_length singletonD
-        by metis
-    next
-      fix
-        xa :: nat
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "x \<notin> above a x"
-      thus "xa < length p"
-        using insertI1 less_antisym nth_append_length
-        by metis
-    next
-      fix
-        xa :: nat and
-        xb :: "'a"
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "x \<notin> above a x" and
-        "xb \<in> above (p!xa) x"
-      thus "xb = x"
-        using insertI1 less_antisym nth_append_length
-              p_pos_in_ps singletonD
-        by metis
-    next
-      fix
-        xa :: nat
-      assume
-        "xa < Suc (length p)" and
-        "above ((p@[a])!xa) x = {x}" and
-        "x \<notin> above a x"
-      thus "x \<in> above (p!xa) x"
-        using insertI1 less_antisym nth_append_length p_pos_in_ps
-        by metis
-    next
-      fix
-        xa :: nat and
-        xaa :: "'a"
-      assume
-        "xa < length p" and
-        "above (p!xa) x = {x}" and
-        "xaa \<in> above ((p@[a])!xa) x"
-      thus "xaa = x"
-        by (simp add: nth_append)
-    next
-      fix
-        xa :: nat
-      assume
-        "xa < length p" and
-        "above (p!xa) x = {x}"
-      thus "x \<in> above ((p@[a])!xa) x"
-        by (simp add: nth_append)
-    qed
-    have f1:
-      "finite {n. n < length p \<and> (above (p!n) x = {x})}"
-      by simp
-    have f2:
-      "finite {n. n = length p \<and> (above ([a]!0) x = {x})}"
-      by simp
-    have
-      "card ({i. i < length p \<and> (above (p!i) x = {x})} \<union>
-        {i. i = length p \<and> (above ([a]!0) x = {x})}) =
-          card {i. i < length p \<and> (above (p!i) x = {x})} +
-            card {i. i = length p \<and> (above ([a]!0) x = {x})}"
-      using f1 f2 card_Un_disjoint
-      by blast
-    thus
-      "card {i. i < Suc (length p) \<and> (above ((p@[a])!i) x = {x})} =
-        card {i. i < length p \<and> (above (p!i) x = {x})} +
-          card {i. i = 0 \<and> (above ([a]!i) x = {x})}"
-      using set_prof_eq shift_idx_a
-      by auto
-  qed
-  have pref_count_code_induct:
-    "win_count_code (p@[a]) x =
-      win_count_code p x + win_count_code [a] x"
-  proof (induction p, simp)
-    fix
-      aa :: "'a Preference_Relation" and
-      p :: "'a Profile"
-    assume
-      "win_count_code (p@[a]) x =
-        win_count_code p x + win_count_code [a] x"
-    thus
-      "win_count_code ((aa#p)@[a]) x =
-        win_count_code (aa#p) x + win_count_code [a] x"
-      by simp
-  qed
-  show "win_count (p@[a]) x = win_count_code (p@[a]) x"
-    using pref_count_code_induct pref_count_induct
-          base_case one_ballot_equiv
-    by presburger
-qed
-
-fun prefer_count :: "'a Profile \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> nat" where
-  "prefer_count p x y =
-      card {i::nat. i < length p \<and> (let r = (p!i) in (y \<preceq>\<^sub>r x))}"
-
-fun prefer_count_code :: "'a Profile \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> nat" where
-  "prefer_count_code Nil x y = 0" |
-  "prefer_count_code (p#ps) x y =
-      (if y \<preceq>\<^sub>p x then 1 else 0) + prefer_count_code ps x y"
-
-lemma pref_count_equiv[code]: "prefer_count p x y = prefer_count_code p x y"
-proof (induction p rule: rev_induct, simp)
-  case (snoc a p)
-  fix
-    a :: "'a Preference_Relation" and
-    p :: "'a Profile"
-  assume
-    base_case:
-    "prefer_count p x y = prefer_count_code p x y"
-  have size_one: "length [a] = 1"
-    by simp
-  have p_pos_in_ps:
-    "\<forall>i<length p. p!i = (p@[a])!i"
-    by (simp add: nth_append)
-  have
-    "prefer_count [a] x y =
-      (let q = [a] in
-        card {i::nat. i < length q \<and>
-              (let r = (q!i) in (y \<preceq>\<^sub>r x))})"
-    by simp
-  hence one_ballot_equiv:
-    "prefer_count [a] x y = prefer_count_code [a] x y"
-    using size_one
-    by (simp add: nth_Cons')
-  have pref_count_induct:
-    "prefer_count (p@[a]) x y =
-      prefer_count p x y + prefer_count [a] x y"
-  proof (simp)
-    have
-      "{i. i = 0 \<and> (y, x) \<in> [a]!i} =
-        (if ((y, x) \<in> a) then {0} else {})"
-      by (simp add: Collect_conv_if)
-    hence shift_idx_a:
-      "card {i. i = length p \<and> (y, x) \<in> [a]!0} =
-        card {i. i = 0 \<and> (y, x) \<in> [a]!i}"
-      by simp
-    have set_prof_eq:
-      "{i. i < Suc (length p) \<and> (y, x) \<in> (p@[a])!i} =
-        {i. i < length p \<and> (y, x) \<in> p!i} \<union>
-          {i. i = length p \<and> (y, x) \<in> [a]!0}"
-    proof (safe, simp_all)
-      fix
-        xa :: nat
-      assume
-        "xa < Suc (length p)" and
-        "(y, x) \<in> (p@[a])!xa" and
-        "xa \<noteq> length p"
-      thus "(y, x) \<in> p!xa"
-        using less_antisym p_pos_in_ps
-        by metis
-    next
-      fix
-        xa :: nat
-      assume
-        "xa < Suc (length p)" and
-        "(y, x) \<in> (p@[a])!xa" and
-        "(y, x) \<notin> a"
-      thus "xa < length p"
-        using less_antisym nth_append_length
-        by metis
-    next
-      fix
-        xa :: nat
-      assume
-        "xa < Suc (length p)" and
-        "(y, x) \<in> (p@[a])!xa" and
-        "(y, x) \<notin> a"
-      thus "(y, x) \<in> p!xa"
-        using less_antisym nth_append_length p_pos_in_ps
-        by metis
-    next
-      fix
-        xa :: nat
-      assume
-        "xa < length p" and
-        "(y, x) \<in> p!xa"
-      thus "(y, x) \<in> (p@[a])!xa"
-        using less_antisym p_pos_in_ps
-        by metis
-    qed
-    have f1:
-      "finite {n. n < length p \<and> (y, x) \<in> p!n}"
-      by simp
-    have f2:
-      "finite {n. n = length p \<and> (y, x) \<in> [a]!0}"
-      by simp
-    have
-      "card ({i. i < length p \<and> (y, x) \<in> p!i} \<union>
-        {i. i = length p \<and> (y, x) \<in> [a]!0}) =
-          card {i. i < length p \<and> (y, x) \<in> p!i} +
-            card {i. i = length p \<and> (y, x) \<in> [a]!0}"
-      using f1 f2 card_Un_disjoint
-      by blast
-    thus
-      "card {i. i < Suc (length p) \<and> (y, x) \<in> (p@[a])!i} =
-        card {i. i < length p \<and> (y, x) \<in> p!i} +
-          card {i. i = 0 \<and> (y, x) \<in> [a]!i}"
-      using set_prof_eq shift_idx_a
-      by auto
-  qed
-  have pref_count_code_induct:
-    "prefer_count_code (p@[a]) x y =
-      prefer_count_code p x y + prefer_count_code [a] x y"
-  proof (simp, safe)
-    assume
-      assm: "(y, x) \<in> a"
-    show
-      "prefer_count_code (p@[a]) x y = Suc (prefer_count_code p x y)"
-    proof (induction p, simp_all)
-      show "(y, x) \<in> a"
-        using assm
-        by simp
-    qed
-  next
-    assume
-      assm: "(y, x) \<notin> a"
-    show
-      "prefer_count_code (p@[a]) x y = prefer_count_code p x y"
-    proof (induction p, simp_all, safe)
-      assume
-        "(y, x) \<in> a"
-      thus "False"
-        using assm
-        by simp
-    qed
-  qed
-  show "prefer_count (p@[a]) x y = prefer_count_code (p@[a]) x y"
-    using pref_count_code_induct pref_count_induct
-          base_case one_ballot_equiv
-    by presburger
-qed
-
-lemma set_compr: "{ f x | x . x \<in> S } = f ` S"
-  by auto
-
-lemma pref_count_set_compr: "{prefer_count p x y | y . y \<in> A-{x}} =
-          (prefer_count p x) ` (A-{x})"
-  by auto
-
-lemma pref_count:
-  assumes prof: "profile A p"
-  assumes x_in_A: "x \<in> A"
-  assumes y_in_A: "y \<in> A"
-  assumes neq: "x \<noteq> y"
-  shows "prefer_count p x y = (length p) - (prefer_count p y x)"
-proof -
-  have 00: "card {i::nat. i < length p} = length p"
-    by simp
-  have 10:
-    "{i::nat. i < length p \<and> (let r = (p!i) in (y \<preceq>\<^sub>r x))} =
-        {i::nat. i < length p} -
-          {i::nat. i < length p \<and> \<not> (let r = (p!i) in (y \<preceq>\<^sub>r x))}"
-    by auto
-  have 0: "\<forall> i::nat . i < length p \<longrightarrow> linear_order_on A (p!i)"
-    using prof profile_def
-    by metis
-  hence "\<forall> i::nat . i < length p \<longrightarrow> connex A (p!i)"
-    by (simp add: lin_ord_imp_connex)
-  hence 1: "\<forall> i::nat . i < length p \<longrightarrow>
-              \<not> (let r = (p!i) in (y \<preceq>\<^sub>r x)) \<longrightarrow> (let r = (p!i) in (x \<preceq>\<^sub>r y))"
-    using connex_def x_in_A y_in_A
-    by metis
-  from 0 have
-    "\<forall> i::nat . i < length p \<longrightarrow> antisym  (p!i)"
-    using lin_imp_antisym
-    by metis
-  hence "\<forall> i::nat . i < length p \<longrightarrow> ((y, x) \<in> (p!i) \<longrightarrow> (x, y) \<notin> (p!i))"
-    using antisymD neq
-    by metis
-  hence "\<forall> i::nat . i < length p \<longrightarrow>
-          ((let r = (p!i) in (y \<preceq>\<^sub>r x)) \<longrightarrow> \<not> (let r = (p!i) in (x \<preceq>\<^sub>r y)))"
-    by simp
-  with 1 have
-    "\<forall> i::nat . i < length p \<longrightarrow>
-      \<not> (let r = (p!i) in (y \<preceq>\<^sub>r x)) = (let r = (p!i) in (x \<preceq>\<^sub>r y))"
-    by metis
-  hence 2:
-    "{i::nat. i < length p \<and> \<not> (let r = (p!i) in (y \<preceq>\<^sub>r x))} =
-        {i::nat. i < length p \<and> (let r = (p!i) in (x \<preceq>\<^sub>r y))}"
-    by metis
-  hence 20:
-    "{i::nat. i < length p \<and> (let r = (p!i) in (y \<preceq>\<^sub>r x))} =
-        {i::nat. i < length p} -
-          {i::nat. i < length p \<and> (let r = (p!i) in (x \<preceq>\<^sub>r y))}"
-    using "10" "2"
-    by simp
-  have
-    "{i::nat. i < length p \<and> (let r = (p!i) in (x \<preceq>\<^sub>r y))} \<subseteq>
-        {i::nat. i < length p}"
-    by (simp add: Collect_mono)
-  hence 30:
-    "card ({i::nat. i < length p} -
-        {i::nat. i < length p \<and> (let r = (p!i) in (x \<preceq>\<^sub>r y))}) =
-      (card {i::nat. i < length p}) -
-        card({i::nat. i < length p \<and> (let r = (p!i) in (x \<preceq>\<^sub>r y))})"
-    by (simp add: card_Diff_subset)
-  have "prefer_count p x y =
-          card {i::nat. i < length p \<and> (let r = (p!i) in (y \<preceq>\<^sub>r x))}"
-    by simp
-  also have
-    "... = card({i::nat. i < length p} -
-            {i::nat. i < length p \<and> (let r = (p!i) in (x \<preceq>\<^sub>r y))})"
-    using "20"
-    by simp
-  also have
-    "... = (card {i::nat. i < length p}) -
-              card({i::nat. i < length p \<and> (let r = (p!i) in (x \<preceq>\<^sub>r y))})"
-    using "30"
-    by metis
-  also have
-    "... = length p - (prefer_count p y x)"
-    by simp
-  finally show ?thesis
-    by (simp add: "20" "30")
-qed
-
-lemma pref_count_sym:
-    assumes p1: "prefer_count p a x \<ge> prefer_count p x b"
-    assumes prof: "profile A p"
-    assumes a_in_A: "a \<in> A"
-    assumes b_in_A: "b \<in> A"
-    assumes x_in_A: "x \<in> A"
-    assumes neq1: "a \<noteq> x"
-    assumes neq2: "x \<noteq> b"
-    shows "prefer_count p b x \<ge> prefer_count p x a"
-proof -
-  from prof a_in_A x_in_A neq1 have 0:
-    "prefer_count p a x = (length p) - (prefer_count p x a)"
-    using pref_count
-    by metis
-  moreover from prof x_in_A b_in_A neq2 have 1:
-    "prefer_count p x b = (length p) - (prefer_count p b x)"
-    using pref_count
-    by (metis (mono_tags, lifting))
-  hence 2: "(length p) - (prefer_count p x a) \<ge>
-              (length p) - (prefer_count p b x)"
-    using calculation p1
-    by auto
-  hence 3: "(prefer_count p x a) - (length p) \<le>
-              (prefer_count p b x) - (length p)"
-    using a_in_A diff_is_0_eq diff_le_self neq1
-          pref_count prof x_in_A
-    by (metis (no_types))
-  hence "(prefer_count p x a) \<le> (prefer_count p b x)"
-    using "1" "3" calculation p1
-    by linarith
-  thus ?thesis
-    by linarith
-qed
-
-lemma empty_prof_imp_zero_pref_count:
-  assumes "p = []"
-  shows "\<forall> x y. prefer_count p x y = 0"
-  using assms
-  by simp
-
-lemma empty_prof_imp_zero_pref_count_code:
-  assumes "p = []"
-  shows "\<forall> x y. prefer_count_code p x y = 0"
-  using assms
-  by simp
-
-lemma pref_count_code_incr:
-  assumes
-    "prefer_count_code ps x y = n" and
-    "y \<preceq>\<^sub>p x"
-  shows "prefer_count_code (p#ps) x y = n+1"
-  using assms
-  by simp
-
-lemma pref_count_code_not_smaller_imp_constant:
-  assumes
-    "prefer_count_code ps x y = n" and
-    "\<not>(y \<preceq>\<^sub>p x)"
-  shows "prefer_count_code (p#ps) x y = n"
-  using assms
-  by simp
-
-fun wins :: "'a \<Rightarrow> 'a Profile \<Rightarrow> 'a \<Rightarrow> bool" where
-  "wins x p y =
-    (prefer_count p x y > prefer_count p y x)"
-
-(* Alternative a wins against b implies that b does not win against a. *)
-lemma wins_antisym:
-  assumes "wins a p b"
-  shows "\<not> wins b p a"
-  using assms
-  by simp
-
-lemma wins_irreflex: "\<not> wins w p w"
-  using wins_antisym
-  by metis
-
-subsection \<open>Condorcet Winner\<close>
-
-fun condorcet_winner :: "'a set \<Rightarrow> 'a Profile \<Rightarrow> 'a \<Rightarrow> bool" where
-  "condorcet_winner A p w =
-      (finite_profile A p \<and>  w \<in> A \<and> (\<forall>x \<in> A - {w} . wins w p x))"
-
-lemma cond_winner_unique:
-  assumes winner_c: "condorcet_winner A p c" and
-          winner_w: "condorcet_winner A p w"
-  shows "w = c"
-proof (rule ccontr)
-  assume
-    assumption: "w \<noteq> c"
-  from winner_w
-  have "wins w p c"
-    using assumption insert_Diff insert_iff winner_c
-    by simp
-  hence "\<not> wins c p w"
-    by (simp add: wins_antisym)
-  moreover from winner_c
-  have
-    c_wins_against_w: "wins c p w"
-    using Diff_iff assumption
-          singletonD winner_w
-    by simp
-  ultimately show False
-    by simp
-qed
-
-lemma cond_winner_unique2:
-  assumes winner: "condorcet_winner A p w" and
-          not_w:  "x \<noteq> w" and
-          in_A:   "x \<in> A"
-        shows "\<not> condorcet_winner A p x"
-  using not_w cond_winner_unique winner
-  by metis
-
-lemma cond_winner_unique3:
-  assumes "condorcet_winner A p w"
-  shows "{a \<in> A. condorcet_winner A p a} = {w}"
-proof (safe, simp_all, safe)
-  fix
-    x :: "'a"
-  assume
-    fin_A: "finite A" and
-    prof_A: "profile A p" and
-    x_in_A: "x \<in> A" and
-    x_wins:
-      "\<forall>xa \<in> A - {x}.
-        card {i. i < length p \<and> (x, xa) \<in> p!i} <
-          card {i. i < length p \<and> (xa, x) \<in> p!i}"
-  from assms have assm:
-    "finite_profile A p \<and>  w \<in> A \<and>
-      (\<forall>x \<in> A - {w}.
-        card {i::nat. i < length p \<and> (w, x) \<in> p!i} <
-          card {i::nat. i < length p \<and> (x, w) \<in> p!i})"
-    by simp
-  hence
-    "(\<forall>x \<in> A - {w}.
-      card {i::nat. i < length p \<and> (w, x) \<in> p!i} <
-        card {i::nat. i < length p \<and> (x, w) \<in> p!i})"
-    by simp
-  hence w_beats_x:
-    "x \<noteq> w \<Longrightarrow>
-      card {i::nat. i < length p \<and> (w, x) \<in> p!i} <
-        card {i::nat. i < length p \<and> (x, w) \<in> p!i}"
-    using x_in_A
-    by simp
-  also from assm have
-    "finite_profile A p"
-    by simp
-  moreover from assm have
-    "w \<in> A"
-    by simp
-  hence x_beats_w:
-    "x \<noteq> w \<Longrightarrow>
-      card {i. i < length p \<and> (x, w) \<in> p!i} <
-        card {i. i < length p \<and> (w, x) \<in> p!i}"
-    using x_wins
-    by simp
-  from w_beats_x x_beats_w show
-    "x = w"
-    by linarith
-next
-  fix
-    x :: "'a"
-  from assms show "w \<in> A"
-    by simp
-next
-  fix
-    x :: "'a"
-  from assms show "finite A"
-    by simp
-next
-  fix
-    x :: "'a"
-  from assms show "profile A p"
-    by simp
-next
-  fix
-    x :: "'a"
-  from assms show "w \<in> A"
-    by simp
-next
-  fix
-    x :: "'a" and
-    xa :: "'a"
-  assume
-    xa_in_A: "xa \<in> A" and
-    w_wins:
-      "\<not> card {i. i < length p \<and> (w, xa) \<in> p!i} <
-        card {i. i < length p \<and> (xa, w) \<in> p!i}"
-  from assms have
-    "finite_profile A p \<and>  w \<in> A \<and>
-      (\<forall>x \<in> A - {w} .
-        card {i::nat. i < length p \<and> (w, x) \<in> p!i} <
-          card {i::nat. i < length p \<and> (x, w) \<in> p!i})"
-    by simp
-  thus "xa = w"
-    using xa_in_A w_wins insert_Diff insert_iff
-    by (metis (no_types, lifting))
-qed
-
-subsection \<open>Limited Profile\<close>
-
-(*
-   This function restricts a profile p to a set A and
-   keeps all of A's preferences.
-*)
-fun limit_profile :: "'a set \<Rightarrow> 'a Profile \<Rightarrow> 'a Profile" where
-  "limit_profile A p = map (limit A) p"
-
-lemma limit_prof_trans:
-  assumes
-    "B \<subseteq> A" and
-    "C \<subseteq> B" and
-    "finite_profile A p"
-  shows "limit_profile C p = limit_profile C (limit_profile B p)"
-  using assms
-  by auto
-
-lemma limit_profile_sound:
-  assumes
-    profile: "finite_profile S p" and
-    subset: "A \<subseteq> S"
-  shows "finite_profile A (limit_profile A p)"
-proof (simp)
-  from profile
-  show "finite_profile A (map (limit A) p)"
-    using length_map limit_presv_lin_ord nth_map
-          profile_def subset infinite_super
-    by metis
-qed
-
-lemma limit_prof_presv_size:
-  assumes f_prof: "finite_profile S p" and
-          subset:  "A \<subseteq> S"
-  shows "length p = length (limit_profile A p)"
-  by simp
-
-subsection \<open>Lifting Property\<close>
-
-definition equiv_prof_except_a :: "'a set \<Rightarrow> 'a Profile \<Rightarrow> 'a Profile \<Rightarrow>
-                                        'a \<Rightarrow> bool" where
-  "equiv_prof_except_a A p q a \<equiv>
-    finite_profile A p \<and> finite_profile A q \<and>
-      a \<in> A \<and> length p = length q \<and>
-      (\<forall>i::nat.
-        i < length p \<longrightarrow>
-          equiv_rel_except_a A (p!i) (q!i) a)"
-
-(*
-   An alternative gets lifted from one profile to another iff
-   its ranking increases in at least one ballot, and nothing else changes.
-*)
-definition lifted :: "'a set \<Rightarrow> 'a Profile \<Rightarrow> 'a Profile \<Rightarrow> 'a \<Rightarrow> bool" where
-  "lifted A p q a \<equiv>
-    finite_profile A p \<and> finite_profile A q \<and>
-      a \<in> A \<and> length p = length q \<and>
-      (\<forall>i::nat.
-        (i < length p \<and> \<not>Preference_Relation.lifted A (p!i) (q!i) a) \<longrightarrow>
-          (p!i) = (q!i)) \<and>
-      (\<exists>i::nat. i < length p \<and> Preference_Relation.lifted A (p!i) (q!i) a)"
-
-lemma lifted_imp_equiv_prof_except_a:
-  assumes lifted: "lifted A p q a"
-  shows "equiv_prof_except_a A p q a"
-proof -
-  have
-    "\<forall>i::nat. i < length p \<longrightarrow>
-      equiv_rel_except_a A (p!i) (q!i) a"
-  proof
-    fix i :: nat
-    show
-      "i < length p \<longrightarrow>
-        equiv_rel_except_a A (p!i) (q!i) a"
-    proof
-      assume i_ok: "i < length p"
-      show "equiv_rel_except_a A (p!i) (q!i) a"
-        using lifted_def i_ok lifted profile_def trivial_equiv_rel
-              lifted_imp_equiv_rel_except_a
-        by metis
-    qed
-  qed
-  thus ?thesis
-    using lifted_def lifted equiv_prof_except_a_def
-    by metis
-qed
-
-lemma negl_diff_imp_eq_limit_prof:
-  assumes
-    change: "equiv_prof_except_a S p q a" and
-    subset: "A \<subseteq> S" and
-    notInA: "a \<notin> A"
-  shows "limit_profile A p = limit_profile A q"
-proof -
-  have
-    "\<forall>i::nat. i < length p \<longrightarrow>
-      equiv_rel_except_a S (p!i) (q!i) a"
-    using change equiv_prof_except_a_def
-    by metis
-  hence "\<forall>i::nat. i < length p \<longrightarrow> limit A (p!i) = limit A (q!i)"
-    using notInA negl_diff_imp_eq_limit subset
-    by metis
-  hence "map (limit A) p = map (limit A) q"
-    using change equiv_prof_except_a_def
-          length_map nth_equalityI nth_map
-    by (metis (mono_tags, lifting))
-  thus ?thesis
-    by simp
-qed
-
-lemma limit_prof_eq_or_lifted:
-  assumes
-    lifted: "lifted S p q a" and
-    subset: "A \<subseteq> S"
-  shows
-    "limit_profile A p = limit_profile A q \<or>
-        lifted A (limit_profile A p) (limit_profile A q) a"
-proof cases
-  assume inA: "a \<in> A"
-  have
-    "\<forall>i::nat. i < length p \<longrightarrow>
-        (Preference_Relation.lifted S (p!i) (q!i) a \<or> (p!i) = (q!i))"
-    using lifted_def lifted
-    by metis
-  hence one:
-    "\<forall>i::nat. i < length p \<longrightarrow>
-         (Preference_Relation.lifted A (limit A (p!i)) (limit A (q!i)) a \<or>
-           (limit A (p!i)) = (limit A (q!i)))"
-    using limit_lifted_imp_eq_or_lifted subset
-    by metis
-  thus ?thesis
-  proof cases
-    assume "\<forall>i::nat. i < length p \<longrightarrow> (limit A (p!i)) = (limit A (q!i))"
-    thus ?thesis
-      using lifted_def length_map lifted
-            limit_profile.simps nth_equalityI nth_map
-      by (metis (mono_tags, lifting))
-  next
-    assume assm:
-      "\<not>(\<forall>i::nat. i < length p \<longrightarrow> (limit A (p!i)) = (limit A (q!i)))"
-    let ?p = "limit_profile A p"
-    let ?q = "limit_profile A q"
-    have "profile A ?p \<and> profile A ?q"
-      using lifted_def lifted limit_profile_sound subset
-      by metis
-    moreover have "length ?p = length ?q"
-      using lifted_def lifted
-      by fastforce
-    moreover have
-      "\<exists>i::nat. i < length ?p \<and> Preference_Relation.lifted A (?p!i) (?q!i) a"
-      using assm lifted_def length_map lifted
-            limit_profile.simps nth_map one
-      by (metis (no_types, lifting))
-    moreover have
-      "\<forall>i::nat.
-        (i < length ?p \<and> \<not>Preference_Relation.lifted A (?p!i) (?q!i) a) \<longrightarrow>
-          (?p!i) = (?q!i)"
-      using lifted_def length_map lifted
-            limit_profile.simps nth_map one
-      by metis
-    ultimately have "lifted A ?p ?q a"
-      using lifted_def inA lifted rev_finite_subset subset
-      by (metis (no_types, lifting))
-    thus ?thesis
-      by simp
-  qed
-next
-  assume "a \<notin> A"
-  thus ?thesis
-    using lifted negl_diff_imp_eq_limit_prof subset
-          lifted_imp_equiv_prof_except_a
-    by metis
-qed
-
-end
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/framework.pl b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/framework.pl
index f61df48..127f7c3 100644
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/framework.pl
+++ b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/framework.pl
@@ -1,5 +1,5 @@
 %%%%%%%%%%%%%%%%%%%%
-%% Generated by ViRAGe 0.1.0 at 2021-07-12 15:25:59.
+%% Generated by ViRAGe 0.1.0 at 2023-09-26 04:35:52 MESZ.
 %%%%%%%%%%%%%%%%%%%%
 %% To add new composition rules, go to the bottom of this file.
 %% A rule definition consists of three parts. The first one represents its origin,
@@ -12,7 +12,7 @@
 %% % example_rule_without_meaning
 %% property_a(X) :- property_b(X).
 %%%%%%%%%%%%%%%%%%%%
-% ==== src/test/resources/verifiedVotingRuleConstruction/theories/ - Verified_Voting_Rule_Construction
+% ==== src/test/resources/verifiedVotingRuleConstruction/theories/ROOT - Verified_Voting_Rule_Construction
 %
 % === COMPONENT_TYPE
 % == Rel.rel(?'a)
@@ -20,54 +20,54 @@
 % elimination_set(Evaluation_Function,Nat.nat,Nat.nat,Nat.nat,HOL.bool,Set.set(?'a),Profile)
 % == HOL.bool
 % == Electoral_Module
-% elimination_module(Evaluation_Function,Nat.nat,Nat.nat,Nat.nat,HOL.bool)
-% leq_average_eliminator(Evaluation_Function)
-% loop_composition(Electoral_Module,Termination_Condition)
-% minimax()
-% revision_composition(Electoral_Module)
-% leq_eliminator(Evaluation_Function,Nat.nat)
-% copeland()
-% nanson_baldwin_rule()
-% condorcet()
-% min_eliminator(Evaluation_Function)
-% sequential_composition(Electoral_Module,Electoral_Module)
-% blacks_rule()
-% minimax_rule()
-% maximum_parallel_composition(Electoral_Module,Electoral_Module)
 % iterelect(Electoral_Module)
-% max_eliminator(Evaluation_Function)
-% drop_module(Nat.nat,Preference_Relation)
-% condorcet'()
-% schwartz_rule()
-% borda()
+% pairwise_majority_rule'()
+% loop_comp_helper(Electoral_Module,Electoral_Module,Termination_Condition)
 % pass_module(Nat.nat,Preference_Relation)
-% less_eliminator(Evaluation_Function,Nat.nat)
+% schwartz_rule()
+% drop_module(Nat.nat,Preference_Relation)
+% sequential_composition(Electoral_Module,Electoral_Module)
+% elimination_module(Evaluation_Function,Nat.nat,Nat.nat,Nat.nat,HOL.bool)
 % borda_rule()
+% borda()
+% copeland_rule()
 % pairwise_majority_rule()
+% loop_composition(Electoral_Module,Termination_Condition)
 % plurality()
-% pairwise_majority_rule'()
+% leq_eliminator(Evaluation_Function,Nat.nat)
 % less_average_eliminator(Evaluation_Function)
-% smc(Preference_Relation)
-% classic_nanson_rule()
-% defer_module()
-% copeland_rule()
-% elector(Electoral_Module)
-% loop_comp_helper(Electoral_Module,Electoral_Module,Termination_Condition)
+% leq_average_eliminator(Evaluation_Function)
+% condorcet()
 % elect_module()
-% parallel_composition(Electoral_Module,Electoral_Module,Aggregator)
+% max_eliminator(Evaluation_Function)
+% maximum_parallel_composition(Electoral_Module,Electoral_Module)
 % iter(Electoral_Module)
+% parallel_composition(Electoral_Module,Electoral_Module,Aggregator)
+% minimax_rule()
+% elector(Electoral_Module)
+% defer_module()
+% minimax()
+% less_eliminator(Evaluation_Function,Nat.nat)
+% revision_composition(Electoral_Module)
+% blacks_rule()
+% classic_nanson_rule()
+% smc(Preference_Relation)
+% min_eliminator(Evaluation_Function)
+% condorcet'()
+% nanson_baldwin_rule()
+% copeland()
 % == Nat.nat
-% prefer_count_code(Profile,?'a,?'a)
+% average(Evaluation_Function,Set.set(?'a),Profile)
 % win_count_code(Profile,?'a)
-% win_count(Profile,?'a)
 % prefer_count(Profile,?'a,?'a)
 % rank(Preference_Relation,?'a)
-% average(Evaluation_Function,Set.set(?'a),Profile)
+% prefer_count_code(Profile,?'a,?'a)
+% win_count(Profile,?'a)
 % == Termination_Condition
-% set_equals_partition(Set.set(?'a))
-% disjoint3()
 % defer_equal_condition(Nat.nat)
+% set_equals_partition(Set.set(?'a))
 % well_formed(Set.set(?'a))
+% disjoint3()
 % == Aggregator
 % max_aggregator()
 % == ?'a
@@ -78,16 +78,17 @@
 % == Profile
 % limit_profile(Set.set(?'a),Profile)
 % == Evaluation_Function
-% condorcet_score()
-% borda_score()
 % copeland_score()
+% condorcet_score()
 % minimax_score()
+% borda_score()
 %
 % === PROPERTY
 % defer_condorcet_consistency(Electoral_Module)
 % wins(?'a,Profile,?'a)
 % is_less_preferred_than(?'a,Preference_Relation,?'a)
 % aggregator(Aggregator)
+% linear_order(Preference_Relation)
 % condorcet_compatibility(Electoral_Module)
 % defer_invariant_monotonicity(Electoral_Module)
 % indep_of_alt(Electoral_Module,Set.set(?'a),?'a)
@@ -99,6 +100,7 @@
 % condorcet_rating(Evaluation_Function)
 % invariant_monotonicity(Electoral_Module)
 % agg_conservative(Aggregator)
+% trans(Preference_Relation)
 % electoral_module(Electoral_Module)
 % connex(Set.set(?'a),Preference_Relation)
 % defers(Nat.nat,Electoral_Module)
@@ -110,6 +112,7 @@
 % defer_lift_invariance(Electoral_Module)
 % monotonicity(Electoral_Module)
 % prof_geq_result(Electoral_Module,Set.set(?'a),Profile,Profile,?'a)
+% custom_greater(Nat.nat,Nat.nat)
 % equiv_prof_except_a(Set.set(?'a),Profile,Profile,?'a)
 % homogeneity(Electoral_Module)
 % prof_leq_result(Electoral_Module,Set.set(?'a),Profile,Profile,?'a)
@@ -120,11 +123,13 @@
 % disjoint_compatibility(Electoral_Module,Electoral_Module)
 % elects(Nat.nat,Electoral_Module)
 % condorcet_consistency(Electoral_Module)
+% finite_profile(Set.set(?'a),Profile)
 % equiv_rel_except_a(Set.set(?'a),Preference_Relation,Preference_Relation,?'a)
 % condorcet_winner(Set.set(?'a),Profile,?'a)
 % linear_order(Preference_Relation)
 % trans(Preference_Relation)
 % finite_profile(Set.set(?'a),Profile)
+% custom_greater(Nat.nat,Nat.nat)
 %
 % === COMPOSITION_RULE
 % = Aggregator_Facts.thy
@@ -143,6 +148,26 @@
 % Preference_Relation.lin_imp_antisym
 antisym(R) :- linear_order_on(_,R).
 
+% = ASSUMPTION
+% condorcet_compatibility_alias_0
+condorcet_compatibility(iter(X)) :- condorcet_compatibility(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% condorcet_compatibility_alias_1
+condorcet_compatibility(iterelect(X)) :- condorcet_compatibility(elector(iter(X))).
+
+% = ASSUMPTION
+% condorcet_compatibility_alias_2
+condorcet_compatibility(maximum_parallel_composition(X,Y)) :- condorcet_compatibility(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% condorcet_compatibility_alias_3
+condorcet_compatibility(smc(X)) :- condorcet_compatibility(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% condorcet_compatibility_alias_4
+condorcet_compatibility(elector(X)) :- condorcet_compatibility(sequential_composition(X,elect_module)).
+
 % = Condorcet_Rules.thy
 % Condorcet_Rules.cr_eval_imp_ccomp_max_elim
 condorcet_compatibility(max_eliminator(E)) :- finite_profile(_,_),condorcet_rating(E).
@@ -163,6 +188,26 @@
 % Condorcet_Rules.dcc_imp_cc_elector
 condorcet_consistency(elector(M)) :- defer_condorcet_consistency(M).
 
+% = ASSUMPTION
+% condorcet_consistency_alias_0
+condorcet_consistency(iter(X)) :- condorcet_consistency(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% condorcet_consistency_alias_1
+condorcet_consistency(iterelect(X)) :- condorcet_consistency(elector(iter(X))).
+
+% = ASSUMPTION
+% condorcet_consistency_alias_2
+condorcet_consistency(maximum_parallel_composition(X,Y)) :- condorcet_consistency(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% condorcet_consistency_alias_3
+condorcet_consistency(smc(X)) :- condorcet_consistency(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% condorcet_consistency_alias_4
+condorcet_consistency(elector(X)) :- condorcet_consistency(sequential_composition(X,elect_module)).
+
 % = Condorcet_Facts.thy
 % Condorcet_Facts.minimax_score_cond_rating
 condorcet_rating(minimax_score).
@@ -187,6 +232,26 @@
 % Preference_Relation.lin_ord_imp_connex
 connex(A,R) :- linear_order_on(A,R).
 
+% = ASSUMPTION
+% decrementing_alias_0
+decrementing(iter(X)) :- decrementing(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% decrementing_alias_1
+decrementing(iterelect(X)) :- decrementing(elector(iter(X))).
+
+% = ASSUMPTION
+% decrementing_alias_2
+decrementing(maximum_parallel_composition(X,Y)) :- decrementing(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% decrementing_alias_3
+decrementing(smc(X)) :- decrementing(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% decrementing_alias_4
+decrementing(elector(X)) :- decrementing(sequential_composition(X,elect_module)).
+
 % = Condorcet_Facts.thy
 % Condorcet_Facts.copeland_is_dcc
 defer_condorcet_consistency(copeland).
@@ -203,14 +268,74 @@
 % Condorcet_Rules.cr_eval_imp_dcc_max_elim
 defer_condorcet_consistency(max_eliminator(E)) :- condorcet_rating(E).
 
+% = ASSUMPTION
+% defer_condorcet_consistency_alias_0
+defer_condorcet_consistency(iter(X)) :- defer_condorcet_consistency(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% defer_condorcet_consistency_alias_1
+defer_condorcet_consistency(iterelect(X)) :- defer_condorcet_consistency(elector(iter(X))).
+
+% = ASSUMPTION
+% defer_condorcet_consistency_alias_2
+defer_condorcet_consistency(maximum_parallel_composition(X,Y)) :- defer_condorcet_consistency(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% defer_condorcet_consistency_alias_3
+defer_condorcet_consistency(smc(X)) :- defer_condorcet_consistency(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% defer_condorcet_consistency_alias_4
+defer_condorcet_consistency(elector(X)) :- defer_condorcet_consistency(sequential_composition(X,elect_module)).
+
 % = Condorcet_Rules.thy
 % Condorcet_Rules.ccomp_and_dd_imp_dcc
 defer_condorcet_consistency(M) :- condorcet_compatibility(M),defer_deciding(M).
 
+% = ASSUMPTION
+% defer_deciding_alias_0
+defer_deciding(iter(X)) :- defer_deciding(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% defer_deciding_alias_1
+defer_deciding(iterelect(X)) :- defer_deciding(elector(iter(X))).
+
+% = ASSUMPTION
+% defer_deciding_alias_2
+defer_deciding(maximum_parallel_composition(X,Y)) :- defer_deciding(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% defer_deciding_alias_3
+defer_deciding(smc(X)) :- defer_deciding(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% defer_deciding_alias_4
+defer_deciding(elector(X)) :- defer_deciding(sequential_composition(X,elect_module)).
+
 % = Monotonicity_Rules.thy
 % Monotonicity_Rules.rev_comp_def_inv_mono
 defer_invariant_monotonicity(revision_composition(M)) :- invariant_monotonicity(M).
 
+% = ASSUMPTION
+% defer_invariant_monotonicity_alias_0
+defer_invariant_monotonicity(iter(X)) :- defer_invariant_monotonicity(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% defer_invariant_monotonicity_alias_1
+defer_invariant_monotonicity(iterelect(X)) :- defer_invariant_monotonicity(elector(iter(X))).
+
+% = ASSUMPTION
+% defer_invariant_monotonicity_alias_2
+defer_invariant_monotonicity(maximum_parallel_composition(X,Y)) :- defer_invariant_monotonicity(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% defer_invariant_monotonicity_alias_3
+defer_invariant_monotonicity(smc(X)) :- defer_invariant_monotonicity(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% defer_invariant_monotonicity_alias_4
+defer_invariant_monotonicity(elector(X)) :- defer_invariant_monotonicity(sequential_composition(X,elect_module)).
+
 % = Monotonicity_Facts.thy
 % Monotonicity_Facts.def_mod_def_lift_inv
 defer_lift_invariance(defer_module).
@@ -227,6 +352,26 @@
 % Monotonicity_Facts.drop_mod_def_lift_inv
 defer_lift_invariance(drop_module(_,R)) :- linear_order(R).
 
+% = ASSUMPTION
+% defer_lift_invariance_alias_0
+defer_lift_invariance(iter(X)) :- defer_lift_invariance(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% defer_lift_invariance_alias_1
+defer_lift_invariance(iterelect(X)) :- defer_lift_invariance(elector(iter(X))).
+
+% = ASSUMPTION
+% defer_lift_invariance_alias_2
+defer_lift_invariance(maximum_parallel_composition(X,Y)) :- defer_lift_invariance(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% defer_lift_invariance_alias_3
+defer_lift_invariance(smc(X)) :- defer_lift_invariance(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% defer_lift_invariance_alias_4
+defer_lift_invariance(elector(X)) :- defer_lift_invariance(sequential_composition(X,elect_module)).
+
 % = Monotonicity_Rules.thy
 % Monotonicity_Rules.seq_comp_presv_def_lift_inv
 defer_lift_invariance(sequential_composition(M,N)) :- defer_lift_invariance(M),defer_lift_invariance(N).
@@ -247,6 +392,26 @@
 % Monotonicity_Rules.dl_inv_imp_def_mono
 defer_monotonicity(M) :- defer_lift_invariance(M).
 
+% = ASSUMPTION
+% defer_monotonicity_alias_0
+defer_monotonicity(iter(X)) :- defer_monotonicity(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% defer_monotonicity_alias_1
+defer_monotonicity(iterelect(X)) :- defer_monotonicity(elector(iter(X))).
+
+% = ASSUMPTION
+% defer_monotonicity_alias_2
+defer_monotonicity(maximum_parallel_composition(X,Y)) :- defer_monotonicity(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% defer_monotonicity_alias_3
+defer_monotonicity(smc(X)) :- defer_monotonicity(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% defer_monotonicity_alias_4
+defer_monotonicity(elector(X)) :- defer_monotonicity(sequential_composition(X,elect_module)).
+
 % = Result_Facts.thy
 % Result_Facts.pass_zero_mod_def_zero
 defers(0,pass_module(0,R)) :- linear_order(R).
@@ -259,6 +424,26 @@
 % Result_Facts.pass_two_mod_def_two
 defers(2,pass_module(2,R)) :- linear_order(R).
 
+% = ASSUMPTION
+% defers_alias_0
+defers(ALIAS_VAR_0,iter(X)) :- defers(ALIAS_VAR_0,loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% defers_alias_1
+defers(ALIAS_VAR_0,iterelect(X)) :- defers(ALIAS_VAR_0,elector(iter(X))).
+
+% = ASSUMPTION
+% defers_alias_2
+defers(ALIAS_VAR_0,maximum_parallel_composition(X,Y)) :- defers(ALIAS_VAR_0,parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% defers_alias_3
+defers(ALIAS_VAR_0,smc(X)) :- defers(ALIAS_VAR_0,iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% defers_alias_4
+defers(ALIAS_VAR_0,elector(X)) :- defers(ALIAS_VAR_0,sequential_composition(X,elect_module)).
+
 % = Result_Rules.thy
 % Result_Rules.seq_comp_def_one
 defers(1,sequential_composition(M,N)) :- non_blocking(M),non_electing(M),defers(1,N).
@@ -275,6 +460,46 @@
 % Disjoint_Compatibility_Rules.disj_compat_comm
 disjoint_compatibility(N,M) :- disjoint_compatibility(M,N).
 
+% = ASSUMPTION
+% disjoint_compatibility_alias_0
+disjoint_compatibility(iter(X),ALIAS_VAR_1) :- disjoint_compatibility(loop_composition(X,defer_equal_condition(1)),ALIAS_VAR_1).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_1
+disjoint_compatibility(iterelect(X),ALIAS_VAR_1) :- disjoint_compatibility(elector(iter(X)),ALIAS_VAR_1).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_2
+disjoint_compatibility(maximum_parallel_composition(X,Y),ALIAS_VAR_1) :- disjoint_compatibility(parallel_composition(X,Y,max_aggregator),ALIAS_VAR_1).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_3
+disjoint_compatibility(smc(X),ALIAS_VAR_1) :- disjoint_compatibility(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X))),ALIAS_VAR_1).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_4
+disjoint_compatibility(elector(X),ALIAS_VAR_1) :- disjoint_compatibility(sequential_composition(X,elect_module),ALIAS_VAR_1).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_0
+disjoint_compatibility(ALIAS_VAR_0,iter(X)) :- disjoint_compatibility(ALIAS_VAR_0,loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_1
+disjoint_compatibility(ALIAS_VAR_0,iterelect(X)) :- disjoint_compatibility(ALIAS_VAR_0,elector(iter(X))).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_2
+disjoint_compatibility(ALIAS_VAR_0,maximum_parallel_composition(X,Y)) :- disjoint_compatibility(ALIAS_VAR_0,parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_3
+disjoint_compatibility(ALIAS_VAR_0,smc(X)) :- disjoint_compatibility(ALIAS_VAR_0,iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% disjoint_compatibility_alias_4
+disjoint_compatibility(ALIAS_VAR_0,elector(X)) :- disjoint_compatibility(ALIAS_VAR_0,sequential_composition(X,elect_module)).
+
 % = Result_Facts.thy
 % Result_Facts.plurality_electing
 electing(plurality).
@@ -287,6 +512,26 @@
 % Sequential_Majority_Comparison.smc_electing
 electing(smc(X)) :- linear_order(X).
 
+% = ASSUMPTION
+% electing_alias_0
+electing(iter(X)) :- electing(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% electing_alias_1
+electing(iterelect(X)) :- electing(elector(iter(X))).
+
+% = ASSUMPTION
+% electing_alias_2
+electing(maximum_parallel_composition(X,Y)) :- electing(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% electing_alias_3
+electing(smc(X)) :- electing(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% electing_alias_4
+electing(elector(X)) :- electing(sequential_composition(X,elect_module)).
+
 % = Result_Rules.thy
 % Result_Rules.elector_electing
 electing(elector(M)) :- electoral_module(M),non_blocking(M).
@@ -359,6 +604,26 @@
 % Revision_Composition.rev_comp_sound
 electoral_module(revision_composition(M)) :- electoral_module(M).
 
+% = ASSUMPTION
+% electoral_module_alias_0
+electoral_module(iter(X)) :- electoral_module(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% electoral_module_alias_1
+electoral_module(iterelect(X)) :- electoral_module(elector(iter(X))).
+
+% = ASSUMPTION
+% electoral_module_alias_2
+electoral_module(maximum_parallel_composition(X,Y)) :- electoral_module(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% electoral_module_alias_3
+electoral_module(smc(X)) :- electoral_module(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% electoral_module_alias_4
+electoral_module(elector(X)) :- electoral_module(sequential_composition(X,elect_module)).
+
 % = Composite_Structures.thy
 % Composite_Structures.max_par_comp_sound
 electoral_module(maximum_parallel_composition(M,N)) :- electoral_module(M),electoral_module(N).
@@ -371,6 +636,46 @@
 % Parallel_Composition.par_comp_sound
 electoral_module(parallel_composition(M,N,A)) :- electoral_module(M),electoral_module(N),aggregator(A).
 
+% = ASSUMPTION
+% elects_alias_0
+elects(ALIAS_VAR_0,iter(X)) :- elects(ALIAS_VAR_0,loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% elects_alias_1
+elects(ALIAS_VAR_0,iterelect(X)) :- elects(ALIAS_VAR_0,elector(iter(X))).
+
+% = ASSUMPTION
+% elects_alias_2
+elects(ALIAS_VAR_0,maximum_parallel_composition(X,Y)) :- elects(ALIAS_VAR_0,parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% elects_alias_3
+elects(ALIAS_VAR_0,smc(X)) :- elects(ALIAS_VAR_0,iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% elects_alias_4
+elects(ALIAS_VAR_0,elector(X)) :- elects(ALIAS_VAR_0,sequential_composition(X,elect_module)).
+
+% = ASSUMPTION
+% eliminates_alias_0
+eliminates(ALIAS_VAR_0,iter(X)) :- eliminates(ALIAS_VAR_0,loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% eliminates_alias_1
+eliminates(ALIAS_VAR_0,iterelect(X)) :- eliminates(ALIAS_VAR_0,elector(iter(X))).
+
+% = ASSUMPTION
+% eliminates_alias_2
+eliminates(ALIAS_VAR_0,maximum_parallel_composition(X,Y)) :- eliminates(ALIAS_VAR_0,parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% eliminates_alias_3
+eliminates(ALIAS_VAR_0,smc(X)) :- eliminates(ALIAS_VAR_0,iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% eliminates_alias_4
+eliminates(ALIAS_VAR_0,elector(X)) :- eliminates(ALIAS_VAR_0,sequential_composition(X,elect_module)).
+
 % = Result_Rules.thy
 % Result_Rules.par_comp_elim_one
 eliminates(1,maximum_parallel_composition(M,N)) :- defers(1,M),non_electing(M),rejects(2,N),disjoint_compatibility(M,N).
@@ -391,14 +696,78 @@
 % Preference_Relation.lifted_imp_equiv_rel_except_a
 equiv_rel_except_a(A,R,S,A) :- lifted(A,R,S,A).
 
+% = ASSUMPTION
+% finite_profile_intro
+finite_profile(_,_).
+
 % = Monotonicity_Rules.thy
 % Monotonicity_Rules.lifted_imp_fin_prof
 finite_profile(A,P) :- lifted(A,P,_,A).
 
+% = ASSUMPTION
+% homogeneity_alias_0
+homogeneity(iter(X)) :- homogeneity(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% homogeneity_alias_1
+homogeneity(iterelect(X)) :- homogeneity(elector(iter(X))).
+
+% = ASSUMPTION
+% homogeneity_alias_2
+homogeneity(maximum_parallel_composition(X,Y)) :- homogeneity(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% homogeneity_alias_3
+homogeneity(smc(X)) :- homogeneity(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% homogeneity_alias_4
+homogeneity(elector(X)) :- homogeneity(sequential_composition(X,elect_module)).
+
+% = ASSUMPTION
+% indep_of_alt_alias_0
+indep_of_alt(iter(X),ALIAS_VAR_1,ALIAS_VAR_2) :- indep_of_alt(loop_composition(X,defer_equal_condition(1)),ALIAS_VAR_1,ALIAS_VAR_2).
+
+% = ASSUMPTION
+% indep_of_alt_alias_1
+indep_of_alt(iterelect(X),ALIAS_VAR_1,ALIAS_VAR_2) :- indep_of_alt(elector(iter(X)),ALIAS_VAR_1,ALIAS_VAR_2).
+
+% = ASSUMPTION
+% indep_of_alt_alias_2
+indep_of_alt(maximum_parallel_composition(X,Y),ALIAS_VAR_1,ALIAS_VAR_2) :- indep_of_alt(parallel_composition(X,Y,max_aggregator),ALIAS_VAR_1,ALIAS_VAR_2).
+
+% = ASSUMPTION
+% indep_of_alt_alias_3
+indep_of_alt(smc(X),ALIAS_VAR_1,ALIAS_VAR_2) :- indep_of_alt(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X))),ALIAS_VAR_1,ALIAS_VAR_2).
+
+% = ASSUMPTION
+% indep_of_alt_alias_4
+indep_of_alt(elector(X),ALIAS_VAR_1,ALIAS_VAR_2) :- indep_of_alt(sequential_composition(X,elect_module),ALIAS_VAR_1,ALIAS_VAR_2).
+
 % = Monotonicity_Facts.thy
 % Monotonicity_Facts.plurality_inv_mono
 invariant_monotonicity(plurality).
 
+% = ASSUMPTION
+% invariant_monotonicity_alias_0
+invariant_monotonicity(iter(X)) :- invariant_monotonicity(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% invariant_monotonicity_alias_1
+invariant_monotonicity(iterelect(X)) :- invariant_monotonicity(elector(iter(X))).
+
+% = ASSUMPTION
+% invariant_monotonicity_alias_2
+invariant_monotonicity(maximum_parallel_composition(X,Y)) :- invariant_monotonicity(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% invariant_monotonicity_alias_3
+invariant_monotonicity(smc(X)) :- invariant_monotonicity(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% invariant_monotonicity_alias_4
+invariant_monotonicity(elector(X)) :- invariant_monotonicity(sequential_composition(X,elect_module)).
+
 % = ASSUMPTION
 % is_less_preferred_than_intro
 is_less_preferred_than(_,_,_).
@@ -419,6 +788,10 @@
 % limited_intro
 limited(_,_).
 
+% = ASSUMPTION
+% linear_order_intro
+linear_order(_).
+
 % = Preference_Relation.thy
 % Preference_Relation.connex_antsym_and_trans_imp_lin_ord
 linear_order_on(A,R) :- connex(A,R),antisym(R),trans(R).
@@ -427,10 +800,70 @@
 % Electoral_Module.mod_contains_result_comm
 mod_contains_result(N,M,A,P,A) :- mod_contains_result(M,N,A,P,A).
 
+% = ASSUMPTION
+% mod_contains_result_alias_0
+mod_contains_result(iter(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(loop_composition(X,defer_equal_condition(1)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_1
+mod_contains_result(iterelect(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(elector(iter(X)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_2
+mod_contains_result(maximum_parallel_composition(X,Y),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(parallel_composition(X,Y,max_aggregator),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_3
+mod_contains_result(smc(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X))),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_4
+mod_contains_result(elector(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(sequential_composition(X,elect_module),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_0
+mod_contains_result(ALIAS_VAR_0,iter(X),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(ALIAS_VAR_0,loop_composition(X,defer_equal_condition(1)),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_1
+mod_contains_result(ALIAS_VAR_0,iterelect(X),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(ALIAS_VAR_0,elector(iter(X)),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_2
+mod_contains_result(ALIAS_VAR_0,maximum_parallel_composition(X,Y),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(ALIAS_VAR_0,parallel_composition(X,Y,max_aggregator),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_3
+mod_contains_result(ALIAS_VAR_0,smc(X),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(ALIAS_VAR_0,iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X))),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% mod_contains_result_alias_4
+mod_contains_result(ALIAS_VAR_0,elector(X),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- mod_contains_result(ALIAS_VAR_0,sequential_composition(X,elect_module),ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
 % = Sequential_Majority_Comparison.thy
 % Sequential_Majority_Comparison.smc_monotone
 monotonicity(smc(X)) :- linear_order(X).
 
+% = ASSUMPTION
+% monotonicity_alias_0
+monotonicity(iter(X)) :- monotonicity(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% monotonicity_alias_1
+monotonicity(iterelect(X)) :- monotonicity(elector(iter(X))).
+
+% = ASSUMPTION
+% monotonicity_alias_2
+monotonicity(maximum_parallel_composition(X,Y)) :- monotonicity(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% monotonicity_alias_3
+monotonicity(smc(X)) :- monotonicity(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% monotonicity_alias_4
+monotonicity(elector(X)) :- monotonicity(sequential_composition(X,elect_module)).
+
 % = Monotonicity_Rules.thy
 % Monotonicity_Rules.seq_comp_mono
 monotonicity(sequential_composition(M,N)) :- defer_lift_invariance(M),non_electing(M),defers(1,M),electing(N).
@@ -439,6 +872,26 @@
 % Result_Rules.electing_imp_non_blocking
 non_blocking(M) :- electing(M).
 
+% = ASSUMPTION
+% non_blocking_alias_0
+non_blocking(iter(X)) :- non_blocking(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% non_blocking_alias_1
+non_blocking(iterelect(X)) :- non_blocking(elector(iter(X))).
+
+% = ASSUMPTION
+% non_blocking_alias_2
+non_blocking(maximum_parallel_composition(X,Y)) :- non_blocking(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% non_blocking_alias_3
+non_blocking(smc(X)) :- non_blocking(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% non_blocking_alias_4
+non_blocking(elector(X)) :- non_blocking(sequential_composition(X,elect_module)).
+
 % = Result_Rules.thy
 % Result_Rules.seq_comp_presv_non_blocking
 non_blocking(sequential_composition(M,N)) :- non_blocking(M),non_blocking(N).
@@ -491,6 +944,26 @@
 % Result_Facts.leq_avg_elim_non_electing
 non_electing(leq_average_eliminator(_)) :- finite_profile(_,_).
 
+% = ASSUMPTION
+% non_electing_alias_0
+non_electing(iter(X)) :- non_electing(loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% non_electing_alias_1
+non_electing(iterelect(X)) :- non_electing(elector(iter(X))).
+
+% = ASSUMPTION
+% non_electing_alias_2
+non_electing(maximum_parallel_composition(X,Y)) :- non_electing(parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% non_electing_alias_3
+non_electing(smc(X)) :- non_electing(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% non_electing_alias_4
+non_electing(elector(X)) :- non_electing(sequential_composition(X,elect_module)).
+
 % = Result_Rules.thy
 % Result_Rules.seq_comp_presv_non_electing
 non_electing(sequential_composition(M,N)) :- non_electing(M),non_electing(N).
@@ -503,6 +976,66 @@
 % Result_Rules.conserv_agg_presv_non_electing
 non_electing(parallel_composition(M,N,A)) :- non_electing(M),non_electing(N),agg_conservative(A).
 
+% = ASSUMPTION
+% prof_contains_result_alias_0
+prof_contains_result(iter(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_contains_result(loop_composition(X,defer_equal_condition(1)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_contains_result_alias_1
+prof_contains_result(iterelect(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_contains_result(elector(iter(X)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_contains_result_alias_2
+prof_contains_result(maximum_parallel_composition(X,Y),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_contains_result(parallel_composition(X,Y,max_aggregator),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_contains_result_alias_3
+prof_contains_result(smc(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_contains_result(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X))),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_contains_result_alias_4
+prof_contains_result(elector(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_contains_result(sequential_composition(X,elect_module),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_geq_result_alias_0
+prof_geq_result(iter(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_geq_result(loop_composition(X,defer_equal_condition(1)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_geq_result_alias_1
+prof_geq_result(iterelect(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_geq_result(elector(iter(X)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_geq_result_alias_2
+prof_geq_result(maximum_parallel_composition(X,Y),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_geq_result(parallel_composition(X,Y,max_aggregator),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_geq_result_alias_3
+prof_geq_result(smc(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_geq_result(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X))),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_geq_result_alias_4
+prof_geq_result(elector(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_geq_result(sequential_composition(X,elect_module),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_leq_result_alias_0
+prof_leq_result(iter(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_leq_result(loop_composition(X,defer_equal_condition(1)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_leq_result_alias_1
+prof_leq_result(iterelect(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_leq_result(elector(iter(X)),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_leq_result_alias_2
+prof_leq_result(maximum_parallel_composition(X,Y),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_leq_result(parallel_composition(X,Y,max_aggregator),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_leq_result_alias_3
+prof_leq_result(smc(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_leq_result(iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X))),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
+% = ASSUMPTION
+% prof_leq_result_alias_4
+prof_leq_result(elector(X),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4) :- prof_leq_result(sequential_composition(X,elect_module),ALIAS_VAR_1,ALIAS_VAR_2,ALIAS_VAR_3,ALIAS_VAR_4).
+
 % = ASSUMPTION
 % profile_intro
 profile(_,_).
@@ -519,10 +1052,34 @@
 % Result_Facts.drop_zero_mod_rej_zero
 rejects(0,drop_module(0,R)) :- linear_order(R).
 
+% = ASSUMPTION
+% rejects_alias_0
+rejects(ALIAS_VAR_0,iter(X)) :- rejects(ALIAS_VAR_0,loop_composition(X,defer_equal_condition(1))).
+
+% = ASSUMPTION
+% rejects_alias_1
+rejects(ALIAS_VAR_0,iterelect(X)) :- rejects(ALIAS_VAR_0,elector(iter(X))).
+
+% = ASSUMPTION
+% rejects_alias_2
+rejects(ALIAS_VAR_0,maximum_parallel_composition(X,Y)) :- rejects(ALIAS_VAR_0,parallel_composition(X,Y,max_aggregator)).
+
+% = ASSUMPTION
+% rejects_alias_3
+rejects(ALIAS_VAR_0,smc(X)) :- rejects(ALIAS_VAR_0,iterelect(maximum_parallel_composition(sequential_composition(pass_module(2,X),sequential_composition(revision_composition(plurality),pass_module(1,X))),drop_module(2,X)))).
+
+% = ASSUMPTION
+% rejects_alias_4
+rejects(ALIAS_VAR_0,elector(X)) :- rejects(ALIAS_VAR_0,sequential_composition(X,elect_module)).
+
 % = Sequential_Composition.thy
 % Sequential_Composition.seq_comp_presv_alts
 set_equals_partition(A,sequential_composition(M,N,A,P)) :- electoral_module(M),electoral_module(N),finite_profile(A,P).
 
+% = ASSUMPTION
+% trans_intro
+trans(_).
+
 % = Preference_Relation.thy
 % Preference_Relation.lin_imp_trans
 trans(R) :- linear_order_on(_,R).
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document.pdf b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document.pdf
index 118aab4..4ab7431 100644
Binary files a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document.pdf and b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document.pdf differ
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator.tex
deleted file mode 100644
index b743c4d..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator.tex
+++ /dev/null
@@ -1,133 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Aggregator}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-\isanewline
-%
-\isamarkupsection{Aggregator%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Aggregator\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Result{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-An aggregator gets two partitions (results of electoral modules) as input and
-output another partition. They are used to aggregate results of parallel
-composed electoral modules.
-They are commutative, i.e., the order of the aggregated modules does not affect
-the resulting aggregation. Moreover, they are conservative in the sense that
-the resulting decisions are subsets of the two given partitions' decisions.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Aggregator\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ aggregator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}aggregator\ agg\ {\isasymequiv}\isanewline
-\ \ \ \ {\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Properties%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Aggregator.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%13=6%:%
-%:%15=9%:%
-%:%31=11%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%48=21%:%
-%:%57=23%:%
-%:%67=25%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%82=33%:%
-%:%98=36%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator_Facts.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator_Facts.tex
deleted file mode 100644
index bbc60bb..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator_Facts.tex
+++ /dev/null
@@ -1,258 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Aggregator{\isacharunderscore}{\kern0pt}Facts}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Aggregator{\isacharunderscore}{\kern0pt}Facts\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Aggregator{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Maximum{\isacharunderscore}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}comm{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}commutative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}commutative{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}aggregator\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}consv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ subset{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Composition_Rules/Aggregator_Facts.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%20=5%:%
-%:%23=6%:%
-%:%24=7%:%
-%:%25=8%:%
-%:%26=8%:%
-%:%29=9%:%
-%:%33=9%:%
-%:%34=9%:%
-%:%35=10%:%
-%:%36=10%:%
-%:%37=11%:%
-%:%38=11%:%
-%:%39=12%:%
-%:%40=12%:%
-%:%41=13%:%
-%:%42=13%:%
-%:%43=14%:%
-%:%44=14%:%
-%:%45=15%:%
-%:%46=16%:%
-%:%47=17%:%
-%:%48=18%:%
-%:%49=19%:%
-%:%50=20%:%
-%:%51=21%:%
-%:%52=22%:%
-%:%53=22%:%
-%:%54=23%:%
-%:%55=24%:%
-%:%56=25%:%
-%:%57=25%:%
-%:%58=26%:%
-%:%64=26%:%
-%:%67=27%:%
-%:%68=28%:%
-%:%69=29%:%
-%:%70=30%:%
-%:%71=30%:%
-%:%78=31%:%
-%:%79=31%:%
-%:%80=32%:%
-%:%81=32%:%
-%:%82=33%:%
-%:%84=35%:%
-%:%85=36%:%
-%:%86=36%:%
-%:%87=37%:%
-%:%88=37%:%
-%:%89=38%:%
-%:%90=38%:%
-%:%91=39%:%
-%:%93=41%:%
-%:%94=42%:%
-%:%95=42%:%
-%:%96=43%:%
-%:%97=43%:%
-%:%98=43%:%
-%:%99=44%:%
-%:%101=46%:%
-%:%102=47%:%
-%:%103=47%:%
-%:%104=48%:%
-%:%105=48%:%
-%:%106=48%:%
-%:%107=49%:%
-%:%110=52%:%
-%:%111=53%:%
-%:%112=53%:%
-%:%113=54%:%
-%:%114=54%:%
-%:%115=54%:%
-%:%116=55%:%
-%:%118=57%:%
-%:%119=58%:%
-%:%120=58%:%
-%:%121=59%:%
-%:%122=59%:%
-%:%123=59%:%
-%:%124=60%:%
-%:%128=64%:%
-%:%129=65%:%
-%:%130=65%:%
-%:%131=66%:%
-%:%132=66%:%
-%:%133=67%:%
-%:%134=67%:%
-%:%135=68%:%
-%:%141=68%:%
-%:%146=69%:%
-%:%151=70%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator_Properties.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator_Properties.tex
deleted file mode 100644
index 4ff5f97..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Aggregator_Properties.tex
+++ /dev/null
@@ -1,74 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Aggregator{\isacharunderscore}{\kern0pt}Properties}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Aggregator{\isacharunderscore}{\kern0pt}Properties\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}commutative\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}commutative\ agg\ {\isasymequiv}\isanewline
-\ \ \ \ aggregator\ agg\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}conservative\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ agg\ {\isasymequiv}\isanewline
-\ \ \ \ aggregator\ agg\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ e{\isadigit{1}}\ e{\isadigit{2}}\ d{\isadigit{1}}\ d{\isadigit{2}}\ r{\isadigit{1}}\ r{\isadigit{2}}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}r{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}agg\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Properties/Aggregator_Properties.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%19=4%:%
-%:%22=5%:%
-%:%23=6%:%
-%:%24=6%:%
-%:%25=7%:%
-%:%27=9%:%
-%:%28=10%:%
-%:%29=11%:%
-%:%30=11%:%
-%:%31=12%:%
-%:%37=18%:%
-%:%40=19%:%
-%:%45=20%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Blacks_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Blacks_Rule.tex
deleted file mode 100644
index 1a139c4..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Blacks_Rule.tex
+++ /dev/null
@@ -1,103 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Blacks{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Black's Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Blacks{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ Pairwise{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \ \ \ \ \ \ \ \ Borda{\isacharunderscore}{\kern0pt}Rule\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is Black's voting rule. It is composed of a function that determines
-the Condorcet winner, i.e., the Pairwise Majority rule, and the Borda rule.
-Whenever there exists no Condorcet winner, it elects the choice made by the
-Borda rule, otherwise the Condorcet winner is elected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ blacks{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}blacks{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule\ {\isasymtriangleright}\ borda{\isacharunderscore}{\kern0pt}rule{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Voting_Rules/Blacks_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%54=20%:%
-%:%64=22%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%69=24%:%
-%:%74=25%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Borda_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Borda_Rule.tex
deleted file mode 100644
index b4d0fc7..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Borda_Rule.tex
+++ /dev/null
@@ -1,110 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Borda{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupchapter{Voting Rules%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Borda Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Borda{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Borda rule. On each ballot, each alternative is assigned a score
-that depends on how many alternatives are ranked below. The sum of all such
-scores for an alternative is hence called their Borda score. The alternative
-with the highest Borda score is elected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ borda{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ borda\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Voting_Rules/Borda_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%17=9%:%
-%:%33=11%:%
-%:%34=11%:%
-%:%35=12%:%
-%:%36=13%:%
-%:%37=14%:%
-%:%38=15%:%
-%:%47=18%:%
-%:%48=19%:%
-%:%49=20%:%
-%:%50=21%:%
-%:%59=23%:%
-%:%69=25%:%
-%:%70=25%:%
-%:%71=26%:%
-%:%74=27%:%
-%:%79=28%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Classic_Nanson_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Classic_Nanson_Rule.tex
deleted file mode 100644
index 2062cfd..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Classic_Nanson_Rule.tex
+++ /dev/null
@@ -1,107 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Classic{\isacharunderscore}{\kern0pt}Nanson{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Classic Nanson Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Classic{\isacharunderscore}{\kern0pt}Nanson{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the classic Nanson's voting rule, i.e., the rule that was originally
-invented by Nanson, but not the Nanson-Baldwin rule. The idea is similar,
-however, as alternatives with a Borda score less or equal than the average
-Borda score are excluded. The Borda scores of the remaining alternatives
-are hence adjusted to the new set of (still) eligible alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ classic{\isacharunderscore}{\kern0pt}nanson{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}classic{\isacharunderscore}{\kern0pt}nanson{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Voting_Rules/Classic_Nanson_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%55=21%:%
-%:%65=23%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%71=26%:%
-%:%76=27%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Composite_Elimination_Modules.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Composite_Elimination_Modules.tex
deleted file mode 100644
index d7b2dff..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Composite_Elimination_Modules.tex
+++ /dev/null
@@ -1,1019 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Borda Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-This is the Borda module used by the Borda rule. The Borda rule is a voting
-rule, where on each ballot, each alternative is assigned a score that depends
-on how many alternatives are ranked below. The sum of all such scores for an
-alternative is hence called their Borda score. The alternative with the highest
-Borda score is elected. The module implemented herein only rejects the
-alternatives not elected by the voting rule, and defers the alternatives that
-would be elected by the full voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ borda{\isacharunderscore}{\kern0pt}score\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ borda\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}borda\ A\ p\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Condorcet Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-This is the Condorcet module used by the Condorcet (voting) rule. The Condorcet
-rule is a voting rule that implements the Condorcet criterion, i.e., it elects
-the Condorcet winner if it exists, otherwise a tie remains between all
-alternatives. The module implemented herein only rejects the alternatives not
-elected by the voting rule, and defers the alternatives that would be elected
-by the full voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}score\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ x{\isacharparenright}{\kern0pt}\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ condorcet\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Copeland Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-This is the Copeland module used by the Copeland voting rule. The Copeland
-rule elects the alternatives with the highest difference between the amount
-of simple-majority wins and the amount of simple-majority losses. The module
-implemented herein only rejects the alternatives not elected by the voting
-rule, and defers the alternatives that would be elected by the full voting
-rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}score\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ x\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ y\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ copeland\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland\ A\ p\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}eliminator\ copeland{\isacharunderscore}{\kern0pt}score\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}M\ {\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ M\ {\isachardot}{\kern0pt}\ True{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ M{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}singleton\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ amount{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}A{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ {\isasymnot}\ wins\ x\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}irreflex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}M\ {\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ M\ {\isachardot}{\kern0pt}\ False{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ amount{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disjunct{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isasyminter}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ union{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finiteness{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finiteness{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ finiteness{\isadigit{1}}\ finiteness{\isadigit{2}}\ disjunct\ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}disjoint\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ union\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ amount{\isadigit{1}}\ amount{\isadigit{2}}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}loss{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}eq\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}iff\ wins{\isacharunderscore}{\kern0pt}antisym\ winner\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}copeland{\isacharunderscore}{\kern0pt}score{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}score\ w\ A\ p\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}loss{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \ \ \ \ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\ winner\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ a\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}loss{\isacharunderscore}{\kern0pt}count\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}\ {\isacharminus}{\kern0pt}\ {\isadigit{0}}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ f{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ loser{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}{\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner\ loser\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}wins\ w\ p\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ l\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ l\ p\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}irreflex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ M\ f\ {\isachardot}{\kern0pt}\ finite\ M\ {\isasymlongrightarrow}\ card\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ M\ {\isachardot}{\kern0pt}\ f\ x{\isacharbraceright}{\kern0pt}\ {\isasymle}\ card\ M{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ finite{\isacharunderscore}{\kern0pt}Diff\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{3}}\ {\isadigit{4}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{5}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isasymle}\isanewline
-\ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card{\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card{\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharcomma}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ loser{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{5}}\ {\isadigit{2}}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Minimax Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-This is the Minimax module used by the Minimax voting rule. The Minimax rule
-elects the alternatives with the highest Minimax score. The module implemented
-herein only rejects the alternatives not elected by the voting rule, and defers
-the alternatives that would be elected by the full voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}score\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}score\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ minimax\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax\ A\ p\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}eliminator\ minimax{\isacharunderscore}{\kern0pt}score\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemma%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}minimax{\isacharunderscore}{\kern0pt}score{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}score\ l\ A\ p\ {\isasymle}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isacharquery}{\kern0pt}set\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isacharquery}{\kern0pt}lscore\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}score\ l\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prof\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}set\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}lscore\ {\isacharequal}{\kern0pt}\ Min\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}lscore\ {\isasymin}\ {\isacharquery}{\kern0pt}set\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ {\isasymin}\ {\isacharquery}{\kern0pt}set{\isachardot}{\kern0pt}\ {\isacharquery}{\kern0pt}lscore\ {\isasymle}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ local{\isachardot}{\kern0pt}finite\ not{\isacharunderscore}{\kern0pt}empty\ Min{\isacharunderscore}{\kern0pt}le\ Min{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Composites/Composite_Elimination_Modules.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%16=6%:%
-%:%30=8%:%
-%:%42=11%:%
-%:%43=12%:%
-%:%44=13%:%
-%:%45=14%:%
-%:%46=15%:%
-%:%47=16%:%
-%:%48=17%:%
-%:%57=19%:%
-%:%67=21%:%
-%:%68=21%:%
-%:%69=22%:%
-%:%70=23%:%
-%:%71=24%:%
-%:%72=24%:%
-%:%73=25%:%
-%:%80=27%:%
-%:%92=30%:%
-%:%93=31%:%
-%:%94=32%:%
-%:%95=33%:%
-%:%96=34%:%
-%:%97=35%:%
-%:%106=37%:%
-%:%116=39%:%
-%:%117=39%:%
-%:%118=40%:%
-%:%119=41%:%
-%:%120=42%:%
-%:%121=43%:%
-%:%122=43%:%
-%:%123=44%:%
-%:%130=46%:%
-%:%142=49%:%
-%:%143=50%:%
-%:%144=51%:%
-%:%145=52%:%
-%:%146=53%:%
-%:%147=54%:%
-%:%156=56%:%
-%:%166=58%:%
-%:%167=58%:%
-%:%168=59%:%
-%:%169=60%:%
-%:%170=61%:%
-%:%171=62%:%
-%:%172=62%:%
-%:%173=63%:%
-%:%180=65%:%
-%:%190=68%:%
-%:%191=68%:%
-%:%192=69%:%
-%:%193=70%:%
-%:%200=71%:%
-%:%201=71%:%
-%:%202=72%:%
-%:%203=72%:%
-%:%204=73%:%
-%:%205=73%:%
-%:%206=74%:%
-%:%207=74%:%
-%:%208=75%:%
-%:%209=75%:%
-%:%210=76%:%
-%:%211=76%:%
-%:%212=77%:%
-%:%213=77%:%
-%:%214=78%:%
-%:%215=78%:%
-%:%216=79%:%
-%:%217=79%:%
-%:%218=80%:%
-%:%219=80%:%
-%:%220=81%:%
-%:%221=81%:%
-%:%222=82%:%
-%:%223=82%:%
-%:%224=83%:%
-%:%225=83%:%
-%:%226=84%:%
-%:%227=84%:%
-%:%228=85%:%
-%:%229=85%:%
-%:%230=86%:%
-%:%231=86%:%
-%:%232=87%:%
-%:%233=87%:%
-%:%234=88%:%
-%:%235=88%:%
-%:%236=89%:%
-%:%237=90%:%
-%:%238=90%:%
-%:%239=91%:%
-%:%240=91%:%
-%:%241=92%:%
-%:%242=92%:%
-%:%243=93%:%
-%:%244=93%:%
-%:%245=94%:%
-%:%246=94%:%
-%:%247=95%:%
-%:%248=95%:%
-%:%249=96%:%
-%:%250=96%:%
-%:%251=97%:%
-%:%252=97%:%
-%:%253=98%:%
-%:%254=98%:%
-%:%255=99%:%
-%:%256=99%:%
-%:%257=100%:%
-%:%258=100%:%
-%:%259=101%:%
-%:%260=101%:%
-%:%261=102%:%
-%:%262=103%:%
-%:%263=103%:%
-%:%264=104%:%
-%:%265=104%:%
-%:%266=105%:%
-%:%267=106%:%
-%:%268=107%:%
-%:%269=107%:%
-%:%270=108%:%
-%:%271=108%:%
-%:%272=109%:%
-%:%273=109%:%
-%:%274=110%:%
-%:%275=110%:%
-%:%276=111%:%
-%:%277=111%:%
-%:%278=112%:%
-%:%279=112%:%
-%:%280=113%:%
-%:%281=113%:%
-%:%282=114%:%
-%:%283=114%:%
-%:%284=115%:%
-%:%285=115%:%
-%:%286=116%:%
-%:%287=117%:%
-%:%288=118%:%
-%:%289=118%:%
-%:%290=119%:%
-%:%291=119%:%
-%:%292=120%:%
-%:%293=120%:%
-%:%294=121%:%
-%:%295=122%:%
-%:%296=122%:%
-%:%297=123%:%
-%:%298=123%:%
-%:%299=124%:%
-%:%300=124%:%
-%:%301=125%:%
-%:%302=125%:%
-%:%303=126%:%
-%:%309=126%:%
-%:%312=127%:%
-%:%313=128%:%
-%:%314=129%:%
-%:%315=129%:%
-%:%316=130%:%
-%:%317=131%:%
-%:%320=132%:%
-%:%324=132%:%
-%:%325=132%:%
-%:%326=133%:%
-%:%327=134%:%
-%:%328=134%:%
-%:%333=134%:%
-%:%336=135%:%
-%:%337=136%:%
-%:%338=137%:%
-%:%339=137%:%
-%:%340=138%:%
-%:%341=139%:%
-%:%344=140%:%
-%:%348=140%:%
-%:%349=140%:%
-%:%350=141%:%
-%:%351=141%:%
-%:%352=142%:%
-%:%353=142%:%
-%:%354=143%:%
-%:%355=144%:%
-%:%356=145%:%
-%:%357=145%:%
-%:%358=146%:%
-%:%359=147%:%
-%:%360=147%:%
-%:%361=148%:%
-%:%362=148%:%
-%:%363=149%:%
-%:%364=149%:%
-%:%365=150%:%
-%:%366=150%:%
-%:%367=151%:%
-%:%368=151%:%
-%:%369=152%:%
-%:%370=152%:%
-%:%371=153%:%
-%:%372=153%:%
-%:%373=154%:%
-%:%374=154%:%
-%:%375=155%:%
-%:%376=155%:%
-%:%377=156%:%
-%:%378=156%:%
-%:%379=157%:%
-%:%380=157%:%
-%:%381=158%:%
-%:%382=158%:%
-%:%383=159%:%
-%:%384=159%:%
-%:%385=160%:%
-%:%391=160%:%
-%:%394=161%:%
-%:%395=165%:%
-%:%396=166%:%
-%:%397=166%:%
-%:%398=167%:%
-%:%399=168%:%
-%:%400=169%:%
-%:%401=170%:%
-%:%402=171%:%
-%:%409=172%:%
-%:%410=172%:%
-%:%411=173%:%
-%:%412=173%:%
-%:%413=174%:%
-%:%414=174%:%
-%:%415=175%:%
-%:%416=175%:%
-%:%417=176%:%
-%:%418=176%:%
-%:%419=177%:%
-%:%420=177%:%
-%:%421=178%:%
-%:%422=178%:%
-%:%423=179%:%
-%:%424=179%:%
-%:%425=180%:%
-%:%426=180%:%
-%:%427=180%:%
-%:%428=181%:%
-%:%429=182%:%
-%:%430=183%:%
-%:%431=183%:%
-%:%432=184%:%
-%:%433=184%:%
-%:%434=185%:%
-%:%435=185%:%
-%:%436=186%:%
-%:%437=186%:%
-%:%438=187%:%
-%:%439=187%:%
-%:%440=188%:%
-%:%441=188%:%
-%:%442=189%:%
-%:%443=189%:%
-%:%444=189%:%
-%:%445=190%:%
-%:%446=191%:%
-%:%447=192%:%
-%:%448=192%:%
-%:%449=193%:%
-%:%450=193%:%
-%:%451=194%:%
-%:%452=194%:%
-%:%453=195%:%
-%:%454=195%:%
-%:%455=196%:%
-%:%456=196%:%
-%:%457=197%:%
-%:%458=197%:%
-%:%459=198%:%
-%:%460=198%:%
-%:%461=199%:%
-%:%462=199%:%
-%:%463=200%:%
-%:%464=200%:%
-%:%465=201%:%
-%:%466=201%:%
-%:%467=202%:%
-%:%468=202%:%
-%:%469=203%:%
-%:%470=203%:%
-%:%471=204%:%
-%:%486=206%:%
-%:%498=209%:%
-%:%499=210%:%
-%:%500=211%:%
-%:%501=212%:%
-%:%510=214%:%
-%:%520=216%:%
-%:%521=216%:%
-%:%522=217%:%
-%:%523=218%:%
-%:%524=219%:%
-%:%525=220%:%
-%:%526=220%:%
-%:%527=221%:%
-%:%534=223%:%
-%:%544=225%:%
-%:%545=225%:%
-%:%546=226%:%
-%:%547=227%:%
-%:%548=228%:%
-%:%549=229%:%
-%:%550=230%:%
-%:%551=231%:%
-%:%558=232%:%
-%:%559=232%:%
-%:%560=233%:%
-%:%561=233%:%
-%:%562=234%:%
-%:%563=235%:%
-%:%564=236%:%
-%:%565=236%:%
-%:%566=237%:%
-%:%567=237%:%
-%:%568=238%:%
-%:%569=238%:%
-%:%570=239%:%
-%:%571=239%:%
-%:%572=240%:%
-%:%573=240%:%
-%:%574=241%:%
-%:%575=241%:%
-%:%576=242%:%
-%:%577=242%:%
-%:%578=243%:%
-%:%579=243%:%
-%:%580=244%:%
-%:%581=244%:%
-%:%582=245%:%
-%:%583=245%:%
-%:%584=246%:%
-%:%585=246%:%
-%:%586=247%:%
-%:%587=247%:%
-%:%588=248%:%
-%:%589=248%:%
-%:%590=249%:%
-%:%591=249%:%
-%:%592=250%:%
-%:%593=250%:%
-%:%594=251%:%
-%:%595=251%:%
-%:%596=252%:%
-%:%596=253%:%
-%:%597=254%:%
-%:%598=254%:%
-%:%599=255%:%
-%:%600=255%:%
-%:%601=256%:%
-%:%602=256%:%
-%:%603=257%:%
-%:%604=257%:%
-%:%605=258%:%
-%:%606=258%:%
-%:%607=259%:%
-%:%608=259%:%
-%:%609=260%:%
-%:%610=260%:%
-%:%611=261%:%
-%:%612=261%:%
-%:%613=262%:%
-%:%619=262%:%
-%:%624=263%:%
-%:%629=264%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Composite_Structures.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Composite_Structures.tex
deleted file mode 100644
index fe948a3..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Composite_Structures.tex
+++ /dev/null
@@ -1,1266 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Composite{\isacharunderscore}{\kern0pt}Structures}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Composite{\isacharunderscore}{\kern0pt}Structures\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Maximum{\isacharunderscore}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}Equal{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Sequential{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Parallel{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Loop{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Aggregator{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Aggregator{\isacharunderscore}{\kern0pt}Facts{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Elect Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-The elect composition sequences an electoral module and the elect
-module. It finalizes the module's decision as it simply elects all their
-non-rejected alternatives. Thereby, any such elect-composed module induces
-a proper voting rule in the social choice sense, as all alternatives are either
-rejected or elected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ elector\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elector\ m\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ elect{\isacharunderscore}{\kern0pt}module{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ elector{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Defer One Loop Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-This is a family of loop compositions. It uses the same module in sequence
-until either no new decisions are made or only one alternative is remaining
-in the defer-set. The second family herein uses the above family and
-subsequently elects the remaining alternative.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ iter\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}iter\ m\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}let\ t\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isadigit{1}}\ in\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}loop\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}{\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d\ {\isasymequiv}\ iter\ m{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ iterelect\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}iterelect\ m\ {\isacharequal}{\kern0pt}\ elector\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsection{Maximum Parallel Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\begin{isamarkuptext}%
-This is a family of parallel compositions. It composes a new electoral module
-from two electoral modules combined with the maximum aggregator. Therein, the
-two modules each make a decision and then a partition is returned where every
-alternative receives the maximum result of the two input partitions. This means
-that, if any alternative is elected by at least one of the modules, then it
-gets elected, if any non-elected alternative is deferred by at least one of the
-modules, then it gets deferred, only alternatives rejected by both modules get
-rejected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition\ m\ n\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}let\ a\ {\isacharequal}{\kern0pt}\ max{\isacharunderscore}{\kern0pt}aggregator\ in\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}parallel\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}\isakeyword{infix}\ {\isachardoublequoteopen}{\isasymparallel}\isactrlsub {\isasymup}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition\ m\ n{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}m\ mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}result{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ A\ p\ x\ {\isasymor}\isanewline
-\ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ x\ {\isasymin}\ e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ Un{\isacharunderscore}{\kern0pt}iff\ a{\isadigit{1}}\ empty{\isacharunderscore}{\kern0pt}iff\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ DiffD{\isadigit{1}}\ DiffD{\isadigit{2}}\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardot}{\kern0pt}simps\ Un{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}conv\ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ prod{\isachardot}{\kern0pt}collapse\ f{\isacharunderscore}{\kern0pt}prof\ sndI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Int{\isacharunderscore}{\kern0pt}iff\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}a{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ a{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ in{\isacharunderscore}{\kern0pt}A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ f{\isacharunderscore}{\kern0pt}prof\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ fst\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ case{\isacharunderscore}{\kern0pt}prod{\isacharunderscore}{\kern0pt}unfold\ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ surjective{\isacharunderscore}{\kern0pt}pairing\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}iff\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}consv\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\ not{\isacharunderscore}{\kern0pt}a{\isadigit{1}}\ not{\isacharunderscore}{\kern0pt}a{\isadigit{2}}\ Int{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}both{\isacharunderscore}{\kern0pt}reject{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ fst\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ case{\isacharunderscore}{\kern0pt}prodI{\isadigit{2}}\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ split{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ prod{\isachardot}{\kern0pt}collapse\ split{\isacharunderscore}{\kern0pt}beta\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}iff\ a\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set\ module{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n\ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ surjective{\isacharunderscore}{\kern0pt}pairing\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p\ {\isasymand}\ x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ elect\ m\ A\ p\ {\isasymand}\ x\ {\isasymnotin}\ defer\ m\ A\ p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymnotin}\ elect\ n\ A\ p\ {\isasymand}\ x\ {\isasymnotin}\ defer\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ empty{\isacharunderscore}{\kern0pt}iff\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ DiffD{\isadigit{1}}\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardot}{\kern0pt}simps\ Un{\isacharunderscore}{\kern0pt}iff\ a\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ prod{\isachardot}{\kern0pt}simps\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ old{\isachardot}{\kern0pt}prod{\isachardot}{\kern0pt}inject\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ prod{\isachardot}{\kern0pt}collapse\ parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Set{\isachardot}{\kern0pt}set{\isacharunderscore}{\kern0pt}insert\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}insert\isanewline
-\ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}result\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}both{\isacharunderscore}{\kern0pt}reject\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ rejected\isanewline
-\ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ n\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ rejected\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}result\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}both{\isacharunderscore}{\kern0pt}reject\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ rejected\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\isanewline
-\ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ rejected\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}intersect{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ A\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n\ A\ p\ in\isanewline
-\ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dcompat{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}by{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ DiffI\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ x\ y{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}sum{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ S\ {\isacharplus}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ compatible\ \isacommand{obtain}\isamarkupfalse%
-\ A\ \isakeyword{where}\ A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ x\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ x\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ y\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ compatible\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}representation{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}intersect\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ x\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ subsets{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ S\ {\isasymand}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ finite\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}subset\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ S\ {\isacharplus}{\kern0pt}\ n\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}Int\ reject{\isacharunderscore}{\kern0pt}representation\ reject{\isacharunderscore}{\kern0pt}sum\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ S{\isachardot}{\kern0pt}\ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ x\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ y\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ subsets\ subset{\isacharunderscore}{\kern0pt}eq\ sup{\isachardot}{\kern0pt}absorb{\isacharunderscore}{\kern0pt}iff{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isachardot}{\kern0pt}cobounded{\isadigit{1}}\ sup{\isacharunderscore}{\kern0pt}left{\isacharunderscore}{\kern0pt}commute\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}x\ {\isasymparallel}\isactrlsub {\isasymup}\ y{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Composites/Composite_Structures.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%16=6%:%
-%:%17=7%:%
-%:%18=8%:%
-%:%19=9%:%
-%:%20=10%:%
-%:%21=11%:%
-%:%22=12%:%
-%:%23=13%:%
-%:%37=15%:%
-%:%49=18%:%
-%:%50=19%:%
-%:%51=20%:%
-%:%52=21%:%
-%:%53=22%:%
-%:%62=24%:%
-%:%72=26%:%
-%:%73=26%:%
-%:%74=27%:%
-%:%81=29%:%
-%:%91=31%:%
-%:%92=31%:%
-%:%93=32%:%
-%:%94=33%:%
-%:%97=34%:%
-%:%101=34%:%
-%:%102=34%:%
-%:%116=36%:%
-%:%128=39%:%
-%:%129=40%:%
-%:%130=41%:%
-%:%131=42%:%
-%:%140=44%:%
-%:%150=46%:%
-%:%151=46%:%
-%:%152=47%:%
-%:%154=49%:%
-%:%155=50%:%
-%:%156=51%:%
-%:%157=51%:%
-%:%158=52%:%
-%:%159=53%:%
-%:%160=54%:%
-%:%161=55%:%
-%:%162=56%:%
-%:%163=56%:%
-%:%164=57%:%
-%:%171=59%:%
-%:%183=62%:%
-%:%184=63%:%
-%:%185=64%:%
-%:%186=65%:%
-%:%187=66%:%
-%:%188=67%:%
-%:%189=68%:%
-%:%190=69%:%
-%:%199=71%:%
-%:%209=73%:%
-%:%210=73%:%
-%:%211=74%:%
-%:%212=75%:%
-%:%213=76%:%
-%:%214=77%:%
-%:%215=78%:%
-%:%216=78%:%
-%:%217=79%:%
-%:%218=80%:%
-%:%225=82%:%
-%:%235=84%:%
-%:%236=84%:%
-%:%237=85%:%
-%:%238=86%:%
-%:%239=87%:%
-%:%240=88%:%
-%:%243=89%:%
-%:%247=89%:%
-%:%248=89%:%
-%:%249=90%:%
-%:%250=90%:%
-%:%264=92%:%
-%:%274=94%:%
-%:%275=94%:%
-%:%276=95%:%
-%:%277=96%:%
-%:%278=97%:%
-%:%279=98%:%
-%:%280=99%:%
-%:%281=100%:%
-%:%282=101%:%
-%:%283=102%:%
-%:%290=103%:%
-%:%291=103%:%
-%:%292=104%:%
-%:%293=104%:%
-%:%294=105%:%
-%:%295=105%:%
-%:%296=106%:%
-%:%298=108%:%
-%:%299=109%:%
-%:%300=109%:%
-%:%301=110%:%
-%:%302=110%:%
-%:%303=111%:%
-%:%304=111%:%
-%:%305=112%:%
-%:%306=112%:%
-%:%307=113%:%
-%:%308=113%:%
-%:%309=114%:%
-%:%310=115%:%
-%:%311=116%:%
-%:%312=116%:%
-%:%313=117%:%
-%:%314=117%:%
-%:%315=118%:%
-%:%316=118%:%
-%:%317=119%:%
-%:%318=119%:%
-%:%319=120%:%
-%:%320=120%:%
-%:%321=121%:%
-%:%322=121%:%
-%:%323=122%:%
-%:%324=122%:%
-%:%325=123%:%
-%:%326=123%:%
-%:%327=124%:%
-%:%328=125%:%
-%:%329=126%:%
-%:%330=127%:%
-%:%331=128%:%
-%:%332=129%:%
-%:%333=129%:%
-%:%334=130%:%
-%:%335=130%:%
-%:%336=131%:%
-%:%337=131%:%
-%:%338=132%:%
-%:%339=132%:%
-%:%340=132%:%
-%:%341=133%:%
-%:%342=134%:%
-%:%343=134%:%
-%:%344=135%:%
-%:%345=136%:%
-%:%346=136%:%
-%:%347=137%:%
-%:%348=137%:%
-%:%349=138%:%
-%:%351=140%:%
-%:%352=141%:%
-%:%353=141%:%
-%:%354=142%:%
-%:%355=143%:%
-%:%356=143%:%
-%:%357=144%:%
-%:%358=144%:%
-%:%359=145%:%
-%:%361=147%:%
-%:%362=148%:%
-%:%363=148%:%
-%:%364=149%:%
-%:%365=149%:%
-%:%366=150%:%
-%:%367=150%:%
-%:%368=151%:%
-%:%369=152%:%
-%:%370=153%:%
-%:%371=154%:%
-%:%372=155%:%
-%:%373=156%:%
-%:%374=157%:%
-%:%375=157%:%
-%:%376=158%:%
-%:%377=158%:%
-%:%378=159%:%
-%:%384=159%:%
-%:%387=160%:%
-%:%388=161%:%
-%:%389=161%:%
-%:%390=162%:%
-%:%391=163%:%
-%:%392=164%:%
-%:%393=165%:%
-%:%394=166%:%
-%:%395=167%:%
-%:%396=168%:%
-%:%403=169%:%
-%:%404=169%:%
-%:%405=170%:%
-%:%406=170%:%
-%:%407=171%:%
-%:%408=172%:%
-%:%409=173%:%
-%:%410=173%:%
-%:%411=174%:%
-%:%412=174%:%
-%:%413=175%:%
-%:%414=175%:%
-%:%415=176%:%
-%:%417=178%:%
-%:%418=179%:%
-%:%419=179%:%
-%:%420=180%:%
-%:%421=181%:%
-%:%422=181%:%
-%:%423=182%:%
-%:%424=182%:%
-%:%425=183%:%
-%:%427=185%:%
-%:%428=186%:%
-%:%429=186%:%
-%:%430=187%:%
-%:%431=187%:%
-%:%432=188%:%
-%:%433=188%:%
-%:%434=189%:%
-%:%435=190%:%
-%:%436=191%:%
-%:%437=191%:%
-%:%438=192%:%
-%:%439=192%:%
-%:%440=193%:%
-%:%441=193%:%
-%:%442=193%:%
-%:%443=194%:%
-%:%444=195%:%
-%:%445=196%:%
-%:%446=196%:%
-%:%447=197%:%
-%:%448=197%:%
-%:%449=198%:%
-%:%450=198%:%
-%:%451=199%:%
-%:%452=200%:%
-%:%453=201%:%
-%:%454=201%:%
-%:%455=202%:%
-%:%456=202%:%
-%:%457=203%:%
-%:%458=203%:%
-%:%459=204%:%
-%:%460=204%:%
-%:%461=205%:%
-%:%462=206%:%
-%:%463=207%:%
-%:%464=208%:%
-%:%465=209%:%
-%:%466=209%:%
-%:%467=210%:%
-%:%468=210%:%
-%:%469=211%:%
-%:%470=211%:%
-%:%471=211%:%
-%:%472=212%:%
-%:%473=212%:%
-%:%474=213%:%
-%:%480=213%:%
-%:%483=214%:%
-%:%484=215%:%
-%:%485=215%:%
-%:%486=216%:%
-%:%487=217%:%
-%:%488=218%:%
-%:%489=219%:%
-%:%490=220%:%
-%:%491=221%:%
-%:%492=222%:%
-%:%495=223%:%
-%:%499=223%:%
-%:%500=223%:%
-%:%501=224%:%
-%:%502=225%:%
-%:%503=226%:%
-%:%504=227%:%
-%:%505=228%:%
-%:%506=228%:%
-%:%511=228%:%
-%:%514=229%:%
-%:%515=230%:%
-%:%516=230%:%
-%:%517=231%:%
-%:%518=232%:%
-%:%519=233%:%
-%:%520=234%:%
-%:%521=235%:%
-%:%522=236%:%
-%:%523=237%:%
-%:%526=238%:%
-%:%530=238%:%
-%:%531=238%:%
-%:%532=239%:%
-%:%533=240%:%
-%:%534=240%:%
-%:%539=240%:%
-%:%542=241%:%
-%:%543=242%:%
-%:%544=242%:%
-%:%545=243%:%
-%:%546=244%:%
-%:%547=245%:%
-%:%548=246%:%
-%:%549=247%:%
-%:%550=248%:%
-%:%551=249%:%
-%:%554=250%:%
-%:%558=250%:%
-%:%559=250%:%
-%:%560=251%:%
-%:%561=252%:%
-%:%562=253%:%
-%:%563=254%:%
-%:%564=254%:%
-%:%569=254%:%
-%:%572=255%:%
-%:%573=256%:%
-%:%574=256%:%
-%:%575=257%:%
-%:%576=258%:%
-%:%577=259%:%
-%:%578=260%:%
-%:%579=261%:%
-%:%580=262%:%
-%:%581=263%:%
-%:%584=264%:%
-%:%588=264%:%
-%:%589=264%:%
-%:%590=265%:%
-%:%591=266%:%
-%:%592=266%:%
-%:%597=266%:%
-%:%600=267%:%
-%:%601=268%:%
-%:%602=268%:%
-%:%603=269%:%
-%:%604=270%:%
-%:%605=271%:%
-%:%606=272%:%
-%:%607=273%:%
-%:%608=274%:%
-%:%609=275%:%
-%:%616=276%:%
-%:%617=276%:%
-%:%618=277%:%
-%:%619=277%:%
-%:%620=278%:%
-%:%621=279%:%
-%:%622=280%:%
-%:%623=280%:%
-%:%624=281%:%
-%:%625=281%:%
-%:%626=282%:%
-%:%627=283%:%
-%:%628=284%:%
-%:%629=284%:%
-%:%630=285%:%
-%:%631=285%:%
-%:%632=286%:%
-%:%633=286%:%
-%:%634=287%:%
-%:%635=288%:%
-%:%636=289%:%
-%:%637=289%:%
-%:%638=290%:%
-%:%639=290%:%
-%:%640=291%:%
-%:%642=293%:%
-%:%643=294%:%
-%:%644=294%:%
-%:%645=295%:%
-%:%646=295%:%
-%:%647=296%:%
-%:%648=296%:%
-%:%649=297%:%
-%:%655=297%:%
-%:%658=298%:%
-%:%659=299%:%
-%:%660=299%:%
-%:%661=300%:%
-%:%662=301%:%
-%:%663=302%:%
-%:%664=303%:%
-%:%665=304%:%
-%:%668=307%:%
-%:%671=308%:%
-%:%675=308%:%
-%:%676=308%:%
-%:%677=309%:%
-%:%678=310%:%
-%:%679=310%:%
-%:%684=310%:%
-%:%687=311%:%
-%:%688=312%:%
-%:%689=312%:%
-%:%690=313%:%
-%:%691=314%:%
-%:%692=315%:%
-%:%693=316%:%
-%:%694=317%:%
-%:%701=318%:%
-%:%702=318%:%
-%:%703=319%:%
-%:%704=319%:%
-%:%705=319%:%
-%:%706=320%:%
-%:%710=324%:%
-%:%711=325%:%
-%:%712=325%:%
-%:%713=326%:%
-%:%714=326%:%
-%:%715=327%:%
-%:%716=327%:%
-%:%717=328%:%
-%:%718=328%:%
-%:%719=329%:%
-%:%720=330%:%
-%:%721=330%:%
-%:%722=331%:%
-%:%723=331%:%
-%:%724=332%:%
-%:%725=332%:%
-%:%726=333%:%
-%:%727=333%:%
-%:%728=334%:%
-%:%729=334%:%
-%:%730=335%:%
-%:%731=335%:%
-%:%732=336%:%
-%:%733=336%:%
-%:%734=337%:%
-%:%735=337%:%
-%:%736=338%:%
-%:%737=338%:%
-%:%738=339%:%
-%:%739=339%:%
-%:%740=340%:%
-%:%741=340%:%
-%:%742=341%:%
-%:%744=343%:%
-%:%745=344%:%
-%:%746=344%:%
-%:%747=345%:%
-%:%748=345%:%
-%:%749=346%:%
-%:%750=346%:%
-%:%751=347%:%
-%:%752=347%:%
-%:%753=348%:%
-%:%754=348%:%
-%:%755=349%:%
-%:%756=349%:%
-%:%757=350%:%
-%:%758=350%:%
-%:%759=351%:%
-%:%760=352%:%
-%:%761=352%:%
-%:%762=353%:%
-%:%763=353%:%
-%:%764=354%:%
-%:%765=354%:%
-%:%766=355%:%
-%:%767=355%:%
-%:%768=356%:%
-%:%774=356%:%
-%:%779=357%:%
-%:%784=358%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Consistency.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Consistency.tex
deleted file mode 100644
index 58090d9..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Consistency.tex
+++ /dev/null
@@ -1,63 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Condorcet{\isacharunderscore}{\kern0pt}Consistency}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Condorcet{\isacharunderscore}{\kern0pt}Consistency\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}e\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ e{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Properties/Condorcet_Consistency.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%19=4%:%
-%:%22=5%:%
-%:%23=6%:%
-%:%24=6%:%
-%:%25=7%:%
-%:%31=13%:%
-%:%34=14%:%
-%:%39=15%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Facts.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Facts.tex
deleted file mode 100644
index c47d28f..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Facts.tex
+++ /dev/null
@@ -1,795 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Condorcet{\isacharunderscore}{\kern0pt}Facts}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Condorcet{\isacharunderscore}{\kern0pt}Facts\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Properties{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Consistency{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Condorcet{\isacharunderscore}{\kern0pt}Rules\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ condorcet{\isacharunderscore}{\kern0pt}score{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymnot}\ condorcet{\isacharunderscore}{\kern0pt}rating\ f\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ rs\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ rs\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}aa{\isachardot}{\kern0pt}\ {\isasymnot}\ f\ {\isacharparenleft}{\kern0pt}aa{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ A\ rs\ {\isacharless}{\kern0pt}\ f\ a\ A\ rs\ {\isasymand}\ a\ {\isasymnoteq}\ aa\ {\isasymand}\ aa\ {\isasymin}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}rating\ f\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs\ a{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ rs\ a\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ f\ aa\ A\ rs\ {\isacharless}{\kern0pt}\ f\ a\ A\ rs\ {\isasymor}\ a\ {\isacharequal}{\kern0pt}\ aa\ {\isasymor}\ aa\ {\isasymnotin}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique\ condorcet{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\ zero{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}cr{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ copeland{\isacharunderscore}{\kern0pt}score{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ copeland{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ l{\isacharbraceright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharless}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ winner\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ w\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}copeland{\isacharunderscore}{\kern0pt}score\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ winner\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ l{\isacharbraceright}{\kern0pt}\ {\isasymle}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isacharminus}{\kern0pt}{\isadigit{2}}\ {\isacharless}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}eq\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}singleton\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ diff{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ empty{\isacharunderscore}{\kern0pt}iff\ finite{\isacharunderscore}{\kern0pt}Diff\ insertE\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\ neq{\isadigit{0}}{\isacharunderscore}{\kern0pt}conv\ one{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}numeral{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ semiring{\isacharunderscore}{\kern0pt}norm{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ winner\ zero{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}diff\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ l\ p\ y{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ wins\ y\ p\ l{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ A\ {\isacharminus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ le{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ condorcet{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cscore{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}rating{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}cond{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}condorcet\ A\ p\ {\isasymequiv}\ max{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cscore{\isacharunderscore}{\kern0pt}dcc\ cond{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}cond\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ copeland{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cplscore{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ copeland{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ copeland{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}cr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ copel{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}copel{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}copeland\ A\ p\ {\isasymequiv}\ max{\isacharunderscore}{\kern0pt}eliminator\ copeland{\isacharunderscore}{\kern0pt}score\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}cplscore{\isacharunderscore}{\kern0pt}dcc\ copel{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}copel\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ minimax{\isacharunderscore}{\kern0pt}score{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\ minimax{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\ prefer{\isacharunderscore}{\kern0pt}count{\isachardot}{\kern0pt}simps{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}l\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ l{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymge}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymge}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ prof\ winner\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w\ \ {\isasymge}\ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}minimax{\isacharunderscore}{\kern0pt}score\ minimax{\isacharunderscore}{\kern0pt}score{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}without{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prof\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\ finite{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}\ n\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Min{\isacharunderscore}{\kern0pt}in\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ n\ \isakeyword{where}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ Min\ {\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ y\ {\isacharbar}{\kern0pt}y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isadigit{6}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffE\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{6}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isasymnoteq}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{6}}\ prof\ winner\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ n\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ prof\ n{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ n{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \ \ \ \ \ l{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}w\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}sym\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{7}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ n\ w\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ l\ w\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ l{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{3}}{\isadigit{0}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{4}}{\isadigit{0}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{7}}{\isadigit{0}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}wins\ l\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ l{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}without{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ wins{\isacharunderscore}{\kern0pt}antisym\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ minimax{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}mmaxscore{\isacharunderscore}{\kern0pt}dcc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ minimax{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ minimax{\isacharunderscore}{\kern0pt}score{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}rating{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ mmax{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}mmax{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}minimax\ A\ p\ {\isasymequiv}\ max{\isacharunderscore}{\kern0pt}eliminator\ minimax{\isacharunderscore}{\kern0pt}score\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}mmaxscore{\isacharunderscore}{\kern0pt}dcc\ mmax{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}mmax\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Composition_Rules/Condorcet_Facts.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%16=6%:%
-%:%17=7%:%
-%:%22=7%:%
-%:%25=8%:%
-%:%26=9%:%
-%:%27=10%:%
-%:%28=10%:%
-%:%35=11%:%
-%:%36=11%:%
-%:%37=12%:%
-%:%38=12%:%
-%:%39=13%:%
-%:%46=20%:%
-%:%47=21%:%
-%:%48=21%:%
-%:%49=22%:%
-%:%50=22%:%
-%:%51=23%:%
-%:%52=23%:%
-%:%53=24%:%
-%:%54=24%:%
-%:%55=25%:%
-%:%56=25%:%
-%:%57=26%:%
-%:%63=26%:%
-%:%66=27%:%
-%:%67=28%:%
-%:%68=29%:%
-%:%69=29%:%
-%:%72=30%:%
-%:%76=30%:%
-%:%77=30%:%
-%:%78=31%:%
-%:%79=31%:%
-%:%80=32%:%
-%:%81=32%:%
-%:%82=33%:%
-%:%83=34%:%
-%:%84=35%:%
-%:%85=36%:%
-%:%86=37%:%
-%:%87=37%:%
-%:%88=38%:%
-%:%89=39%:%
-%:%90=40%:%
-%:%91=41%:%
-%:%92=41%:%
-%:%93=42%:%
-%:%94=43%:%
-%:%95=44%:%
-%:%96=44%:%
-%:%97=45%:%
-%:%98=45%:%
-%:%99=45%:%
-%:%100=46%:%
-%:%101=47%:%
-%:%102=48%:%
-%:%103=48%:%
-%:%104=49%:%
-%:%105=49%:%
-%:%106=50%:%
-%:%107=50%:%
-%:%108=50%:%
-%:%109=51%:%
-%:%110=52%:%
-%:%111=53%:%
-%:%112=53%:%
-%:%113=54%:%
-%:%114=54%:%
-%:%115=55%:%
-%:%116=55%:%
-%:%117=56%:%
-%:%118=56%:%
-%:%119=57%:%
-%:%120=58%:%
-%:%121=59%:%
-%:%122=60%:%
-%:%123=61%:%
-%:%124=61%:%
-%:%125=62%:%
-%:%126=62%:%
-%:%127=63%:%
-%:%128=64%:%
-%:%129=65%:%
-%:%130=65%:%
-%:%131=66%:%
-%:%132=66%:%
-%:%133=67%:%
-%:%134=67%:%
-%:%135=68%:%
-%:%136=68%:%
-%:%137=69%:%
-%:%138=69%:%
-%:%139=70%:%
-%:%140=70%:%
-%:%141=71%:%
-%:%147=71%:%
-%:%150=72%:%
-%:%151=73%:%
-%:%152=73%:%
-%:%159=74%:%
-%:%160=74%:%
-%:%161=75%:%
-%:%162=75%:%
-%:%163=76%:%
-%:%164=77%:%
-%:%165=77%:%
-%:%166=78%:%
-%:%167=78%:%
-%:%168=79%:%
-%:%169=79%:%
-%:%170=80%:%
-%:%171=81%:%
-%:%172=81%:%
-%:%173=82%:%
-%:%174=82%:%
-%:%175=82%:%
-%:%176=83%:%
-%:%177=83%:%
-%:%178=84%:%
-%:%179=84%:%
-%:%180=85%:%
-%:%186=85%:%
-%:%189=86%:%
-%:%190=87%:%
-%:%191=87%:%
-%:%198=88%:%
-%:%199=88%:%
-%:%200=89%:%
-%:%201=89%:%
-%:%202=90%:%
-%:%203=91%:%
-%:%204=91%:%
-%:%205=92%:%
-%:%206=92%:%
-%:%207=93%:%
-%:%208=93%:%
-%:%209=94%:%
-%:%210=95%:%
-%:%211=95%:%
-%:%212=96%:%
-%:%213=96%:%
-%:%214=97%:%
-%:%215=97%:%
-%:%216=98%:%
-%:%217=98%:%
-%:%218=99%:%
-%:%219=99%:%
-%:%220=100%:%
-%:%226=100%:%
-%:%229=101%:%
-%:%230=102%:%
-%:%231=102%:%
-%:%238=103%:%
-%:%239=103%:%
-%:%240=104%:%
-%:%241=104%:%
-%:%242=105%:%
-%:%243=106%:%
-%:%244=107%:%
-%:%245=108%:%
-%:%246=109%:%
-%:%247=109%:%
-%:%248=110%:%
-%:%249=111%:%
-%:%250=112%:%
-%:%251=113%:%
-%:%252=113%:%
-%:%253=114%:%
-%:%256=117%:%
-%:%257=118%:%
-%:%258=118%:%
-%:%259=119%:%
-%:%260=119%:%
-%:%261=120%:%
-%:%264=123%:%
-%:%265=124%:%
-%:%266=124%:%
-%:%267=125%:%
-%:%270=128%:%
-%:%271=129%:%
-%:%272=129%:%
-%:%273=130%:%
-%:%274=130%:%
-%:%275=131%:%
-%:%276=132%:%
-%:%277=133%:%
-%:%278=133%:%
-%:%279=134%:%
-%:%280=134%:%
-%:%281=135%:%
-%:%282=135%:%
-%:%283=136%:%
-%:%284=136%:%
-%:%285=137%:%
-%:%286=137%:%
-%:%287=138%:%
-%:%288=138%:%
-%:%289=139%:%
-%:%290=140%:%
-%:%291=140%:%
-%:%292=141%:%
-%:%293=141%:%
-%:%294=159%:%
-%:%295=160%:%
-%:%296=160%:%
-%:%297=161%:%
-%:%298=161%:%
-%:%299=162%:%
-%:%300=162%:%
-%:%301=163%:%
-%:%302=163%:%
-%:%303=164%:%
-%:%304=164%:%
-%:%305=165%:%
-%:%306=165%:%
-%:%307=166%:%
-%:%308=166%:%
-%:%309=167%:%
-%:%310=167%:%
-%:%311=168%:%
-%:%312=168%:%
-%:%313=169%:%
-%:%314=169%:%
-%:%315=170%:%
-%:%316=170%:%
-%:%317=171%:%
-%:%318=171%:%
-%:%319=172%:%
-%:%320=173%:%
-%:%321=174%:%
-%:%322=174%:%
-%:%323=175%:%
-%:%324=175%:%
-%:%325=176%:%
-%:%326=176%:%
-%:%327=176%:%
-%:%328=177%:%
-%:%329=178%:%
-%:%330=179%:%
-%:%331=180%:%
-%:%332=180%:%
-%:%333=181%:%
-%:%334=181%:%
-%:%335=182%:%
-%:%336=182%:%
-%:%337=183%:%
-%:%338=183%:%
-%:%339=184%:%
-%:%340=184%:%
-%:%341=185%:%
-%:%342=185%:%
-%:%343=186%:%
-%:%344=186%:%
-%:%345=187%:%
-%:%346=187%:%
-%:%347=188%:%
-%:%348=188%:%
-%:%349=189%:%
-%:%350=189%:%
-%:%351=190%:%
-%:%352=190%:%
-%:%353=191%:%
-%:%354=191%:%
-%:%355=192%:%
-%:%356=192%:%
-%:%357=193%:%
-%:%358=193%:%
-%:%359=194%:%
-%:%360=194%:%
-%:%361=195%:%
-%:%362=195%:%
-%:%363=196%:%
-%:%364=196%:%
-%:%365=197%:%
-%:%366=198%:%
-%:%367=198%:%
-%:%368=199%:%
-%:%369=199%:%
-%:%370=200%:%
-%:%371=200%:%
-%:%372=201%:%
-%:%373=201%:%
-%:%374=202%:%
-%:%375=202%:%
-%:%376=203%:%
-%:%377=203%:%
-%:%378=204%:%
-%:%379=204%:%
-%:%380=205%:%
-%:%381=205%:%
-%:%382=206%:%
-%:%383=206%:%
-%:%384=207%:%
-%:%385=208%:%
-%:%386=208%:%
-%:%387=209%:%
-%:%388=209%:%
-%:%389=210%:%
-%:%395=210%:%
-%:%398=211%:%
-%:%399=212%:%
-%:%400=212%:%
-%:%407=213%:%
-%:%408=213%:%
-%:%409=214%:%
-%:%410=214%:%
-%:%411=215%:%
-%:%412=216%:%
-%:%413=216%:%
-%:%414=217%:%
-%:%415=217%:%
-%:%416=218%:%
-%:%417=218%:%
-%:%418=219%:%
-%:%419=220%:%
-%:%420=220%:%
-%:%421=221%:%
-%:%422=221%:%
-%:%423=222%:%
-%:%424=222%:%
-%:%425=223%:%
-%:%426=223%:%
-%:%427=224%:%
-%:%428=224%:%
-%:%429=225%:%
-%:%435=225%:%
-%:%438=226%:%
-%:%439=227%:%
-%:%442=228%:%
-%:%447=229%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Properties.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Properties.tex
deleted file mode 100644
index d89a4f0..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Properties.tex
+++ /dev/null
@@ -1,93 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Condorcet{\isacharunderscore}{\kern0pt}Properties}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Condorcet{\isacharunderscore}{\kern0pt}Properties\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}compatibility\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymand}\ finite\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}w\ {\isasymnotin}\ reject\ m\ A\ p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}l{\isachardot}{\kern0pt}\ {\isasymnot}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ l\ {\isasymlongrightarrow}\ l\ {\isasymnotin}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}w\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}l{\isachardot}{\kern0pt}\ {\isasymnot}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ l\ {\isasymlongrightarrow}\ l\ {\isasymin}\ reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ f\ {\isasymequiv}\isanewline
-\ \ \ \ {\isasymforall}A\ p\ w\ {\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}l\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ l\ {\isasymnoteq}\ w\ {\isasymlongrightarrow}\ f\ l\ A\ p\ {\isacharless}{\kern0pt}\ f\ w\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymand}\ finite\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Properties/Condorcet_Properties.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%20=5%:%
-%:%23=6%:%
-%:%24=7%:%
-%:%25=7%:%
-%:%26=8%:%
-%:%32=14%:%
-%:%33=15%:%
-%:%34=19%:%
-%:%35=20%:%
-%:%36=20%:%
-%:%37=21%:%
-%:%39=23%:%
-%:%40=24%:%
-%:%41=25%:%
-%:%42=25%:%
-%:%43=26%:%
-%:%49=32%:%
-%:%52=33%:%
-%:%57=34%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Rules.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Rules.tex
deleted file mode 100644
index 47d61c8..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Condorcet_Rules.tex
+++ /dev/null
@@ -1,1954 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Condorcet{\isacharunderscore}{\kern0pt}Rules}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Condorcet{\isacharunderscore}{\kern0pt}Rules\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Properties{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Consistency{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Sequential{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}e\ w\ A\ p\ {\isacharequal}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}set\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isacharquery}{\kern0pt}eMax\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isacharquery}{\kern0pt}eW\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}e\ w\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}set\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}eW\ {\isasymin}\ {\isacharquery}{\kern0pt}set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectI\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ e\ {\isasymin}\ {\isacharquery}{\kern0pt}set\ {\isachardot}{\kern0pt}\ e\ {\isasymle}\ {\isacharquery}{\kern0pt}eW{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\ eq{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}order\ rating\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}eW\ {\isasymin}\ {\isacharquery}{\kern0pt}set\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}set{\isachardot}{\kern0pt}\ a\ {\isasymle}\ {\isacharquery}{\kern0pt}eW{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{4}}\ Max{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ linA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}l\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ loser{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnoteq}\ l{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}e\ l\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}\ a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}e\ l\ A\ p\ {\isacharless}{\kern0pt}\ e\ w\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}rating{\isacharunderscore}{\kern0pt}def\ linA\ loser\ rating\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}e\ w\ A\ p\ {\isacharequal}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ a\ A\ p\ {\isacharbar}{\kern0pt}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val\ f{\isacharunderscore}{\kern0pt}prof\ rating\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}ccomp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p\ w\ x{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ finite\ A\ {\isasymLongrightarrow}\ w\ {\isasymin}\ A\ {\isasymLongrightarrow}\ e\ w\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isasymLongrightarrow}\ e\ x\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rating\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p\ w\ x{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ profile\ A\ p\ {\isasymLongrightarrow}\ w\ {\isasymin}\ A\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite\ A\ {\isasymLongrightarrow}\ e\ w\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\ {\isasymin}\ A\ {\isasymLongrightarrow}\ e\ x\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ dcc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def{\isacharcomma}{\kern0pt}\ auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ elect{\isacharunderscore}{\kern0pt}module{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}A\ p\ w\ x{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ finite\ A\ {\isasymLongrightarrow}\ profile\ A\ p\ {\isasymLongrightarrow}\ w\ {\isasymin}\ A\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ x\ {\isasymin}\ elect\ m\ A\ p\ {\isasymLongrightarrow}\ x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ x\ {\isasymin}\ elect\ m\ A\ p\ {\isasymLongrightarrow}\ x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subset{\isacharunderscore}{\kern0pt}eq\ finite\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ fst{\isacharunderscore}{\kern0pt}conv\ insert{\isacharunderscore}{\kern0pt}Diff\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ order{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ subset{\isacharunderscore}{\kern0pt}Compl{\isacharunderscore}{\kern0pt}singleton\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ xa{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}xa{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ empty{\isacharunderscore}{\kern0pt}iff\ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\ DiffI\isanewline
-\ \ \ \ \ \ \ \ \ \ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ insert{\isacharunderscore}{\kern0pt}iff\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ insert{\isacharunderscore}{\kern0pt}iff\ eq{\isacharunderscore}{\kern0pt}snd{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ finite\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subsetD\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ IntI\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ dcc\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ IntI\isanewline
-\ \ \ \ \ \ \ \ \ \ Diff{\isacharunderscore}{\kern0pt}empty\ Diff{\isacharunderscore}{\kern0pt}iff\ finite\ prof{\isacharunderscore}{\kern0pt}A\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ condorcet{\isacharunderscore}{\kern0pt}winner{\isachardot}{\kern0pt}simps\ dcc\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ ccomp{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}dd{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}winner{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ ccomp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ dd{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}deciding\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ dd\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ c{\isacharunderscore}{\kern0pt}win{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ equals{\isadigit{0}}D\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ insert{\isacharunderscore}{\kern0pt}subset\ c{\isacharunderscore}{\kern0pt}win\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}w\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}l\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ l\ {\isasymnoteq}\ w\ {\isasymand}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}l{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnotin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ c{\isacharunderscore}{\kern0pt}win\ equals{\isadigit{0}}D\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer\ not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}elect\ \isacommand{have}\isamarkupfalse%
-\ one{\isacharunderscore}{\kern0pt}side{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}w\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ccomp\ condorcet{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ c{\isacharunderscore}{\kern0pt}win\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ ccomp\ \isacommand{have}\isamarkupfalse%
-\ other{\isacharunderscore}{\kern0pt}side{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ c{\isacharunderscore}{\kern0pt}win\ winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ one{\isacharunderscore}{\kern0pt}side{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ ccomp{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}dd{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ ccomp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}compatibility\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ dd{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}deciding\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def{\isacharcomma}{\kern0pt}\ auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finiteness{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ assm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm\ finiteness\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ dd\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ dd\ ccomp\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ ccomp{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}dd{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}winner\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ dd\ defer{\isacharunderscore}{\kern0pt}deciding{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ wins\ d\ p\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finiteness\ prof{\isacharunderscore}{\kern0pt}A\ winner\ Collect{\isacharunderscore}{\kern0pt}cong\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ d\ {\isacharless}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ d\ x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}d\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ d{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ m\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}d{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}d{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}helper{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharparenleft}{\kern0pt}Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}e\ w\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val\isanewline
-\ \ \ \ \ \ \ \ \ \ rating\ winner\ f{\isacharunderscore}{\kern0pt}prof\ max\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ e\ x\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ y\ A\ p\ {\isacharbar}{\kern0pt}y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ rating\ winner\ f{\isacharunderscore}{\kern0pt}prof\ not{\isacharunderscore}{\kern0pt}max\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ rating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}rating\ e{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}trsh\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}Max\ {\isacharbraceleft}{\kern0pt}e\ y\ A\ p\ {\isacharbar}{\kern0pt}\ y{\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}trsh{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ rating\ winner\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharquery}{\kern0pt}trsh\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cr{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}dcc{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}helper{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharquery}{\kern0pt}trsh\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ {\isacharquery}{\kern0pt}trsh\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ calculation\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}\ A\ p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}d\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ d{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ winner\ Collect{\isacharunderscore}{\kern0pt}cong\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite\ winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p\ {\isasymand}\ w\ {\isasymin}\ A\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isasymor}\ wins\ w\ p\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}trsh\ {\isacharequal}{\kern0pt}\ e\ w\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rating\ winner\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}eval{\isacharunderscore}{\kern0pt}val{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ False\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymlongleftrightarrow}\isanewline
-\ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}auto{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m\ {\isasymLongrightarrow}\ electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ cc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ assm{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymLongrightarrow}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ cc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ prefer{\isacharunderscore}{\kern0pt}count\ p\ w\ x\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ wins{\isachardot}{\kern0pt}elims\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ assm{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ {\isasymforall}A\ p\ w{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isacharunderscore}{\kern0pt}def\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p\ w{\isachardot}{\kern0pt}\ finite\ A\ {\isasymand}\ profile\ A\ p\ {\isasymand}\ w\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ condorcet{\isacharunderscore}{\kern0pt}consistency\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm{\isadigit{0}}\ e{\isacharunderscore}{\kern0pt}mod\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Composition_Rules/Condorcet_Rules.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%16=6%:%
-%:%17=7%:%
-%:%18=8%:%
-%:%23=8%:%
-%:%26=9%:%
-%:%27=13%:%
-%:%28=14%:%
-%:%29=14%:%
-%:%30=15%:%
-%:%31=16%:%
-%:%32=17%:%
-%:%33=18%:%
-%:%34=19%:%
-%:%41=20%:%
-%:%42=20%:%
-%:%43=21%:%
-%:%43=24%:%
-%:%44=25%:%
-%:%45=25%:%
-%:%46=26%:%
-%:%47=27%:%
-%:%48=28%:%
-%:%49=29%:%
-%:%50=29%:%
-%:%51=29%:%
-%:%52=30%:%
-%:%53=30%:%
-%:%54=31%:%
-%:%55=32%:%
-%:%56=32%:%
-%:%57=33%:%
-%:%58=33%:%
-%:%59=34%:%
-%:%60=34%:%
-%:%61=35%:%
-%:%62=36%:%
-%:%63=36%:%
-%:%64=37%:%
-%:%65=37%:%
-%:%66=38%:%
-%:%67=38%:%
-%:%68=39%:%
-%:%69=40%:%
-%:%70=40%:%
-%:%71=41%:%
-%:%72=41%:%
-%:%73=42%:%
-%:%74=43%:%
-%:%75=43%:%
-%:%76=44%:%
-%:%77=45%:%
-%:%78=45%:%
-%:%79=45%:%
-%:%80=46%:%
-%:%81=47%:%
-%:%82=47%:%
-%:%83=48%:%
-%:%84=48%:%
-%:%85=48%:%
-%:%86=49%:%
-%:%87=49%:%
-%:%88=50%:%
-%:%94=50%:%
-%:%97=51%:%
-%:%98=57%:%
-%:%99=58%:%
-%:%100=58%:%
-%:%101=59%:%
-%:%102=60%:%
-%:%103=61%:%
-%:%104=62%:%
-%:%105=63%:%
-%:%106=64%:%
-%:%107=65%:%
-%:%114=66%:%
-%:%115=66%:%
-%:%116=67%:%
-%:%117=67%:%
-%:%118=68%:%
-%:%119=68%:%
-%:%120=69%:%
-%:%121=69%:%
-%:%122=70%:%
-%:%123=70%:%
-%:%124=70%:%
-%:%125=71%:%
-%:%126=71%:%
-%:%127=72%:%
-%:%128=72%:%
-%:%129=73%:%
-%:%130=73%:%
-%:%131=73%:%
-%:%132=74%:%
-%:%133=74%:%
-%:%134=75%:%
-%:%140=75%:%
-%:%143=76%:%
-%:%144=78%:%
-%:%145=79%:%
-%:%146=79%:%
-%:%147=80%:%
-%:%148=81%:%
-%:%149=82%:%
-%:%150=83%:%
-%:%151=84%:%
-%:%154=85%:%
-%:%158=85%:%
-%:%159=85%:%
-%:%160=86%:%
-%:%161=86%:%
-%:%162=87%:%
-%:%163=87%:%
-%:%164=88%:%
-%:%166=90%:%
-%:%167=91%:%
-%:%168=91%:%
-%:%169=92%:%
-%:%170=92%:%
-%:%171=93%:%
-%:%172=93%:%
-%:%173=94%:%
-%:%179=100%:%
-%:%180=101%:%
-%:%181=101%:%
-%:%182=102%:%
-%:%188=102%:%
-%:%191=103%:%
-%:%192=104%:%
-%:%193=105%:%
-%:%194=105%:%
-%:%195=106%:%
-%:%196=107%:%
-%:%203=108%:%
-%:%204=108%:%
-%:%205=109%:%
-%:%206=110%:%
-%:%207=110%:%
-%:%208=111%:%
-%:%209=111%:%
-%:%210=112%:%
-%:%211=113%:%
-%:%212=113%:%
-%:%213=114%:%
-%:%214=114%:%
-%:%215=115%:%
-%:%216=115%:%
-%:%217=116%:%
-%:%221=120%:%
-%:%222=121%:%
-%:%223=121%:%
-%:%224=122%:%
-%:%225=122%:%
-%:%226=123%:%
-%:%227=124%:%
-%:%228=125%:%
-%:%229=126%:%
-%:%230=127%:%
-%:%231=127%:%
-%:%232=128%:%
-%:%233=129%:%
-%:%234=130%:%
-%:%235=130%:%
-%:%236=131%:%
-%:%239=134%:%
-%:%240=135%:%
-%:%241=135%:%
-%:%242=136%:%
-%:%243=137%:%
-%:%244=137%:%
-%:%245=138%:%
-%:%246=138%:%
-%:%247=139%:%
-%:%248=139%:%
-%:%249=140%:%
-%:%250=140%:%
-%:%251=141%:%
-%:%252=142%:%
-%:%253=143%:%
-%:%254=144%:%
-%:%255=145%:%
-%:%256=146%:%
-%:%257=146%:%
-%:%258=147%:%
-%:%259=148%:%
-%:%260=149%:%
-%:%261=150%:%
-%:%262=151%:%
-%:%264=153%:%
-%:%265=154%:%
-%:%266=154%:%
-%:%267=155%:%
-%:%268=155%:%
-%:%269=156%:%
-%:%270=156%:%
-%:%271=157%:%
-%:%272=157%:%
-%:%273=158%:%
-%:%274=158%:%
-%:%275=159%:%
-%:%276=160%:%
-%:%277=160%:%
-%:%278=161%:%
-%:%279=161%:%
-%:%280=162%:%
-%:%281=162%:%
-%:%282=163%:%
-%:%283=164%:%
-%:%284=165%:%
-%:%285=166%:%
-%:%286=167%:%
-%:%287=167%:%
-%:%288=168%:%
-%:%289=169%:%
-%:%290=170%:%
-%:%291=171%:%
-%:%293=173%:%
-%:%294=174%:%
-%:%295=175%:%
-%:%296=175%:%
-%:%297=176%:%
-%:%298=176%:%
-%:%299=177%:%
-%:%300=177%:%
-%:%301=178%:%
-%:%302=178%:%
-%:%303=179%:%
-%:%304=179%:%
-%:%305=180%:%
-%:%306=181%:%
-%:%307=182%:%
-%:%308=182%:%
-%:%309=183%:%
-%:%310=183%:%
-%:%311=184%:%
-%:%312=184%:%
-%:%313=185%:%
-%:%314=186%:%
-%:%315=187%:%
-%:%316=188%:%
-%:%317=189%:%
-%:%318=190%:%
-%:%319=190%:%
-%:%320=191%:%
-%:%321=192%:%
-%:%322=193%:%
-%:%323=194%:%
-%:%324=195%:%
-%:%325=196%:%
-%:%327=198%:%
-%:%328=199%:%
-%:%329=200%:%
-%:%330=201%:%
-%:%331=201%:%
-%:%332=202%:%
-%:%333=202%:%
-%:%334=203%:%
-%:%335=203%:%
-%:%336=204%:%
-%:%337=204%:%
-%:%338=205%:%
-%:%339=205%:%
-%:%340=206%:%
-%:%341=207%:%
-%:%342=208%:%
-%:%343=208%:%
-%:%344=209%:%
-%:%345=209%:%
-%:%346=210%:%
-%:%347=210%:%
-%:%348=211%:%
-%:%349=212%:%
-%:%350=213%:%
-%:%351=214%:%
-%:%352=215%:%
-%:%353=215%:%
-%:%354=216%:%
-%:%355=217%:%
-%:%356=218%:%
-%:%357=219%:%
-%:%358=220%:%
-%:%359=221%:%
-%:%361=223%:%
-%:%362=224%:%
-%:%364=226%:%
-%:%365=227%:%
-%:%366=227%:%
-%:%367=228%:%
-%:%368=228%:%
-%:%369=229%:%
-%:%370=229%:%
-%:%371=230%:%
-%:%372=230%:%
-%:%373=230%:%
-%:%374=231%:%
-%:%375=231%:%
-%:%376=232%:%
-%:%377=232%:%
-%:%378=233%:%
-%:%379=233%:%
-%:%380=233%:%
-%:%381=234%:%
-%:%382=234%:%
-%:%383=235%:%
-%:%384=236%:%
-%:%385=237%:%
-%:%386=237%:%
-%:%387=238%:%
-%:%388=238%:%
-%:%389=239%:%
-%:%390=239%:%
-%:%391=240%:%
-%:%392=241%:%
-%:%393=242%:%
-%:%394=243%:%
-%:%395=244%:%
-%:%396=244%:%
-%:%397=245%:%
-%:%398=246%:%
-%:%399=247%:%
-%:%400=248%:%
-%:%401=249%:%
-%:%403=251%:%
-%:%404=252%:%
-%:%405=252%:%
-%:%406=253%:%
-%:%407=253%:%
-%:%408=254%:%
-%:%409=254%:%
-%:%410=255%:%
-%:%411=255%:%
-%:%412=256%:%
-%:%413=256%:%
-%:%414=257%:%
-%:%415=258%:%
-%:%416=258%:%
-%:%417=259%:%
-%:%418=259%:%
-%:%419=260%:%
-%:%420=260%:%
-%:%421=261%:%
-%:%422=262%:%
-%:%423=263%:%
-%:%424=264%:%
-%:%425=265%:%
-%:%426=265%:%
-%:%427=266%:%
-%:%428=267%:%
-%:%429=268%:%
-%:%430=269%:%
-%:%431=270%:%
-%:%432=271%:%
-%:%434=273%:%
-%:%435=274%:%
-%:%436=274%:%
-%:%437=275%:%
-%:%438=275%:%
-%:%439=276%:%
-%:%440=276%:%
-%:%441=277%:%
-%:%442=277%:%
-%:%443=278%:%
-%:%444=278%:%
-%:%445=279%:%
-%:%446=280%:%
-%:%447=280%:%
-%:%448=281%:%
-%:%449=281%:%
-%:%450=282%:%
-%:%451=282%:%
-%:%452=283%:%
-%:%453=284%:%
-%:%454=285%:%
-%:%455=286%:%
-%:%456=287%:%
-%:%457=287%:%
-%:%458=288%:%
-%:%459=289%:%
-%:%460=290%:%
-%:%461=291%:%
-%:%462=292%:%
-%:%463=293%:%
-%:%465=295%:%
-%:%466=296%:%
-%:%467=296%:%
-%:%468=297%:%
-%:%469=297%:%
-%:%470=298%:%
-%:%471=298%:%
-%:%472=299%:%
-%:%473=299%:%
-%:%474=300%:%
-%:%475=300%:%
-%:%476=301%:%
-%:%477=302%:%
-%:%478=302%:%
-%:%479=303%:%
-%:%480=303%:%
-%:%481=304%:%
-%:%482=304%:%
-%:%483=305%:%
-%:%484=306%:%
-%:%485=307%:%
-%:%486=308%:%
-%:%487=309%:%
-%:%488=309%:%
-%:%489=310%:%
-%:%490=311%:%
-%:%491=312%:%
-%:%492=313%:%
-%:%493=314%:%
-%:%494=315%:%
-%:%495=316%:%
-%:%497=318%:%
-%:%498=319%:%
-%:%499=319%:%
-%:%500=320%:%
-%:%501=320%:%
-%:%502=321%:%
-%:%503=321%:%
-%:%504=322%:%
-%:%505=322%:%
-%:%506=323%:%
-%:%507=323%:%
-%:%508=324%:%
-%:%509=325%:%
-%:%510=325%:%
-%:%511=326%:%
-%:%517=326%:%
-%:%520=327%:%
-%:%521=328%:%
-%:%522=328%:%
-%:%523=329%:%
-%:%524=330%:%
-%:%525=331%:%
-%:%526=332%:%
-%:%533=333%:%
-%:%534=333%:%
-%:%535=334%:%
-%:%536=334%:%
-%:%537=335%:%
-%:%538=335%:%
-%:%539=335%:%
-%:%540=336%:%
-%:%541=337%:%
-%:%542=337%:%
-%:%543=338%:%
-%:%544=338%:%
-%:%545=339%:%
-%:%546=339%:%
-%:%547=340%:%
-%:%548=341%:%
-%:%549=341%:%
-%:%550=342%:%
-%:%551=342%:%
-%:%552=343%:%
-%:%553=343%:%
-%:%554=344%:%
-%:%555=344%:%
-%:%556=345%:%
-%:%557=346%:%
-%:%558=346%:%
-%:%559=347%:%
-%:%560=347%:%
-%:%561=348%:%
-%:%562=348%:%
-%:%563=349%:%
-%:%564=350%:%
-%:%565=350%:%
-%:%566=351%:%
-%:%567=351%:%
-%:%568=351%:%
-%:%569=352%:%
-%:%570=352%:%
-%:%571=353%:%
-%:%572=353%:%
-%:%573=354%:%
-%:%574=354%:%
-%:%575=355%:%
-%:%576=355%:%
-%:%577=356%:%
-%:%578=356%:%
-%:%579=357%:%
-%:%580=357%:%
-%:%581=358%:%
-%:%582=358%:%
-%:%583=359%:%
-%:%584=359%:%
-%:%585=360%:%
-%:%586=360%:%
-%:%587=361%:%
-%:%588=361%:%
-%:%589=361%:%
-%:%590=362%:%
-%:%591=363%:%
-%:%592=363%:%
-%:%593=364%:%
-%:%594=365%:%
-%:%595=365%:%
-%:%596=366%:%
-%:%597=366%:%
-%:%598=366%:%
-%:%599=367%:%
-%:%600=367%:%
-%:%601=368%:%
-%:%602=368%:%
-%:%603=369%:%
-%:%604=369%:%
-%:%605=370%:%
-%:%606=370%:%
-%:%607=371%:%
-%:%613=371%:%
-%:%616=372%:%
-%:%617=373%:%
-%:%618=373%:%
-%:%619=374%:%
-%:%620=375%:%
-%:%621=376%:%
-%:%628=377%:%
-%:%629=377%:%
-%:%630=378%:%
-%:%631=378%:%
-%:%632=379%:%
-%:%633=379%:%
-%:%634=380%:%
-%:%635=380%:%
-%:%636=381%:%
-%:%637=381%:%
-%:%638=382%:%
-%:%639=382%:%
-%:%640=383%:%
-%:%641=384%:%
-%:%642=385%:%
-%:%643=386%:%
-%:%644=386%:%
-%:%645=387%:%
-%:%646=388%:%
-%:%647=389%:%
-%:%648=390%:%
-%:%650=392%:%
-%:%651=393%:%
-%:%652=393%:%
-%:%653=394%:%
-%:%654=394%:%
-%:%655=395%:%
-%:%656=395%:%
-%:%657=396%:%
-%:%658=396%:%
-%:%659=397%:%
-%:%662=400%:%
-%:%663=401%:%
-%:%664=401%:%
-%:%665=402%:%
-%:%666=403%:%
-%:%667=403%:%
-%:%668=403%:%
-%:%669=404%:%
-%:%670=405%:%
-%:%671=405%:%
-%:%672=406%:%
-%:%673=407%:%
-%:%674=407%:%
-%:%675=408%:%
-%:%676=409%:%
-%:%677=409%:%
-%:%678=409%:%
-%:%679=410%:%
-%:%680=410%:%
-%:%681=411%:%
-%:%682=411%:%
-%:%683=412%:%
-%:%684=413%:%
-%:%685=413%:%
-%:%686=413%:%
-%:%687=414%:%
-%:%688=414%:%
-%:%689=415%:%
-%:%690=416%:%
-%:%691=416%:%
-%:%692=417%:%
-%:%693=417%:%
-%:%694=417%:%
-%:%695=418%:%
-%:%696=418%:%
-%:%697=419%:%
-%:%698=419%:%
-%:%699=420%:%
-%:%700=420%:%
-%:%701=421%:%
-%:%702=421%:%
-%:%703=422%:%
-%:%704=422%:%
-%:%705=423%:%
-%:%706=423%:%
-%:%707=424%:%
-%:%708=424%:%
-%:%709=425%:%
-%:%710=425%:%
-%:%711=426%:%
-%:%712=426%:%
-%:%713=427%:%
-%:%714=427%:%
-%:%715=428%:%
-%:%718=431%:%
-%:%719=432%:%
-%:%720=432%:%
-%:%721=433%:%
-%:%722=433%:%
-%:%723=434%:%
-%:%724=434%:%
-%:%725=435%:%
-%:%729=439%:%
-%:%730=440%:%
-%:%731=440%:%
-%:%732=441%:%
-%:%733=441%:%
-%:%734=442%:%
-%:%739=447%:%
-%:%740=448%:%
-%:%741=448%:%
-%:%742=449%:%
-%:%743=449%:%
-%:%744=450%:%
-%:%749=455%:%
-%:%750=456%:%
-%:%751=456%:%
-%:%752=457%:%
-%:%758=457%:%
-%:%761=458%:%
-%:%762=459%:%
-%:%763=459%:%
-%:%764=460%:%
-%:%765=461%:%
-%:%766=462%:%
-%:%767=463%:%
-%:%768=464%:%
-%:%775=465%:%
-%:%776=465%:%
-%:%777=466%:%
-%:%778=466%:%
-%:%779=467%:%
-%:%780=468%:%
-%:%781=469%:%
-%:%782=469%:%
-%:%783=470%:%
-%:%784=470%:%
-%:%785=471%:%
-%:%786=472%:%
-%:%787=472%:%
-%:%788=473%:%
-%:%789=473%:%
-%:%790=474%:%
-%:%791=474%:%
-%:%792=475%:%
-%:%793=476%:%
-%:%794=476%:%
-%:%795=477%:%
-%:%796=478%:%
-%:%797=479%:%
-%:%798=479%:%
-%:%799=480%:%
-%:%800=480%:%
-%:%801=481%:%
-%:%802=482%:%
-%:%803=482%:%
-%:%804=483%:%
-%:%810=483%:%
-%:%813=484%:%
-%:%814=488%:%
-%:%815=489%:%
-%:%816=489%:%
-%:%817=490%:%
-%:%818=491%:%
-%:%821=492%:%
-%:%825=492%:%
-%:%826=492%:%
-%:%827=493%:%
-%:%828=493%:%
-%:%829=494%:%
-%:%830=494%:%
-%:%831=495%:%
-%:%832=496%:%
-%:%833=497%:%
-%:%834=498%:%
-%:%835=498%:%
-%:%836=499%:%
-%:%837=500%:%
-%:%838=501%:%
-%:%839=501%:%
-%:%840=502%:%
-%:%841=502%:%
-%:%842=503%:%
-%:%845=506%:%
-%:%846=507%:%
-%:%847=507%:%
-%:%848=508%:%
-%:%849=508%:%
-%:%850=509%:%
-%:%851=509%:%
-%:%852=510%:%
-%:%853=510%:%
-%:%854=511%:%
-%:%855=511%:%
-%:%856=512%:%
-%:%857=512%:%
-%:%858=512%:%
-%:%859=513%:%
-%:%860=514%:%
-%:%861=514%:%
-%:%862=515%:%
-%:%863=515%:%
-%:%864=516%:%
-%:%865=516%:%
-%:%866=517%:%
-%:%869=520%:%
-%:%870=521%:%
-%:%871=521%:%
-%:%872=522%:%
-%:%873=522%:%
-%:%874=523%:%
-%:%875=523%:%
-%:%876=523%:%
-%:%877=524%:%
-%:%878=524%:%
-%:%879=525%:%
-%:%880=525%:%
-%:%881=526%:%
-%:%882=526%:%
-%:%883=526%:%
-%:%884=527%:%
-%:%885=527%:%
-%:%886=528%:%
-%:%887=528%:%
-%:%888=529%:%
-%:%889=529%:%
-%:%890=529%:%
-%:%891=530%:%
-%:%892=530%:%
-%:%893=531%:%
-%:%894=531%:%
-%:%895=532%:%
-%:%896=532%:%
-%:%897=532%:%
-%:%898=533%:%
-%:%901=536%:%
-%:%902=537%:%
-%:%903=537%:%
-%:%904=538%:%
-%:%905=538%:%
-%:%906=539%:%
-%:%907=539%:%
-%:%908=539%:%
-%:%909=540%:%
-%:%910=540%:%
-%:%911=541%:%
-%:%912=541%:%
-%:%913=542%:%
-%:%914=542%:%
-%:%915=543%:%
-%:%916=543%:%
-%:%917=544%:%
-%:%918=544%:%
-%:%919=545%:%
-%:%920=545%:%
-%:%921=546%:%
-%:%922=546%:%
-%:%923=547%:%
-%:%924=548%:%
-%:%925=548%:%
-%:%926=549%:%
-%:%927=549%:%
-%:%928=550%:%
-%:%929=550%:%
-%:%930=551%:%
-%:%931=552%:%
-%:%932=552%:%
-%:%933=553%:%
-%:%934=553%:%
-%:%935=554%:%
-%:%936=554%:%
-%:%937=555%:%
-%:%938=555%:%
-%:%939=556%:%
-%:%940=556%:%
-%:%941=557%:%
-%:%942=557%:%
-%:%943=558%:%
-%:%944=558%:%
-%:%945=559%:%
-%:%946=559%:%
-%:%947=560%:%
-%:%948=560%:%
-%:%949=561%:%
-%:%955=561%:%
-%:%958=562%:%
-%:%959=563%:%
-%:%960=563%:%
-%:%961=564%:%
-%:%965=568%:%
-%:%972=569%:%
-%:%973=569%:%
-%:%974=570%:%
-%:%975=570%:%
-%:%976=571%:%
-%:%977=571%:%
-%:%978=572%:%
-%:%979=572%:%
-%:%980=573%:%
-%:%981=573%:%
-%:%982=574%:%
-%:%983=574%:%
-%:%984=575%:%
-%:%985=576%:%
-%:%986=577%:%
-%:%987=578%:%
-%:%988=578%:%
-%:%989=579%:%
-%:%990=580%:%
-%:%991=580%:%
-%:%992=581%:%
-%:%993=582%:%
-%:%994=582%:%
-%:%995=583%:%
-%:%996=583%:%
-%:%997=584%:%
-%:%998=584%:%
-%:%999=585%:%
-%:%1000=586%:%
-%:%1001=587%:%
-%:%1002=588%:%
-%:%1003=588%:%
-%:%1004=588%:%
-%:%1005=589%:%
-%:%1007=591%:%
-%:%1008=592%:%
-%:%1009=592%:%
-%:%1010=593%:%
-%:%1011=593%:%
-%:%1012=594%:%
-%:%1013=594%:%
-%:%1014=594%:%
-%:%1015=595%:%
-%:%1018=598%:%
-%:%1019=599%:%
-%:%1020=599%:%
-%:%1021=600%:%
-%:%1022=600%:%
-%:%1023=601%:%
-%:%1024=601%:%
-%:%1025=602%:%
-%:%1026=602%:%
-%:%1027=603%:%
-%:%1030=606%:%
-%:%1031=607%:%
-%:%1032=607%:%
-%:%1033=608%:%
-%:%1034=608%:%
-%:%1035=609%:%
-%:%1036=609%:%
-%:%1037=610%:%
-%:%1038=611%:%
-%:%1039=611%:%
-%:%1040=612%:%
-%:%1045=617%:%
-%:%1046=618%:%
-%:%1047=618%:%
-%:%1048=619%:%
-%:%1049=619%:%
-%:%1050=620%:%
-%:%1056=620%:%
-%:%1061=621%:%
-%:%1066=622%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Copeland_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Copeland_Rule.tex
deleted file mode 100644
index ce2aad3..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Copeland_Rule.tex
+++ /dev/null
@@ -1,162 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Copeland{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Copeland Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Copeland{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Facts{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Copeland voting rule. The idea is to elect the alternatives with
-the highest difference between the amount of simple-majority wins and the
-amount of simple-majority losses.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}copeland{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ copeland\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}condorcet{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ copeland{\isacharunderscore}{\kern0pt}rule{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ copeland{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ copeland{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ copeland{\isacharunderscore}{\kern0pt}rule{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Voting_Rules/Copeland_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%35=14%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%55=21%:%
-%:%65=23%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=27%:%
-%:%78=28%:%
-%:%79=28%:%
-%:%80=29%:%
-%:%81=29%:%
-%:%82=30%:%
-%:%83=31%:%
-%:%84=31%:%
-%:%85=32%:%
-%:%86=32%:%
-%:%87=33%:%
-%:%88=33%:%
-%:%89=34%:%
-%:%90=34%:%
-%:%91=35%:%
-%:%92=36%:%
-%:%93=36%:%
-%:%94=37%:%
-%:%100=37%:%
-%:%105=38%:%
-%:%110=39%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Defer_Equal_Condition.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Defer_Equal_Condition.tex
deleted file mode 100644
index f0237b2..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Defer_Equal_Condition.tex
+++ /dev/null
@@ -1,101 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Defer{\isacharunderscore}{\kern0pt}Equal{\isacharunderscore}{\kern0pt}Condition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Defer Equal Condition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Defer{\isacharunderscore}{\kern0pt}Equal{\isacharunderscore}{\kern0pt}Condition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is a family of termination conditions. For a natural number n,
-the according defer-equal condition is true if and only if the given
-result's defer-set contains exactly n elements.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ n\ result\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ result\ in\ card\ d\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Defer_Equal_Condition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%53=19%:%
-%:%63=21%:%
-%:%64=21%:%
-%:%65=22%:%
-%:%68=23%:%
-%:%73=24%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Defer_Module.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Defer_Module.tex
deleted file mode 100644
index d1835b7..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Defer_Module.tex
+++ /dev/null
@@ -1,150 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Defer{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupchapter{Basic Modules%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Defer Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Defer{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The defer module is not concerned about the voter's ballots, and
-simply defers all alternatives.
-It is primarily used for defining an empty loop.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}module\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ defer{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Defer_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%18=10%:%
-%:%34=12%:%
-%:%35=12%:%
-%:%36=13%:%
-%:%37=14%:%
-%:%46=17%:%
-%:%47=18%:%
-%:%48=19%:%
-%:%57=21%:%
-%:%67=23%:%
-%:%68=23%:%
-%:%69=24%:%
-%:%76=26%:%
-%:%86=28%:%
-%:%87=28%:%
-%:%90=29%:%
-%:%94=29%:%
-%:%95=29%:%
-%:%96=30%:%
-%:%97=30%:%
-%:%102=30%:%
-%:%107=31%:%
-%:%112=32%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility.tex
deleted file mode 100644
index d476a67..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility.tex
+++ /dev/null
@@ -1,70 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Disjoint{\isacharunderscore}{\kern0pt}Compatibility}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Disjoint{\isacharunderscore}{\kern0pt}Compatibility\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Indep{\isacharunderscore}{\kern0pt}Of{\isacharunderscore}{\kern0pt}Alt\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}S{\isachardot}{\kern0pt}\ finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Properties/Disjoint_Compatibility.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%20=5%:%
-%:%23=6%:%
-%:%24=11%:%
-%:%25=12%:%
-%:%26=12%:%
-%:%27=13%:%
-%:%28=14%:%
-%:%35=21%:%
-%:%38=22%:%
-%:%43=23%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility_Facts.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility_Facts.tex
deleted file mode 100644
index 6b1e3ef..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility_Facts.tex
+++ /dev/null
@@ -1,275 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isacharunderscore}{\kern0pt}Facts}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isacharunderscore}{\kern0pt}Facts\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Drop{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}pass{\isacharunderscore}{\kern0pt}disj{\isacharunderscore}{\kern0pt}compat{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ S{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{where}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ empty{\isacharunderscore}{\kern0pt}iff\ empty{\isacharunderscore}{\kern0pt}set\ fin\ profile{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p\ q{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ q{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ {\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ {\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ {\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharquery}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Composition_Rules/Disjoint_Compatibility_Facts.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%16=6%:%
-%:%21=6%:%
-%:%24=7%:%
-%:%25=8%:%
-%:%26=9%:%
-%:%27=9%:%
-%:%28=10%:%
-%:%29=11%:%
-%:%32=12%:%
-%:%36=12%:%
-%:%37=12%:%
-%:%38=13%:%
-%:%39=13%:%
-%:%40=14%:%
-%:%41=14%:%
-%:%42=15%:%
-%:%43=15%:%
-%:%44=16%:%
-%:%45=16%:%
-%:%46=17%:%
-%:%47=17%:%
-%:%48=18%:%
-%:%49=18%:%
-%:%50=19%:%
-%:%51=19%:%
-%:%52=20%:%
-%:%53=20%:%
-%:%54=21%:%
-%:%55=21%:%
-%:%56=22%:%
-%:%57=22%:%
-%:%58=23%:%
-%:%59=24%:%
-%:%60=24%:%
-%:%61=25%:%
-%:%62=26%:%
-%:%63=26%:%
-%:%64=27%:%
-%:%65=28%:%
-%:%66=29%:%
-%:%67=29%:%
-%:%68=30%:%
-%:%69=30%:%
-%:%70=31%:%
-%:%71=31%:%
-%:%72=32%:%
-%:%78=38%:%
-%:%79=39%:%
-%:%80=39%:%
-%:%81=40%:%
-%:%82=40%:%
-%:%83=41%:%
-%:%85=43%:%
-%:%86=44%:%
-%:%87=44%:%
-%:%88=45%:%
-%:%89=45%:%
-%:%90=46%:%
-%:%91=46%:%
-%:%92=47%:%
-%:%93=47%:%
-%:%94=48%:%
-%:%95=48%:%
-%:%96=48%:%
-%:%97=49%:%
-%:%98=50%:%
-%:%99=50%:%
-%:%100=51%:%
-%:%101=51%:%
-%:%102=52%:%
-%:%103=52%:%
-%:%104=52%:%
-%:%105=53%:%
-%:%106=54%:%
-%:%107=55%:%
-%:%108=55%:%
-%:%109=56%:%
-%:%110=56%:%
-%:%111=56%:%
-%:%112=57%:%
-%:%113=58%:%
-%:%114=58%:%
-%:%115=59%:%
-%:%116=59%:%
-%:%117=60%:%
-%:%118=60%:%
-%:%119=60%:%
-%:%120=61%:%
-%:%121=62%:%
-%:%122=63%:%
-%:%123=63%:%
-%:%124=64%:%
-%:%125=64%:%
-%:%126=64%:%
-%:%127=65%:%
-%:%133=71%:%
-%:%134=72%:%
-%:%135=72%:%
-%:%136=73%:%
-%:%137=73%:%
-%:%138=74%:%
-%:%144=74%:%
-%:%149=75%:%
-%:%154=76%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility_Rules.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility_Rules.tex
deleted file mode 100644
index 6851942..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Disjoint_Compatibility_Rules.tex
+++ /dev/null
@@ -1,521 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isacharunderscore}{\kern0pt}Rules}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isacharunderscore}{\kern0pt}Rules\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Sequential{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}compat{\isacharunderscore}{\kern0pt}comm{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ n\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}S{\isachardot}{\kern0pt}\ finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ A\ \isakeyword{where}\ old{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}{\isacharparenleft}{\kern0pt}S{\isacharminus}{\kern0pt}A{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ double{\isacharunderscore}{\kern0pt}diff\ order{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ S\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}compat{\isacharunderscore}{\kern0pt}seq{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ module{\isacharunderscore}{\kern0pt}m{\isadigit{2}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}S{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ S{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ modules{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ module{\isacharunderscore}{\kern0pt}m{\isadigit{2}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ A\ \isakeyword{where}\ A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}S\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}A\ {\isasymsubseteq}\ S{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ p\ q{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ A\ {\isasymand}\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ p\ q\ a\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer\ m\ S\ p\ {\isacharequal}{\kern0pt}\ defer\ m\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ a\ b\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ a\ b\ \isacommand{have}\isamarkupfalse%
-\ profiles{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ q{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ DiffD{\isadigit{2}}\ a\ b\ compatible\ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ eq{\isacharunderscore}{\kern0pt}def\ profiles\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}def\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}m{\isadigit{2}}\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ m{\isadigit{2}}\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ q{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ S\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}m\ S\ p\ {\isacharequal}{\kern0pt}\ m\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ a\ b\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ UnI{\isadigit{1}}\ prod{\isachardot}{\kern0pt}sel\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\ modules\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Composition_Rules/Disjoint_Compatibility_Rules.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%20=5%:%
-%:%23=6%:%
-%:%24=7%:%
-%:%25=8%:%
-%:%26=8%:%
-%:%27=9%:%
-%:%28=10%:%
-%:%35=11%:%
-%:%36=11%:%
-%:%37=12%:%
-%:%38=12%:%
-%:%39=13%:%
-%:%44=18%:%
-%:%45=19%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%48=20%:%
-%:%49=21%:%
-%:%50=22%:%
-%:%51=22%:%
-%:%52=23%:%
-%:%57=28%:%
-%:%58=29%:%
-%:%59=29%:%
-%:%60=30%:%
-%:%61=30%:%
-%:%62=31%:%
-%:%63=31%:%
-%:%64=32%:%
-%:%69=37%:%
-%:%70=38%:%
-%:%71=38%:%
-%:%72=39%:%
-%:%73=39%:%
-%:%74=40%:%
-%:%79=45%:%
-%:%80=46%:%
-%:%81=46%:%
-%:%82=47%:%
-%:%83=47%:%
-%:%84=48%:%
-%:%85=48%:%
-%:%86=49%:%
-%:%91=54%:%
-%:%92=55%:%
-%:%93=55%:%
-%:%94=56%:%
-%:%95=56%:%
-%:%96=57%:%
-%:%97=57%:%
-%:%98=57%:%
-%:%99=58%:%
-%:%100=58%:%
-%:%101=59%:%
-%:%102=59%:%
-%:%103=60%:%
-%:%104=60%:%
-%:%105=60%:%
-%:%106=61%:%
-%:%107=61%:%
-%:%108=62%:%
-%:%114=62%:%
-%:%117=63%:%
-%:%118=67%:%
-%:%119=68%:%
-%:%120=68%:%
-%:%121=69%:%
-%:%122=70%:%
-%:%123=71%:%
-%:%124=72%:%
-%:%127=73%:%
-%:%131=73%:%
-%:%132=73%:%
-%:%133=74%:%
-%:%134=74%:%
-%:%135=75%:%
-%:%136=75%:%
-%:%137=76%:%
-%:%138=76%:%
-%:%139=77%:%
-%:%140=77%:%
-%:%141=78%:%
-%:%142=78%:%
-%:%143=79%:%
-%:%144=79%:%
-%:%145=80%:%
-%:%146=80%:%
-%:%147=81%:%
-%:%148=81%:%
-%:%149=82%:%
-%:%150=82%:%
-%:%151=83%:%
-%:%152=83%:%
-%:%153=84%:%
-%:%154=85%:%
-%:%155=85%:%
-%:%156=86%:%
-%:%157=87%:%
-%:%158=87%:%
-%:%159=88%:%
-%:%160=89%:%
-%:%161=89%:%
-%:%162=90%:%
-%:%163=90%:%
-%:%164=91%:%
-%:%165=91%:%
-%:%166=92%:%
-%:%170=96%:%
-%:%171=97%:%
-%:%172=97%:%
-%:%173=98%:%
-%:%174=98%:%
-%:%175=99%:%
-%:%176=99%:%
-%:%177=100%:%
-%:%181=104%:%
-%:%182=105%:%
-%:%183=105%:%
-%:%184=106%:%
-%:%185=106%:%
-%:%186=107%:%
-%:%188=109%:%
-%:%189=110%:%
-%:%190=110%:%
-%:%191=111%:%
-%:%192=111%:%
-%:%193=112%:%
-%:%194=113%:%
-%:%195=114%:%
-%:%196=115%:%
-%:%197=115%:%
-%:%198=116%:%
-%:%199=117%:%
-%:%200=118%:%
-%:%201=118%:%
-%:%202=119%:%
-%:%203=120%:%
-%:%204=120%:%
-%:%205=121%:%
-%:%206=121%:%
-%:%207=122%:%
-%:%208=122%:%
-%:%209=122%:%
-%:%210=123%:%
-%:%211=124%:%
-%:%212=124%:%
-%:%213=125%:%
-%:%214=125%:%
-%:%215=126%:%
-%:%216=126%:%
-%:%217=127%:%
-%:%218=127%:%
-%:%219=128%:%
-%:%220=128%:%
-%:%221=129%:%
-%:%222=129%:%
-%:%223=130%:%
-%:%224=131%:%
-%:%225=132%:%
-%:%226=132%:%
-%:%227=133%:%
-%:%228=134%:%
-%:%229=135%:%
-%:%230=135%:%
-%:%231=136%:%
-%:%232=136%:%
-%:%233=136%:%
-%:%234=137%:%
-%:%235=138%:%
-%:%236=139%:%
-%:%237=139%:%
-%:%238=140%:%
-%:%239=140%:%
-%:%240=140%:%
-%:%241=141%:%
-%:%242=141%:%
-%:%243=142%:%
-%:%244=142%:%
-%:%245=143%:%
-%:%246=143%:%
-%:%247=143%:%
-%:%248=144%:%
-%:%249=145%:%
-%:%250=145%:%
-%:%251=146%:%
-%:%252=146%:%
-%:%253=147%:%
-%:%254=147%:%
-%:%255=148%:%
-%:%256=148%:%
-%:%257=148%:%
-%:%258=149%:%
-%:%259=150%:%
-%:%260=150%:%
-%:%261=151%:%
-%:%262=151%:%
-%:%263=152%:%
-%:%264=152%:%
-%:%265=152%:%
-%:%266=153%:%
-%:%270=157%:%
-%:%271=158%:%
-%:%272=158%:%
-%:%273=159%:%
-%:%274=159%:%
-%:%275=160%:%
-%:%276=160%:%
-%:%277=161%:%
-%:%283=161%:%
-%:%288=162%:%
-%:%293=163%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Drop_Module.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Drop_Module.tex
deleted file mode 100644
index 6e8fdea..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Drop_Module.tex
+++ /dev/null
@@ -1,272 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Drop{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Drop Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Drop{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is a family of electoral modules. For a natural number n and a
-lexicon (linear order) r of all alternatives, the according drop module
-rejects the lexicographically first n alternatives (from A) and
-defers the rest.
-It is primarily used as counterpart to the pass module in a
-parallel composition, in order to segment the alternatives into
-two groups.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}mod\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}modI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Drop_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%48=21%:%
-%:%57=23%:%
-%:%67=25%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%79=31%:%
-%:%89=33%:%
-%:%90=33%:%
-%:%91=34%:%
-%:%92=35%:%
-%:%99=36%:%
-%:%100=36%:%
-%:%101=37%:%
-%:%102=37%:%
-%:%103=38%:%
-%:%104=38%:%
-%:%105=39%:%
-%:%107=41%:%
-%:%108=42%:%
-%:%109=42%:%
-%:%110=43%:%
-%:%111=43%:%
-%:%112=44%:%
-%:%114=46%:%
-%:%115=47%:%
-%:%116=47%:%
-%:%117=48%:%
-%:%118=48%:%
-%:%119=49%:%
-%:%120=50%:%
-%:%121=51%:%
-%:%122=51%:%
-%:%123=52%:%
-%:%124=52%:%
-%:%125=53%:%
-%:%127=55%:%
-%:%128=56%:%
-%:%129=56%:%
-%:%130=57%:%
-%:%131=57%:%
-%:%132=58%:%
-%:%134=60%:%
-%:%135=61%:%
-%:%136=61%:%
-%:%137=62%:%
-%:%138=62%:%
-%:%139=63%:%
-%:%140=63%:%
-%:%141=64%:%
-%:%142=64%:%
-%:%143=64%:%
-%:%144=65%:%
-%:%145=66%:%
-%:%146=67%:%
-%:%147=67%:%
-%:%148=68%:%
-%:%149=68%:%
-%:%150=69%:%
-%:%151=70%:%
-%:%152=71%:%
-%:%153=71%:%
-%:%154=72%:%
-%:%155=72%:%
-%:%156=73%:%
-%:%157=73%:%
-%:%158=74%:%
-%:%159=74%:%
-%:%160=75%:%
-%:%166=75%:%
-%:%171=76%:%
-%:%176=77%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Elect_Module.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Elect_Module.tex
deleted file mode 100644
index a964962..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Elect_Module.tex
+++ /dev/null
@@ -1,147 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Elect{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Elect Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Elect{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The elect module is not concerned about the voter's ballots, and
-just elects all alternatives. It is primarily used in sequence after
-an electoral module that only defers alternatives to finalize the decision,
-thereby inducing a proper voting rule in the social choice sense.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elect{\isacharunderscore}{\kern0pt}module\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Elect_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%54=20%:%
-%:%64=22%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%73=25%:%
-%:%83=27%:%
-%:%84=27%:%
-%:%87=28%:%
-%:%91=28%:%
-%:%92=28%:%
-%:%93=29%:%
-%:%94=29%:%
-%:%99=29%:%
-%:%104=30%:%
-%:%109=31%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Electoral_Module.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Electoral_Module.tex
deleted file mode 100644
index ab4be9a..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Electoral_Module.tex
+++ /dev/null
@@ -1,1952 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Electoral{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-\isanewline
-%
-\isamarkupchapter{Component Types%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Electoral Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Electoral{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Result{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-Electoral modules are the principal component type of the composable modules
-voting framework, as they are a generalization of voting rules in the sense of
-social choice functions.
-These are only the types used for electoral modules. Further restrictions are
-encompassed by the electoral-module predicate.
-
-An electoral module does not need to make final decisions for all alternatives,
-but can instead defer the decision for some or all of them to other modules.
-Hence, electoral modules partition the received (possibly empty) set of
-alternatives into elected, rejected and deferred alternatives. In particular,
-any of those sets, e.g., the set of winning (elected) alternatives, may also
-be left empty, as long as they collectively still hold all the received
-alternatives. Just like a voting rule, an electoral module also receives a
-profile which holds the voters’ preferences, which, unlike a voting rule,
-consider only the (sub-)set of alternatives that the module receives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Definitions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymequiv}\ {\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}modI{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymAnd}A\ p{\isachardot}{\kern0pt}\ {\isasymlbrakk}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ elect\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymequiv}\ elect{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ reject\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isasymequiv}\ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer{\isachardoublequoteclose}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymequiv}\ defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Equivalence Definitions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ reject\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ defer\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}leq{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prof{\isacharunderscore}{\kern0pt}leq{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ reject\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymnotin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prof{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymnotin}\ reject\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ n\ A\ p\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ elect\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ reject\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymlongrightarrow}\ a\ {\isasymin}\ defer\ n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ ele{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ e{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rej{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ def\ ele\ rej\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ UnI{\isadigit{1}}\ asm\ fstI\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ UnI{\isadigit{1}}\ asm\ fstI\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \ \ \ \ \ \ sndI\ subsetD\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ set{\isacharunderscore}{\kern0pt}partit\ partit\ sndI\ subsetD\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ defer\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ asm{\isadigit{3}}\ fst{\isacharunderscore}{\kern0pt}conv\ partit\isanewline
-\ \ \ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}partit\ snd{\isacharunderscore}{\kern0pt}conv\ Un{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ profile\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ prof{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}exhaust{\isacharunderscore}{\kern0pt}sel\ DiffE\ UnCI\ asm{\isadigit{1}}\ asm{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ module\ profile\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}p\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isasymnot}\ disjoint{\isadigit{3}}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D{\isachardot}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}B{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharcomma}{\kern0pt}\ C{\isacharcomma}{\kern0pt}\ D{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ B\ {\isasyminter}\ C\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ B\ {\isasyminter}\ D\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ C\ {\isasyminter}\ D\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ prof{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ disj{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disj{\isadigit{3}}\ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ AA\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ AAa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ AAb\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{where}\isanewline
-\ \ \ \ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ AAa\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ AAb\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ AA\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ AAa\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ AA\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ AAb\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ AAa\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ AAb\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disj{\isadigit{3}}\ eq{\isacharunderscore}{\kern0pt}snd{\isacharunderscore}{\kern0pt}iff\ fstI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ module\ profile\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m\ prof{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}partit\ partit\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}B\ C\ D\ E{\isachardot}{\kern0pt}\ A\ {\isacharequal}{\kern0pt}\ B\ {\isasymand}\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}C{\isacharcomma}{\kern0pt}\ D{\isacharcomma}{\kern0pt}\ E{\isacharparenright}{\kern0pt}\ {\isasymand}\ C\ {\isasymunion}\ D\ {\isasymunion}\ E\ {\isacharequal}{\kern0pt}\ B{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ profile\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\ \isacommand{have}\isamarkupfalse%
-\ prof{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ prof{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}exhaust{\isacharunderscore}{\kern0pt}sel\ DiffE\ UnCI\ asm{\isadigit{1}}\ asm{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ module\ profile\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\ wf{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}p\ set{\isacharunderscore}{\kern0pt}partit\ partit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}supI{\isadigit{1}}\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{1}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ le{\isacharunderscore}{\kern0pt}supI{\isadigit{1}}\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ sup{\isacharunderscore}{\kern0pt}ge{\isadigit{2}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p\ in\isanewline
-\ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ new{\isacharunderscore}{\kern0pt}A\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ new{\isacharunderscore}{\kern0pt}A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ infinite{\isacharunderscore}{\kern0pt}super\isanewline
-\ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound\ module\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ upper{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}bounds{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}result{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ A\ {\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ elec{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymunion}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ alternative{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}elected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ elect\ m\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ DiffI\ e{\isacharunderscore}{\kern0pt}mod\ f{\isacharunderscore}{\kern0pt}prof\ alternative\isanewline
-\ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}elected\ not{\isacharunderscore}{\kern0pt}rejected\isanewline
-\ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}comm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ n\ A\ p\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ m\ A\ p\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ IntI\ assms\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ e{\isacharunderscore}{\kern0pt}mod{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ alternative{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}rejected{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ m\ A\ p\ {\isasymor}\ x\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ alternative\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \ \ e{\isacharunderscore}{\kern0pt}mod\ not{\isacharunderscore}{\kern0pt}rejected\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}alts{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}profs{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}results{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ A\ p\ q\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ input{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ elect\ m\ A\ p{\isachardot}{\kern0pt}\ a\ {\isasymin}\ elect\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ elect\ m\ A\ q{\isachardot}{\kern0pt}\ a\ {\isasymin}\ elect\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ reject\ m\ A\ p{\isachardot}{\kern0pt}\ a\ {\isasymin}\ reject\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ reject\ m\ A\ q{\isachardot}{\kern0pt}\ a\ {\isasymin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ input\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ defer\ m\ A\ p{\isachardot}{\kern0pt}\ a\ {\isasymin}\ defer\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ input\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ defer\ m\ A\ q{\isachardot}{\kern0pt}\ a\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ contra{\isacharunderscore}{\kern0pt}subsetD\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ eq\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ input\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}collapse\ subsetI\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ elect\ n\ A\ q{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ n\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ n\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}n\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}m\ prod{\isachardot}{\kern0pt}collapse\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}reject\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}n\ prod{\isachardot}{\kern0pt}collapse\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ n\ A\ q\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ n\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}defer\ n\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}p\ reject{\isacharunderscore}{\kern0pt}q\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{5}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isacharparenright}{\kern0pt}\ prod{\isacharunderscore}{\kern0pt}eqI{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Electoral_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%13=6%:%
-%:%15=9%:%
-%:%19=11%:%
-%:%35=13%:%
-%:%36=13%:%
-%:%37=14%:%
-%:%38=15%:%
-%:%39=16%:%
-%:%40=17%:%
-%:%41=18%:%
-%:%50=21%:%
-%:%51=22%:%
-%:%52=23%:%
-%:%53=24%:%
-%:%54=25%:%
-%:%55=26%:%
-%:%56=27%:%
-%:%57=28%:%
-%:%58=29%:%
-%:%59=30%:%
-%:%60=31%:%
-%:%61=32%:%
-%:%62=33%:%
-%:%63=34%:%
-%:%64=35%:%
-%:%73=37%:%
-%:%83=40%:%
-%:%84=40%:%
-%:%91=42%:%
-%:%101=52%:%
-%:%102=52%:%
-%:%103=53%:%
-%:%104=54%:%
-%:%105=55%:%
-%:%106=55%:%
-%:%107=56%:%
-%:%108=57%:%
-%:%111=58%:%
-%:%115=58%:%
-%:%116=58%:%
-%:%117=59%:%
-%:%118=59%:%
-%:%123=59%:%
-%:%126=60%:%
-%:%127=64%:%
-%:%128=65%:%
-%:%129=65%:%
-%:%130=66%:%
-%:%131=67%:%
-%:%132=68%:%
-%:%133=69%:%
-%:%134=69%:%
-%:%135=70%:%
-%:%136=71%:%
-%:%137=72%:%
-%:%138=73%:%
-%:%139=73%:%
-%:%140=74%:%
-%:%141=75%:%
-%:%148=97%:%
-%:%158=99%:%
-%:%159=99%:%
-%:%160=100%:%
-%:%161=101%:%
-%:%165=105%:%
-%:%166=106%:%
-%:%167=107%:%
-%:%168=107%:%
-%:%169=108%:%
-%:%170=109%:%
-%:%173=112%:%
-%:%174=113%:%
-%:%175=114%:%
-%:%176=114%:%
-%:%177=115%:%
-%:%178=116%:%
-%:%181=119%:%
-%:%182=120%:%
-%:%183=121%:%
-%:%184=121%:%
-%:%185=122%:%
-%:%186=123%:%
-%:%197=129%:%
-%:%207=131%:%
-%:%208=131%:%
-%:%209=132%:%
-%:%210=133%:%
-%:%211=134%:%
-%:%212=135%:%
-%:%213=136%:%
-%:%216=137%:%
-%:%220=137%:%
-%:%221=137%:%
-%:%222=138%:%
-%:%223=138%:%
-%:%228=138%:%
-%:%231=139%:%
-%:%232=140%:%
-%:%233=140%:%
-%:%234=141%:%
-%:%235=142%:%
-%:%236=143%:%
-%:%237=144%:%
-%:%240=145%:%
-%:%244=145%:%
-%:%245=145%:%
-%:%246=146%:%
-%:%247=146%:%
-%:%252=146%:%
-%:%255=147%:%
-%:%256=148%:%
-%:%257=148%:%
-%:%258=149%:%
-%:%259=150%:%
-%:%260=151%:%
-%:%261=152%:%
-%:%268=153%:%
-%:%269=153%:%
-%:%270=154%:%
-%:%271=154%:%
-%:%272=155%:%
-%:%273=156%:%
-%:%274=156%:%
-%:%275=157%:%
-%:%276=158%:%
-%:%277=158%:%
-%:%278=159%:%
-%:%280=161%:%
-%:%281=162%:%
-%:%282=162%:%
-%:%283=163%:%
-%:%284=163%:%
-%:%285=163%:%
-%:%286=164%:%
-%:%287=165%:%
-%:%288=165%:%
-%:%289=166%:%
-%:%290=166%:%
-%:%291=167%:%
-%:%292=167%:%
-%:%293=168%:%
-%:%294=168%:%
-%:%295=169%:%
-%:%296=169%:%
-%:%297=170%:%
-%:%298=170%:%
-%:%299=171%:%
-%:%300=171%:%
-%:%301=172%:%
-%:%302=173%:%
-%:%303=173%:%
-%:%304=174%:%
-%:%305=175%:%
-%:%306=175%:%
-%:%307=176%:%
-%:%309=178%:%
-%:%310=179%:%
-%:%311=179%:%
-%:%312=180%:%
-%:%313=180%:%
-%:%314=180%:%
-%:%315=181%:%
-%:%316=182%:%
-%:%317=182%:%
-%:%318=183%:%
-%:%319=183%:%
-%:%320=184%:%
-%:%321=184%:%
-%:%322=185%:%
-%:%323=185%:%
-%:%324=186%:%
-%:%325=187%:%
-%:%326=187%:%
-%:%327=188%:%
-%:%328=188%:%
-%:%329=189%:%
-%:%330=189%:%
-%:%331=190%:%
-%:%332=191%:%
-%:%333=191%:%
-%:%334=192%:%
-%:%335=193%:%
-%:%336=193%:%
-%:%337=194%:%
-%:%339=196%:%
-%:%340=197%:%
-%:%341=197%:%
-%:%342=198%:%
-%:%343=198%:%
-%:%344=198%:%
-%:%345=199%:%
-%:%346=200%:%
-%:%347=200%:%
-%:%348=201%:%
-%:%349=201%:%
-%:%350=202%:%
-%:%351=202%:%
-%:%352=203%:%
-%:%353=203%:%
-%:%354=204%:%
-%:%355=204%:%
-%:%356=205%:%
-%:%357=205%:%
-%:%358=206%:%
-%:%359=206%:%
-%:%360=207%:%
-%:%361=208%:%
-%:%362=208%:%
-%:%363=209%:%
-%:%364=210%:%
-%:%365=211%:%
-%:%366=212%:%
-%:%367=212%:%
-%:%368=213%:%
-%:%370=215%:%
-%:%371=216%:%
-%:%372=216%:%
-%:%373=217%:%
-%:%374=217%:%
-%:%375=217%:%
-%:%376=218%:%
-%:%377=219%:%
-%:%378=219%:%
-%:%379=220%:%
-%:%380=220%:%
-%:%381=221%:%
-%:%382=221%:%
-%:%383=222%:%
-%:%384=222%:%
-%:%385=223%:%
-%:%386=224%:%
-%:%387=224%:%
-%:%388=225%:%
-%:%394=225%:%
-%:%397=226%:%
-%:%398=227%:%
-%:%399=227%:%
-%:%400=228%:%
-%:%401=229%:%
-%:%402=230%:%
-%:%403=231%:%
-%:%404=232%:%
-%:%406=234%:%
-%:%413=235%:%
-%:%414=235%:%
-%:%415=236%:%
-%:%416=236%:%
-%:%417=237%:%
-%:%418=238%:%
-%:%419=238%:%
-%:%420=239%:%
-%:%421=240%:%
-%:%422=241%:%
-%:%423=241%:%
-%:%424=242%:%
-%:%426=244%:%
-%:%427=245%:%
-%:%428=245%:%
-%:%429=246%:%
-%:%430=246%:%
-%:%431=246%:%
-%:%432=247%:%
-%:%433=248%:%
-%:%434=248%:%
-%:%435=249%:%
-%:%436=249%:%
-%:%437=250%:%
-%:%438=250%:%
-%:%439=250%:%
-%:%440=251%:%
-%:%441=252%:%
-%:%442=252%:%
-%:%443=253%:%
-%:%444=253%:%
-%:%445=253%:%
-%:%446=254%:%
-%:%447=255%:%
-%:%448=255%:%
-%:%449=256%:%
-%:%450=256%:%
-%:%451=257%:%
-%:%452=257%:%
-%:%453=258%:%
-%:%454=258%:%
-%:%455=259%:%
-%:%456=260%:%
-%:%457=261%:%
-%:%458=261%:%
-%:%459=262%:%
-%:%460=262%:%
-%:%461=263%:%
-%:%462=263%:%
-%:%463=264%:%
-%:%464=265%:%
-%:%465=265%:%
-%:%466=266%:%
-%:%467=267%:%
-%:%468=268%:%
-%:%469=268%:%
-%:%470=269%:%
-%:%472=271%:%
-%:%473=272%:%
-%:%474=272%:%
-%:%475=273%:%
-%:%476=273%:%
-%:%477=274%:%
-%:%479=276%:%
-%:%480=277%:%
-%:%481=277%:%
-%:%482=278%:%
-%:%483=278%:%
-%:%484=278%:%
-%:%485=279%:%
-%:%486=280%:%
-%:%487=280%:%
-%:%488=281%:%
-%:%489=281%:%
-%:%490=281%:%
-%:%491=282%:%
-%:%492=283%:%
-%:%493=283%:%
-%:%494=284%:%
-%:%495=284%:%
-%:%496=285%:%
-%:%497=285%:%
-%:%498=286%:%
-%:%499=287%:%
-%:%500=287%:%
-%:%501=288%:%
-%:%502=288%:%
-%:%503=289%:%
-%:%504=290%:%
-%:%505=290%:%
-%:%506=291%:%
-%:%507=291%:%
-%:%508=292%:%
-%:%509=293%:%
-%:%510=293%:%
-%:%511=294%:%
-%:%512=294%:%
-%:%513=295%:%
-%:%514=295%:%
-%:%515=295%:%
-%:%516=296%:%
-%:%517=297%:%
-%:%518=298%:%
-%:%519=299%:%
-%:%520=300%:%
-%:%524=304%:%
-%:%525=305%:%
-%:%526=305%:%
-%:%527=306%:%
-%:%528=306%:%
-%:%529=307%:%
-%:%530=307%:%
-%:%532=309%:%
-%:%533=310%:%
-%:%534=310%:%
-%:%535=311%:%
-%:%536=311%:%
-%:%537=312%:%
-%:%538=312%:%
-%:%539=313%:%
-%:%540=313%:%
-%:%541=314%:%
-%:%542=315%:%
-%:%543=315%:%
-%:%544=316%:%
-%:%545=316%:%
-%:%546=317%:%
-%:%547=317%:%
-%:%548=318%:%
-%:%549=319%:%
-%:%550=319%:%
-%:%551=320%:%
-%:%552=321%:%
-%:%553=322%:%
-%:%554=322%:%
-%:%555=323%:%
-%:%557=325%:%
-%:%558=326%:%
-%:%559=326%:%
-%:%560=327%:%
-%:%561=327%:%
-%:%562=327%:%
-%:%563=328%:%
-%:%564=329%:%
-%:%565=329%:%
-%:%566=330%:%
-%:%567=330%:%
-%:%568=331%:%
-%:%569=331%:%
-%:%570=331%:%
-%:%571=332%:%
-%:%572=333%:%
-%:%573=333%:%
-%:%574=334%:%
-%:%575=334%:%
-%:%576=334%:%
-%:%577=335%:%
-%:%578=336%:%
-%:%579=336%:%
-%:%580=337%:%
-%:%581=337%:%
-%:%582=338%:%
-%:%583=338%:%
-%:%584=339%:%
-%:%585=339%:%
-%:%586=340%:%
-%:%587=341%:%
-%:%588=342%:%
-%:%589=342%:%
-%:%590=343%:%
-%:%596=343%:%
-%:%599=344%:%
-%:%600=345%:%
-%:%601=345%:%
-%:%602=346%:%
-%:%603=347%:%
-%:%604=348%:%
-%:%605=349%:%
-%:%608=350%:%
-%:%612=350%:%
-%:%613=350%:%
-%:%614=351%:%
-%:%615=351%:%
-%:%620=351%:%
-%:%623=352%:%
-%:%624=353%:%
-%:%625=353%:%
-%:%626=354%:%
-%:%627=355%:%
-%:%628=356%:%
-%:%629=357%:%
-%:%632=358%:%
-%:%636=358%:%
-%:%637=358%:%
-%:%638=359%:%
-%:%639=359%:%
-%:%644=359%:%
-%:%647=360%:%
-%:%648=361%:%
-%:%649=361%:%
-%:%650=362%:%
-%:%651=363%:%
-%:%652=364%:%
-%:%653=365%:%
-%:%656=366%:%
-%:%660=366%:%
-%:%661=366%:%
-%:%662=367%:%
-%:%663=367%:%
-%:%668=367%:%
-%:%671=368%:%
-%:%672=369%:%
-%:%673=369%:%
-%:%674=370%:%
-%:%675=371%:%
-%:%676=372%:%
-%:%677=373%:%
-%:%678=374%:%
-%:%681=375%:%
-%:%685=375%:%
-%:%686=375%:%
-%:%687=376%:%
-%:%688=377%:%
-%:%689=377%:%
-%:%694=377%:%
-%:%697=378%:%
-%:%698=382%:%
-%:%699=383%:%
-%:%700=383%:%
-%:%701=384%:%
-%:%702=385%:%
-%:%703=386%:%
-%:%704=387%:%
-%:%705=388%:%
-%:%707=390%:%
-%:%710=391%:%
-%:%714=391%:%
-%:%715=391%:%
-%:%716=392%:%
-%:%721=392%:%
-%:%724=393%:%
-%:%725=394%:%
-%:%726=394%:%
-%:%727=395%:%
-%:%728=396%:%
-%:%729=397%:%
-%:%730=398%:%
-%:%737=399%:%
-%:%738=399%:%
-%:%739=400%:%
-%:%740=400%:%
-%:%741=400%:%
-%:%742=401%:%
-%:%743=401%:%
-%:%744=402%:%
-%:%745=402%:%
-%:%746=403%:%
-%:%747=403%:%
-%:%748=404%:%
-%:%749=404%:%
-%:%750=405%:%
-%:%751=405%:%
-%:%752=406%:%
-%:%753=406%:%
-%:%754=406%:%
-%:%755=406%:%
-%:%756=407%:%
-%:%757=408%:%
-%:%758=409%:%
-%:%759=409%:%
-%:%760=410%:%
-%:%761=410%:%
-%:%762=411%:%
-%:%763=411%:%
-%:%764=411%:%
-%:%765=412%:%
-%:%766=412%:%
-%:%767=413%:%
-%:%773=413%:%
-%:%776=414%:%
-%:%777=415%:%
-%:%778=415%:%
-%:%779=416%:%
-%:%780=417%:%
-%:%781=418%:%
-%:%782=419%:%
-%:%789=420%:%
-%:%790=420%:%
-%:%791=421%:%
-%:%792=421%:%
-%:%793=421%:%
-%:%794=422%:%
-%:%795=422%:%
-%:%796=423%:%
-%:%797=423%:%
-%:%798=424%:%
-%:%799=425%:%
-%:%800=425%:%
-%:%801=426%:%
-%:%802=426%:%
-%:%803=427%:%
-%:%804=427%:%
-%:%805=428%:%
-%:%806=428%:%
-%:%807=429%:%
-%:%808=429%:%
-%:%809=430%:%
-%:%810=430%:%
-%:%811=430%:%
-%:%812=430%:%
-%:%813=431%:%
-%:%814=432%:%
-%:%815=433%:%
-%:%816=433%:%
-%:%817=434%:%
-%:%818=434%:%
-%:%819=435%:%
-%:%820=435%:%
-%:%821=435%:%
-%:%822=436%:%
-%:%823=436%:%
-%:%824=437%:%
-%:%830=437%:%
-%:%833=438%:%
-%:%834=439%:%
-%:%835=439%:%
-%:%836=440%:%
-%:%837=441%:%
-%:%838=442%:%
-%:%839=443%:%
-%:%846=444%:%
-%:%847=444%:%
-%:%848=445%:%
-%:%849=445%:%
-%:%850=445%:%
-%:%851=446%:%
-%:%852=446%:%
-%:%853=447%:%
-%:%854=447%:%
-%:%855=448%:%
-%:%856=448%:%
-%:%857=449%:%
-%:%858=449%:%
-%:%859=450%:%
-%:%860=450%:%
-%:%861=450%:%
-%:%862=450%:%
-%:%863=451%:%
-%:%864=452%:%
-%:%865=453%:%
-%:%866=453%:%
-%:%867=454%:%
-%:%868=454%:%
-%:%869=455%:%
-%:%870=455%:%
-%:%871=455%:%
-%:%872=456%:%
-%:%873=456%:%
-%:%874=457%:%
-%:%880=457%:%
-%:%883=458%:%
-%:%884=459%:%
-%:%885=459%:%
-%:%886=460%:%
-%:%887=461%:%
-%:%888=462%:%
-%:%889=463%:%
-%:%890=464%:%
-%:%891=465%:%
-%:%892=466%:%
-%:%895=467%:%
-%:%899=467%:%
-%:%900=467%:%
-%:%901=468%:%
-%:%902=469%:%
-%:%903=470%:%
-%:%904=470%:%
-%:%909=470%:%
-%:%912=471%:%
-%:%913=472%:%
-%:%914=472%:%
-%:%915=473%:%
-%:%916=474%:%
-%:%919=475%:%
-%:%923=475%:%
-%:%924=475%:%
-%:%925=476%:%
-%:%926=477%:%
-%:%927=477%:%
-%:%932=477%:%
-%:%935=478%:%
-%:%936=479%:%
-%:%937=479%:%
-%:%938=480%:%
-%:%939=481%:%
-%:%940=482%:%
-%:%941=483%:%
-%:%942=484%:%
-%:%943=485%:%
-%:%946=486%:%
-%:%950=486%:%
-%:%951=486%:%
-%:%952=487%:%
-%:%953=488%:%
-%:%954=488%:%
-%:%959=488%:%
-%:%962=489%:%
-%:%963=490%:%
-%:%964=490%:%
-%:%965=491%:%
-%:%966=492%:%
-%:%967=493%:%
-%:%968=494%:%
-%:%969=495%:%
-%:%976=496%:%
-%:%977=496%:%
-%:%978=497%:%
-%:%979=497%:%
-%:%980=498%:%
-%:%981=498%:%
-%:%982=499%:%
-%:%983=499%:%
-%:%984=500%:%
-%:%985=500%:%
-%:%986=500%:%
-%:%987=501%:%
-%:%988=501%:%
-%:%989=502%:%
-%:%990=503%:%
-%:%991=504%:%
-%:%992=504%:%
-%:%993=505%:%
-%:%994=505%:%
-%:%995=505%:%
-%:%996=506%:%
-%:%997=506%:%
-%:%998=507%:%
-%:%999=507%:%
-%:%1000=508%:%
-%:%1001=508%:%
-%:%1002=508%:%
-%:%1003=509%:%
-%:%1004=509%:%
-%:%1005=510%:%
-%:%1006=511%:%
-%:%1007=512%:%
-%:%1008=512%:%
-%:%1009=513%:%
-%:%1010=513%:%
-%:%1011=513%:%
-%:%1012=514%:%
-%:%1013=514%:%
-%:%1014=515%:%
-%:%1015=515%:%
-%:%1016=516%:%
-%:%1017=516%:%
-%:%1018=516%:%
-%:%1019=517%:%
-%:%1020=517%:%
-%:%1021=518%:%
-%:%1022=519%:%
-%:%1023=520%:%
-%:%1024=520%:%
-%:%1025=521%:%
-%:%1026=521%:%
-%:%1027=521%:%
-%:%1028=522%:%
-%:%1029=522%:%
-%:%1030=523%:%
-%:%1031=523%:%
-%:%1032=524%:%
-%:%1038=524%:%
-%:%1041=525%:%
-%:%1042=526%:%
-%:%1043=526%:%
-%:%1044=527%:%
-%:%1045=528%:%
-%:%1046=529%:%
-%:%1047=530%:%
-%:%1048=531%:%
-%:%1049=532%:%
-%:%1050=533%:%
-%:%1051=534%:%
-%:%1058=535%:%
-%:%1059=535%:%
-%:%1060=536%:%
-%:%1061=536%:%
-%:%1062=537%:%
-%:%1063=538%:%
-%:%1064=538%:%
-%:%1065=539%:%
-%:%1066=539%:%
-%:%1067=540%:%
-%:%1068=540%:%
-%:%1069=541%:%
-%:%1070=542%:%
-%:%1071=542%:%
-%:%1072=543%:%
-%:%1073=543%:%
-%:%1074=544%:%
-%:%1075=544%:%
-%:%1076=545%:%
-%:%1077=546%:%
-%:%1078=546%:%
-%:%1079=547%:%
-%:%1080=547%:%
-%:%1081=548%:%
-%:%1082=548%:%
-%:%1083=548%:%
-%:%1084=549%:%
-%:%1085=550%:%
-%:%1086=550%:%
-%:%1087=551%:%
-%:%1088=551%:%
-%:%1089=552%:%
-%:%1090=552%:%
-%:%1091=553%:%
-%:%1092=554%:%
-%:%1093=554%:%
-%:%1094=555%:%
-%:%1095=555%:%
-%:%1096=556%:%
-%:%1097=556%:%
-%:%1098=556%:%
-%:%1099=557%:%
-%:%1100=558%:%
-%:%1101=558%:%
-%:%1102=559%:%
-%:%1103=559%:%
-%:%1104=560%:%
-%:%1105=560%:%
-%:%1106=561%:%
-%:%1107=562%:%
-%:%1108=562%:%
-%:%1109=563%:%
-%:%1110=564%:%
-%:%1111=564%:%
-%:%1112=565%:%
-%:%1113=565%:%
-%:%1114=566%:%
-%:%1115=567%:%
-%:%1116=567%:%
-%:%1117=568%:%
-%:%1118=569%:%
-%:%1119=569%:%
-%:%1120=570%:%
-%:%1121=570%:%
-%:%1122=570%:%
-%:%1123=571%:%
-%:%1124=571%:%
-%:%1125=572%:%
-%:%1131=572%:%
-%:%1136=573%:%
-%:%1141=574%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Elimination_Module.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Elimination_Module.tex
deleted file mode 100644
index df53134..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Elimination_Module.tex
+++ /dev/null
@@ -1,596 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Elimination{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Elimination Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Elimination{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Evaluation{\isacharunderscore}{\kern0pt}Function{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the elimination module. It rejects a set of alternatives only if these
-are not all alternatives. The alternatives potentially to be rejected are put
-in a so-called elimination set. These are all alternatives that score below
-a preset threshold value that depends on the specific voting rule.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isacharequal}{\kern0pt}\ nat\isanewline
-\isanewline
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Set\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ elimination{\isacharunderscore}{\kern0pt}set\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}nat\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A\ {\isachardot}{\kern0pt}\ r\ {\isacharparenleft}{\kern0pt}e\ a\ A\ p{\isacharparenright}{\kern0pt}\ t\ {\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ elimination{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ A\isanewline
-\ \ \ \ \ \ \ \ then\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}set\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ else\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Common Eliminators%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}less{\isacharunderscore}{\kern0pt}eliminator\ e\ t\ A\ p\ {\isacharequal}{\kern0pt}\ elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ less{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ Threshold{\isacharunderscore}{\kern0pt}Value\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}leq{\isacharunderscore}{\kern0pt}eliminator\ e\ t\ A\ p\ {\isacharequal}{\kern0pt}\ elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}min{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ leq{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}Min\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}\ x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ average\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Threshold{\isacharunderscore}{\kern0pt}Value{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}average\ e\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ {\isacharparenleft}{\kern0pt}card\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}average\ e\ A\ p{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e\ A\ p\ {\isacharequal}{\kern0pt}\ leq{\isacharunderscore}{\kern0pt}eliminator\ e\ {\isacharparenleft}{\kern0pt}average\ e\ A\ p{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ elim{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}unfold\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ t{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ t{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ t{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ t{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ Max\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ Min\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ Min\ {\isacharbraceleft}{\kern0pt}e\ x\ A\ p\ {\isacharbar}{\kern0pt}x{\isachardot}{\kern0pt}\ x\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isacharless}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ e\ a\ A\ p\ {\isasymle}\ {\isacharparenleft}{\kern0pt}{\isasymSum}x{\isasymin}A{\isachardot}{\kern0pt}\ e\ x\ A\ p{\isacharparenright}{\kern0pt}\ div\ card\ A{\isacharbraceright}{\kern0pt}\ {\isasymunion}\ A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ safe\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Elimination_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%55=21%:%
-%:%65=23%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=25%:%
-%:%70=26%:%
-%:%71=27%:%
-%:%72=27%:%
-%:%74=29%:%
-%:%75=30%:%
-%:%76=31%:%
-%:%77=32%:%
-%:%78=32%:%
-%:%79=33%:%
-%:%80=34%:%
-%:%90=39%:%
-%:%100=41%:%
-%:%101=41%:%
-%:%102=42%:%
-%:%103=43%:%
-%:%104=44%:%
-%:%105=45%:%
-%:%106=45%:%
-%:%107=46%:%
-%:%108=47%:%
-%:%109=48%:%
-%:%110=49%:%
-%:%111=49%:%
-%:%112=50%:%
-%:%113=51%:%
-%:%114=52%:%
-%:%115=53%:%
-%:%116=53%:%
-%:%117=54%:%
-%:%118=55%:%
-%:%119=56%:%
-%:%120=57%:%
-%:%121=57%:%
-%:%122=58%:%
-%:%123=59%:%
-%:%124=60%:%
-%:%125=61%:%
-%:%126=61%:%
-%:%127=62%:%
-%:%128=63%:%
-%:%129=64%:%
-%:%130=65%:%
-%:%131=65%:%
-%:%132=66%:%
-%:%133=67%:%
-%:%140=69%:%
-%:%150=71%:%
-%:%151=71%:%
-%:%158=72%:%
-%:%159=72%:%
-%:%160=73%:%
-%:%161=73%:%
-%:%162=74%:%
-%:%163=75%:%
-%:%164=76%:%
-%:%165=76%:%
-%:%166=77%:%
-%:%167=77%:%
-%:%168=78%:%
-%:%169=78%:%
-%:%170=79%:%
-%:%171=79%:%
-%:%172=80%:%
-%:%178=80%:%
-%:%181=81%:%
-%:%182=82%:%
-%:%183=82%:%
-%:%186=83%:%
-%:%190=83%:%
-%:%191=83%:%
-%:%192=84%:%
-%:%193=84%:%
-%:%194=85%:%
-%:%195=85%:%
-%:%196=86%:%
-%:%197=87%:%
-%:%198=88%:%
-%:%199=88%:%
-%:%200=89%:%
-%:%201=90%:%
-%:%202=91%:%
-%:%203=91%:%
-%:%204=92%:%
-%:%210=92%:%
-%:%213=93%:%
-%:%214=94%:%
-%:%215=94%:%
-%:%218=95%:%
-%:%222=95%:%
-%:%223=95%:%
-%:%224=96%:%
-%:%225=96%:%
-%:%226=97%:%
-%:%227=97%:%
-%:%228=98%:%
-%:%229=99%:%
-%:%230=100%:%
-%:%231=100%:%
-%:%232=101%:%
-%:%233=102%:%
-%:%234=103%:%
-%:%235=103%:%
-%:%236=104%:%
-%:%242=104%:%
-%:%245=105%:%
-%:%246=106%:%
-%:%247=106%:%
-%:%250=107%:%
-%:%254=107%:%
-%:%255=107%:%
-%:%256=108%:%
-%:%257=108%:%
-%:%258=109%:%
-%:%259=109%:%
-%:%260=110%:%
-%:%261=111%:%
-%:%262=112%:%
-%:%263=112%:%
-%:%264=113%:%
-%:%265=114%:%
-%:%266=115%:%
-%:%267=115%:%
-%:%268=116%:%
-%:%274=116%:%
-%:%277=117%:%
-%:%278=118%:%
-%:%279=118%:%
-%:%282=119%:%
-%:%286=119%:%
-%:%287=119%:%
-%:%288=120%:%
-%:%289=120%:%
-%:%290=121%:%
-%:%291=121%:%
-%:%292=122%:%
-%:%293=123%:%
-%:%294=124%:%
-%:%295=124%:%
-%:%296=125%:%
-%:%297=126%:%
-%:%298=127%:%
-%:%299=127%:%
-%:%300=128%:%
-%:%306=128%:%
-%:%309=129%:%
-%:%310=130%:%
-%:%311=130%:%
-%:%314=131%:%
-%:%318=131%:%
-%:%319=131%:%
-%:%320=132%:%
-%:%321=132%:%
-%:%322=133%:%
-%:%323=133%:%
-%:%324=134%:%
-%:%325=135%:%
-%:%326=136%:%
-%:%327=136%:%
-%:%328=137%:%
-%:%329=138%:%
-%:%330=139%:%
-%:%331=139%:%
-%:%332=140%:%
-%:%338=140%:%
-%:%341=141%:%
-%:%342=142%:%
-%:%343=142%:%
-%:%346=143%:%
-%:%350=143%:%
-%:%351=143%:%
-%:%352=144%:%
-%:%353=144%:%
-%:%354=145%:%
-%:%355=145%:%
-%:%356=146%:%
-%:%357=147%:%
-%:%358=148%:%
-%:%359=148%:%
-%:%360=149%:%
-%:%361=150%:%
-%:%362=151%:%
-%:%363=151%:%
-%:%364=152%:%
-%:%370=152%:%
-%:%375=153%:%
-%:%380=154%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Evaluation_Function.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Evaluation_Function.tex
deleted file mode 100644
index 6f4dfc7..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Evaluation_Function.tex
+++ /dev/null
@@ -1,97 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Evaluation{\isacharunderscore}{\kern0pt}Function}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Evaluation Function%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Evaluation{\isacharunderscore}{\kern0pt}Function\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Profile{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the evaluation function. From a set of currently eligible alternatives,
-the evaluation function computes a numerical value that is then to be used for
-further (s)election, e.g., by the elimination module.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Evaluation{\isacharunderscore}{\kern0pt}Function\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ \ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Evaluation_Function.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%41=14%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%52=18%:%
-%:%62=20%:%
-%:%63=20%:%
-%:%66=21%:%
-%:%71=22%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Homogeneity.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Homogeneity.tex
deleted file mode 100644
index a0e17c0..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Homogeneity.tex
+++ /dev/null
@@ -1,69 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Homogeneity}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Homogeneity\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\ \isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ times\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}times\ n\ l\ {\isacharequal}{\kern0pt}\ concat\ {\isacharparenleft}{\kern0pt}replicate\ n\ l{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ homogeneity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-{\isachardoublequoteopen}homogeneity\ m\ {\isasymequiv}\isanewline
-\ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}\ A\ p\ n\ {\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ n\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ {\isacharparenleft}{\kern0pt}times\ n\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Properties/Homogeneity.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%19=4%:%
-%:%22=5%:%
-%:%23=6%:%
-%:%24=6%:%
-%:%25=7%:%
-%:%26=8%:%
-%:%27=9%:%
-%:%28=9%:%
-%:%29=10%:%
-%:%33=14%:%
-%:%36=15%:%
-%:%41=16%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Indep_Of_Alt.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Indep_Of_Alt.tex
deleted file mode 100644
index 62775c7..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Indep_Of_Alt.tex
+++ /dev/null
@@ -1,60 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Indep{\isacharunderscore}{\kern0pt}Of{\isacharunderscore}{\kern0pt}Alt}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Indep{\isacharunderscore}{\kern0pt}Of{\isacharunderscore}{\kern0pt}Alt\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ A\ a\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ q{\isachardot}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ q\ a\ {\isasymlongrightarrow}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Properties/Indep_Of_Alt.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%19=4%:%
-%:%22=5%:%
-%:%23=9%:%
-%:%24=10%:%
-%:%25=10%:%
-%:%26=11%:%
-%:%27=12%:%
-%:%30=13%:%
-%:%35=14%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Loop_Composition.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Loop_Composition.tex
deleted file mode 100644
index 72bdd03..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Loop_Composition.tex
+++ /dev/null
@@ -1,806 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Loop{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Loop Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Loop{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Sequential{\isacharunderscore}{\kern0pt}Composition\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The loop composition uses the same module in sequence,
-combined with a termination condition, until either
-  (1) the termination condition is met or
-  (2) no new decisions are made (i.e., a fixed point is reached).%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}termination{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}term{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}inf{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymin}\isanewline
-\ \ \ \ \ \ \ \ measure\ {\isacharparenleft}{\kern0pt}{\isasymlambda}{\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubset}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ acc\ A\ p{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubset}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ P\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ bool\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymlbrakk}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymlbrakk}{\isasymnot}\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}f\ A\ p\ rs\ fa{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fa{\isacharcomma}{\kern0pt}\ f{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isacharunderscore}{\kern0pt}cases{\isadigit{5}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ P\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{2}}\ a{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m\ ta\ acca\ Aa\ pa\ ma{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ acc\ A\ p\ {\isacharequal}{\kern0pt}\ acca\ Aa\ pa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m\ ta\ acca\ Aa\ pa\ ma{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ acc\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ t\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ acc\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ m\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ ta\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ acca\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ Aa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ pa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ ma\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{2}}\ a{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}acc\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymAnd}t\ acc\ A\ p\ m\ ta\ acca\ Aa\ pa\ ma{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}ta\ {\isacharparenleft}{\kern0pt}acca\ Aa\ pa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharparenright}{\kern0pt}\ Aa\ pa\ {\isasymsubset}\ defer\ acca\ Aa\ pa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acca\ Aa\ pa{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}acc{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}acca{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharcomma}{\kern0pt}\ m{\isacharcomma}{\kern0pt}\ t{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}sumC\ {\isacharparenleft}{\kern0pt}acca\ {\isasymtriangleright}\ ma{\isacharcomma}{\kern0pt}\ ma{\isacharcomma}{\kern0pt}\ ta{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ pa{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}r{\isachardot}{\kern0pt}\ wf\ r\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ f\ A\ rs\ fa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ p\ {\isacharparenleft}{\kern0pt}f\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharparenright}{\kern0pt}\ A\ rs\ {\isasymsubset}\ defer\ f\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharparenleft}{\kern0pt}f{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}termination{\isacharunderscore}{\kern0pt}helper\ wf{\isacharunderscore}{\kern0pt}measure\ {\isachardoublequoteopen}termination{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}r\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ Ex\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymlambda}ra{\isachardot}{\kern0pt}\ {\isasymforall}f\ A\ rs\ pa\ fa{\isachardot}{\kern0pt}\ {\isasymexists}ra\ pb\ rb\ pc\ pd\ fb\ Aa\ rsa\ fc\ pe{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ wf\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}p{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ f\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ pa\ {\isacharparenleft}{\kern0pt}f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ wf\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}ra{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ set{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\ {\isacharparenleft}{\kern0pt}pb{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ wf\ rb\ {\isasymand}\ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharparenright}{\kern0pt}\ A\ rs\ {\isasymsubset}\ defer\ f\ A\ rs\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}pc{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ pa{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ f{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ pa{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymin}\ rb\ {\isasymand}\ wf\ rb\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}dom\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}pd{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ Profile{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}defer\ fb\ {\isacharparenleft}{\kern0pt}Aa{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ rsa{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}fb\ {\isasymtriangleright}\ fc{\isacharparenright}{\kern0pt}\ Aa\ rsa\ {\isasymsubset}\ defer\ fb\ Aa\ rsa\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ pe\ {\isacharparenleft}{\kern0pt}fb\ Aa\ rsa{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}fb\ {\isasymtriangleright}\ fc{\isacharcomma}{\kern0pt}\ fc{\isacharcomma}{\kern0pt}\ pe{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ rsa{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ fb{\isacharcomma}{\kern0pt}\ fc{\isacharcomma}{\kern0pt}\ pe{\isacharcomma}{\kern0pt}\ Aa{\isacharcomma}{\kern0pt}\ rsa{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isacharparenright}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ set\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ rr\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymtimes}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition{\isacharparenright}{\kern0pt}\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ set\ {\isasymtimes}\ {\isacharprime}{\kern0pt}a\ Profile{\isacharparenright}{\kern0pt}\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}wf\ rr\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p\ f\ A\ rs\ fa{\isachardot}{\kern0pt}\ p\ {\isacharparenleft}{\kern0pt}f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ defer\ {\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharparenright}{\kern0pt}\ A\ rs\ {\isasymsubset}\ defer\ f\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}f\ {\isasymtriangleright}\ fa{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ f{\isacharcomma}{\kern0pt}\ fa{\isacharcomma}{\kern0pt}\ p{\isacharcomma}{\kern0pt}\ A{\isacharcomma}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymin}\ rr{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}termination{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}helper{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubset}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ then\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}\ else\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{function}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ loop{\isacharunderscore}{\kern0pt}composition\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}module\ A\ p{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ loop{\isacharunderscore}{\kern0pt}composition\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ m\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}fastforce{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ \ {\isachardoublequoteopen}termination{\isachardoublequoteclose}\ wf{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ loop\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymcirclearrowleft}\isactrlsub t\ {\isasymequiv}\ loop{\isacharunderscore}{\kern0pt}composition\ m\ t{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}loop{\isacharunderscore}{\kern0pt}composition\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ then\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}module\ A\ p{\isacharparenright}{\kern0pt}\ else\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ m\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ {\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m\ profile\isanewline
-\ \ \ \ \ \ \ \ \ \ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ m{\isacharunderscore}{\kern0pt}module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}partit\ m{\isacharunderscore}{\kern0pt}module\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}no{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}incr{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}trans\ eq{\isacharunderscore}{\kern0pt}iff\ less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ module{\isacharunderscore}{\kern0pt}m\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Compositional_Structures/Loop_Composition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%35=14%:%
-%:%36=15%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%48=21%:%
-%:%57=23%:%
-%:%67=25%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=28%:%
-%:%72=29%:%
-%:%73=30%:%
-%:%74=31%:%
-%:%75=32%:%
-%:%78=33%:%
-%:%82=33%:%
-%:%83=33%:%
-%:%84=34%:%
-%:%85=34%:%
-%:%90=34%:%
-%:%93=35%:%
-%:%94=39%:%
-%:%95=40%:%
-%:%96=40%:%
-%:%97=41%:%
-%:%98=42%:%
-%:%99=43%:%
-%:%101=45%:%
-%:%102=46%:%
-%:%104=48%:%
-%:%111=49%:%
-%:%112=49%:%
-%:%113=50%:%
-%:%114=50%:%
-%:%115=51%:%
-%:%116=52%:%
-%:%117=53%:%
-%:%118=54%:%
-%:%119=54%:%
-%:%120=55%:%
-%:%123=58%:%
-%:%124=59%:%
-%:%127=62%:%
-%:%128=63%:%
-%:%129=63%:%
-%:%130=64%:%
-%:%131=64%:%
-%:%132=65%:%
-%:%133=65%:%
-%:%134=66%:%
-%:%135=66%:%
-%:%136=66%:%
-%:%137=67%:%
-%:%138=67%:%
-%:%139=68%:%
-%:%140=68%:%
-%:%141=69%:%
-%:%142=69%:%
-%:%143=70%:%
-%:%144=70%:%
-%:%145=71%:%
-%:%151=77%:%
-%:%152=78%:%
-%:%153=78%:%
-%:%154=79%:%
-%:%155=79%:%
-%:%156=80%:%
-%:%157=80%:%
-%:%158=81%:%
-%:%164=87%:%
-%:%165=88%:%
-%:%166=88%:%
-%:%167=89%:%
-%:%168=89%:%
-%:%169=90%:%
-%:%170=91%:%
-%:%171=92%:%
-%:%172=93%:%
-%:%173=94%:%
-%:%174=95%:%
-%:%175=96%:%
-%:%176=97%:%
-%:%177=98%:%
-%:%178=99%:%
-%:%179=100%:%
-%:%180=100%:%
-%:%181=101%:%
-%:%182=102%:%
-%:%183=103%:%
-%:%184=104%:%
-%:%185=105%:%
-%:%186=106%:%
-%:%187=106%:%
-%:%188=107%:%
-%:%189=107%:%
-%:%190=108%:%
-%:%191=108%:%
-%:%192=109%:%
-%:%193=109%:%
-%:%194=110%:%
-%:%195=110%:%
-%:%196=111%:%
-%:%197=111%:%
-%:%198=112%:%
-%:%199=112%:%
-%:%200=113%:%
-%:%201=113%:%
-%:%202=114%:%
-%:%209=121%:%
-%:%210=122%:%
-%:%211=122%:%
-%:%212=123%:%
-%:%218=123%:%
-%:%221=124%:%
-%:%222=124%:%
-%:%229=125%:%
-%:%230=125%:%
-%:%231=126%:%
-%:%232=126%:%
-%:%233=127%:%
-%:%238=132%:%
-%:%239=133%:%
-%:%240=133%:%
-%:%241=134%:%
-%:%242=134%:%
-%:%243=135%:%
-%:%244=135%:%
-%:%245=136%:%
-%:%275=166%:%
-%:%276=167%:%
-%:%277=167%:%
-%:%278=168%:%
-%:%279=168%:%
-%:%280=169%:%
-%:%283=172%:%
-%:%284=173%:%
-%:%288=177%:%
-%:%289=178%:%
-%:%290=178%:%
-%:%291=179%:%
-%:%292=179%:%
-%:%293=180%:%
-%:%294=180%:%
-%:%295=181%:%
-%:%296=181%:%
-%:%297=182%:%
-%:%298=182%:%
-%:%299=183%:%
-%:%305=183%:%
-%:%308=184%:%
-%:%309=185%:%
-%:%310=185%:%
-%:%311=186%:%
-%:%314=189%:%
-%:%317=190%:%
-%:%321=190%:%
-%:%322=190%:%
-%:%327=190%:%
-%:%330=191%:%
-%:%331=192%:%
-%:%332=192%:%
-%:%333=193%:%
-%:%334=194%:%
-%:%335=195%:%
-%:%336=196%:%
-%:%337=197%:%
-%:%338=198%:%
-%:%341=199%:%
-%:%345=199%:%
-%:%346=199%:%
-%:%351=199%:%
-%:%354=200%:%
-%:%355=200%:%
-%:%358=201%:%
-%:%362=201%:%
-%:%363=201%:%
-%:%364=202%:%
-%:%365=202%:%
-%:%370=202%:%
-%:%373=203%:%
-%:%374=204%:%
-%:%375=204%:%
-%:%376=205%:%
-%:%377=206%:%
-%:%378=207%:%
-%:%379=208%:%
-%:%380=209%:%
-%:%381=209%:%
-%:%382=210%:%
-%:%384=212%:%
-%:%387=213%:%
-%:%391=213%:%
-%:%392=213%:%
-%:%397=213%:%
-%:%400=214%:%
-%:%401=215%:%
-%:%402=215%:%
-%:%403=216%:%
-%:%404=217%:%
-%:%405=218%:%
-%:%406=219%:%
-%:%407=220%:%
-%:%408=221%:%
-%:%415=222%:%
-%:%416=222%:%
-%:%417=223%:%
-%:%418=223%:%
-%:%419=224%:%
-%:%420=224%:%
-%:%421=225%:%
-%:%422=225%:%
-%:%423=226%:%
-%:%424=227%:%
-%:%425=228%:%
-%:%426=228%:%
-%:%427=229%:%
-%:%442=231%:%
-%:%452=233%:%
-%:%453=233%:%
-%:%454=234%:%
-%:%455=235%:%
-%:%458=236%:%
-%:%462=236%:%
-%:%463=236%:%
-%:%464=237%:%
-%:%465=238%:%
-%:%466=238%:%
-%:%471=238%:%
-%:%474=239%:%
-%:%475=240%:%
-%:%476=240%:%
-%:%477=241%:%
-%:%478=242%:%
-%:%479=243%:%
-%:%480=244%:%
-%:%481=245%:%
-%:%482=246%:%
-%:%489=247%:%
-%:%490=247%:%
-%:%491=248%:%
-%:%492=248%:%
-%:%493=249%:%
-%:%494=249%:%
-%:%495=250%:%
-%:%496=250%:%
-%:%497=251%:%
-%:%498=252%:%
-%:%499=253%:%
-%:%500=253%:%
-%:%501=254%:%
-%:%516=256%:%
-%:%532=260%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Maximum_Aggregator.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Maximum_Aggregator.tex
deleted file mode 100644
index cd1cf1e..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Maximum_Aggregator.tex
+++ /dev/null
@@ -1,343 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Maximum{\isacharunderscore}{\kern0pt}Aggregator}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Maximum Aggregator%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Maximum{\isacharunderscore}{\kern0pt}Aggregator\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The max(imum) aggregator takes two partitions of an alternative set A as
-input. It returns a partition where every alternative receives the maximum
-result of the two input partitions.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}aggregator\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Aggregator{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharparenleft}{\kern0pt}d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemma%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}aggregator\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}{\isacharcomma}{\kern0pt}\ r{\isadigit{1}}{\isacharcomma}{\kern0pt}\ d{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isadigit{2}}{\isacharcomma}{\kern0pt}\ r{\isadigit{2}}{\isacharcomma}{\kern0pt}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isadigit{1}}\ {\isasyminter}\ r{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}l\ {\isasymin}\ A{\isachardot}{\kern0pt}\ l\ {\isasymnotin}\ e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e{\isadigit{1}}\ {\isasymunion}\ e{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ set{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}aggregator\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ aggregator{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}e{\isadigit{2}}\ {\isasymunion}\ r{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ e{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ r{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ asm{\isadigit{3}}\ asm{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ e{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ d{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{1}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ r{\isadigit{2}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}e{\isadigit{2}}\ {\isasymunion}\ r{\isadigit{2}}\ {\isasymunion}\ d{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ e{\isadigit{1}}\ {\isasymunion}\ r{\isadigit{1}}\ {\isasymunion}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ d{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnotin}\ r{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ d{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ asm{\isadigit{3}}\ asm{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Maximum_Aggregator.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%53=19%:%
-%:%63=21%:%
-%:%64=21%:%
-%:%65=22%:%
-%:%75=27%:%
-%:%85=29%:%
-%:%86=29%:%
-%:%88=31%:%
-%:%95=32%:%
-%:%96=32%:%
-%:%97=33%:%
-%:%98=33%:%
-%:%99=34%:%
-%:%100=34%:%
-%:%101=35%:%
-%:%102=35%:%
-%:%103=35%:%
-%:%104=36%:%
-%:%105=37%:%
-%:%106=37%:%
-%:%107=38%:%
-%:%108=38%:%
-%:%109=38%:%
-%:%110=39%:%
-%:%111=40%:%
-%:%112=41%:%
-%:%113=41%:%
-%:%114=42%:%
-%:%115=42%:%
-%:%116=42%:%
-%:%117=43%:%
-%:%118=44%:%
-%:%119=44%:%
-%:%120=45%:%
-%:%121=45%:%
-%:%122=45%:%
-%:%123=46%:%
-%:%124=46%:%
-%:%125=47%:%
-%:%140=49%:%
-%:%150=51%:%
-%:%151=51%:%
-%:%154=52%:%
-%:%158=52%:%
-%:%159=52%:%
-%:%160=53%:%
-%:%161=53%:%
-%:%162=54%:%
-%:%163=54%:%
-%:%164=55%:%
-%:%165=56%:%
-%:%166=57%:%
-%:%167=58%:%
-%:%168=59%:%
-%:%169=60%:%
-%:%170=61%:%
-%:%171=62%:%
-%:%172=63%:%
-%:%173=63%:%
-%:%174=64%:%
-%:%175=65%:%
-%:%176=66%:%
-%:%177=67%:%
-%:%178=68%:%
-%:%179=68%:%
-%:%180=69%:%
-%:%181=69%:%
-%:%182=70%:%
-%:%183=70%:%
-%:%184=71%:%
-%:%185=71%:%
-%:%186=72%:%
-%:%187=72%:%
-%:%188=73%:%
-%:%189=74%:%
-%:%190=75%:%
-%:%191=76%:%
-%:%192=77%:%
-%:%193=78%:%
-%:%194=79%:%
-%:%195=80%:%
-%:%196=81%:%
-%:%197=81%:%
-%:%198=82%:%
-%:%199=83%:%
-%:%200=84%:%
-%:%201=85%:%
-%:%202=86%:%
-%:%203=86%:%
-%:%204=87%:%
-%:%205=87%:%
-%:%206=88%:%
-%:%207=88%:%
-%:%208=89%:%
-%:%214=89%:%
-%:%219=90%:%
-%:%224=91%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Minimax_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Minimax_Rule.tex
deleted file mode 100644
index 75dc3d2..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Minimax_Rule.tex
+++ /dev/null
@@ -1,160 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Minimax{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Minimax Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Minimax{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Facts{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Minimax voting rule. It elects the alternatives with the highest
-Minimax score.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}minimax{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ minimax\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}condorcet{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ minimax{\isacharunderscore}{\kern0pt}rule{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ minimax{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ minimax{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ minimax{\isacharunderscore}{\kern0pt}rule{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Voting_Rules/Minimax_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%35=14%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%54=20%:%
-%:%64=22%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=26%:%
-%:%77=27%:%
-%:%78=27%:%
-%:%79=28%:%
-%:%80=28%:%
-%:%81=29%:%
-%:%82=30%:%
-%:%83=30%:%
-%:%84=31%:%
-%:%85=31%:%
-%:%86=32%:%
-%:%87=32%:%
-%:%88=33%:%
-%:%89=33%:%
-%:%90=34%:%
-%:%91=35%:%
-%:%92=35%:%
-%:%93=36%:%
-%:%99=36%:%
-%:%104=37%:%
-%:%109=38%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Facts.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Facts.tex
deleted file mode 100644
index 7f79b35..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Facts.tex
+++ /dev/null
@@ -1,923 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Monotonicity{\isacharunderscore}{\kern0pt}Facts}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Monotonicity{\isacharunderscore}{\kern0pt}Facts\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Monotonicity{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Drop{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Plurality{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ defer{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ order\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}dl{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ order\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ allI\ impI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}winner{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}subset{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ sizes{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}length\ p\ {\isacharequal}{\kern0pt}\ length\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffE\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ asm{\isadigit{1}}\ insertCI\ insert{\isacharunderscore}{\kern0pt}absorb\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ profile{\isacharunderscore}{\kern0pt}def\ sizes\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}winner\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}QtoP{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isadigit{3}}\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Collect{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}other{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymge}\ win{\isacharunderscore}{\kern0pt}count\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}mono\ sizes{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ sizes{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}pq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymlongleftrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ this\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ asm{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ profile{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}PtoQ{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}pq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ q\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}cong\ DiffI\ above{\isacharunderscore}{\kern0pt}pq\ above{\isacharunderscore}{\kern0pt}QtoP\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}absorb\ insert{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\ sizes\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isasymnoteq}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}less{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharless}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ strict{\isacharunderscore}{\kern0pt}less\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}other\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ less{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isacharless}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ antisym\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ y\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ x\ {\isasymnotin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasymnotin}\ elect\ plurality\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ elect\ plurality\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ less\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ q\ a\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}win{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ q\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ q\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ q\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}invariant{\isacharunderscore}{\kern0pt}monotonicity\ plurality{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ plurality\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ plurality\ A\ p\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Composition_Rules/Monotonicity_Facts.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%16=6%:%
-%:%17=7%:%
-%:%18=8%:%
-%:%23=8%:%
-%:%26=9%:%
-%:%27=10%:%
-%:%28=10%:%
-%:%31=11%:%
-%:%35=11%:%
-%:%36=11%:%
-%:%37=12%:%
-%:%38=12%:%
-%:%43=12%:%
-%:%46=13%:%
-%:%47=14%:%
-%:%48=15%:%
-%:%49=15%:%
-%:%50=16%:%
-%:%51=17%:%
-%:%54=18%:%
-%:%58=18%:%
-%:%59=18%:%
-%:%64=18%:%
-%:%67=19%:%
-%:%68=20%:%
-%:%69=21%:%
-%:%70=21%:%
-%:%71=22%:%
-%:%72=23%:%
-%:%75=24%:%
-%:%79=24%:%
-%:%80=24%:%
-%:%85=24%:%
-%:%88=25%:%
-%:%89=26%:%
-%:%90=26%:%
-%:%93=29%:%
-%:%100=30%:%
-%:%101=30%:%
-%:%102=31%:%
-%:%103=31%:%
-%:%104=32%:%
-%:%105=33%:%
-%:%106=34%:%
-%:%107=35%:%
-%:%108=36%:%
-%:%109=36%:%
-%:%110=37%:%
-%:%111=38%:%
-%:%112=39%:%
-%:%113=39%:%
-%:%114=40%:%
-%:%115=41%:%
-%:%116=42%:%
-%:%117=42%:%
-%:%118=43%:%
-%:%119=43%:%
-%:%120=44%:%
-%:%123=47%:%
-%:%124=48%:%
-%:%125=48%:%
-%:%126=49%:%
-%:%127=49%:%
-%:%128=50%:%
-%:%129=50%:%
-%:%130=51%:%
-%:%131=52%:%
-%:%132=53%:%
-%:%133=53%:%
-%:%134=54%:%
-%:%135=54%:%
-%:%136=55%:%
-%:%137=55%:%
-%:%138=56%:%
-%:%139=57%:%
-%:%140=58%:%
-%:%141=58%:%
-%:%142=59%:%
-%:%143=59%:%
-%:%144=59%:%
-%:%145=60%:%
-%:%146=61%:%
-%:%147=61%:%
-%:%148=62%:%
-%:%149=62%:%
-%:%150=63%:%
-%:%151=63%:%
-%:%152=63%:%
-%:%153=64%:%
-%:%154=65%:%
-%:%155=65%:%
-%:%156=66%:%
-%:%157=66%:%
-%:%158=67%:%
-%:%159=68%:%
-%:%160=68%:%
-%:%161=69%:%
-%:%162=69%:%
-%:%163=70%:%
-%:%164=70%:%
-%:%165=71%:%
-%:%167=73%:%
-%:%168=74%:%
-%:%169=74%:%
-%:%170=75%:%
-%:%171=76%:%
-%:%172=77%:%
-%:%173=77%:%
-%:%174=78%:%
-%:%175=78%:%
-%:%176=78%:%
-%:%177=79%:%
-%:%179=81%:%
-%:%180=82%:%
-%:%181=82%:%
-%:%182=83%:%
-%:%183=84%:%
-%:%184=84%:%
-%:%185=85%:%
-%:%186=85%:%
-%:%187=86%:%
-%:%189=88%:%
-%:%190=89%:%
-%:%191=89%:%
-%:%192=90%:%
-%:%193=90%:%
-%:%194=91%:%
-%:%195=92%:%
-%:%196=92%:%
-%:%197=93%:%
-%:%198=93%:%
-%:%199=94%:%
-%:%200=95%:%
-%:%201=96%:%
-%:%202=96%:%
-%:%203=97%:%
-%:%204=97%:%
-%:%205=98%:%
-%:%206=98%:%
-%:%207=99%:%
-%:%208=100%:%
-%:%209=101%:%
-%:%210=101%:%
-%:%211=102%:%
-%:%212=102%:%
-%:%213=102%:%
-%:%214=103%:%
-%:%215=104%:%
-%:%216=104%:%
-%:%217=105%:%
-%:%218=105%:%
-%:%219=105%:%
-%:%220=106%:%
-%:%221=107%:%
-%:%222=108%:%
-%:%223=108%:%
-%:%224=109%:%
-%:%225=109%:%
-%:%226=110%:%
-%:%227=110%:%
-%:%228=111%:%
-%:%229=112%:%
-%:%230=113%:%
-%:%231=113%:%
-%:%232=114%:%
-%:%233=114%:%
-%:%234=114%:%
-%:%235=115%:%
-%:%238=118%:%
-%:%239=119%:%
-%:%240=119%:%
-%:%241=120%:%
-%:%242=120%:%
-%:%243=121%:%
-%:%244=121%:%
-%:%245=121%:%
-%:%246=122%:%
-%:%248=124%:%
-%:%249=125%:%
-%:%250=125%:%
-%:%251=126%:%
-%:%252=126%:%
-%:%253=127%:%
-%:%255=129%:%
-%:%256=130%:%
-%:%257=130%:%
-%:%258=131%:%
-%:%260=133%:%
-%:%261=134%:%
-%:%262=134%:%
-%:%263=135%:%
-%:%264=135%:%
-%:%265=135%:%
-%:%266=135%:%
-%:%267=136%:%
-%:%268=137%:%
-%:%269=137%:%
-%:%270=138%:%
-%:%271=138%:%
-%:%272=139%:%
-%:%273=139%:%
-%:%274=139%:%
-%:%275=139%:%
-%:%276=140%:%
-%:%277=141%:%
-%:%278=141%:%
-%:%279=142%:%
-%:%280=142%:%
-%:%281=142%:%
-%:%282=143%:%
-%:%283=144%:%
-%:%284=144%:%
-%:%285=145%:%
-%:%286=145%:%
-%:%287=146%:%
-%:%288=146%:%
-%:%289=147%:%
-%:%290=147%:%
-%:%291=148%:%
-%:%292=148%:%
-%:%293=149%:%
-%:%294=149%:%
-%:%295=149%:%
-%:%296=150%:%
-%:%298=152%:%
-%:%299=153%:%
-%:%300=153%:%
-%:%301=154%:%
-%:%302=154%:%
-%:%303=155%:%
-%:%305=157%:%
-%:%306=158%:%
-%:%307=158%:%
-%:%308=159%:%
-%:%309=160%:%
-%:%310=160%:%
-%:%311=161%:%
-%:%312=161%:%
-%:%313=162%:%
-%:%314=162%:%
-%:%315=163%:%
-%:%316=163%:%
-%:%317=164%:%
-%:%318=165%:%
-%:%319=166%:%
-%:%320=166%:%
-%:%321=167%:%
-%:%322=167%:%
-%:%323=168%:%
-%:%324=168%:%
-%:%325=169%:%
-%:%326=169%:%
-%:%327=170%:%
-%:%328=170%:%
-%:%329=171%:%
-%:%330=171%:%
-%:%331=172%:%
-%:%332=173%:%
-%:%333=173%:%
-%:%334=174%:%
-%:%335=174%:%
-%:%336=175%:%
-%:%337=175%:%
-%:%338=176%:%
-%:%339=177%:%
-%:%340=177%:%
-%:%341=178%:%
-%:%342=178%:%
-%:%343=179%:%
-%:%344=179%:%
-%:%345=180%:%
-%:%346=180%:%
-%:%347=181%:%
-%:%348=181%:%
-%:%349=182%:%
-%:%350=182%:%
-%:%351=183%:%
-%:%352=184%:%
-%:%353=184%:%
-%:%354=185%:%
-%:%355=186%:%
-%:%356=186%:%
-%:%357=187%:%
-%:%358=187%:%
-%:%359=188%:%
-%:%360=189%:%
-%:%361=189%:%
-%:%362=190%:%
-%:%363=190%:%
-%:%364=191%:%
-%:%365=191%:%
-%:%366=192%:%
-%:%367=193%:%
-%:%368=194%:%
-%:%369=194%:%
-%:%370=195%:%
-%:%371=195%:%
-%:%372=196%:%
-%:%373=197%:%
-%:%374=197%:%
-%:%375=198%:%
-%:%376=198%:%
-%:%377=198%:%
-%:%378=199%:%
-%:%379=200%:%
-%:%380=200%:%
-%:%381=201%:%
-%:%382=201%:%
-%:%383=202%:%
-%:%384=202%:%
-%:%385=203%:%
-%:%386=204%:%
-%:%387=204%:%
-%:%388=205%:%
-%:%389=205%:%
-%:%390=206%:%
-%:%391=206%:%
-%:%392=206%:%
-%:%393=207%:%
-%:%394=208%:%
-%:%395=208%:%
-%:%396=209%:%
-%:%397=209%:%
-%:%398=209%:%
-%:%399=210%:%
-%:%400=211%:%
-%:%401=211%:%
-%:%402=212%:%
-%:%403=212%:%
-%:%404=212%:%
-%:%405=213%:%
-%:%406=214%:%
-%:%407=214%:%
-%:%408=215%:%
-%:%409=215%:%
-%:%410=216%:%
-%:%411=216%:%
-%:%412=216%:%
-%:%413=217%:%
-%:%414=218%:%
-%:%415=219%:%
-%:%416=219%:%
-%:%417=220%:%
-%:%418=220%:%
-%:%419=221%:%
-%:%420=221%:%
-%:%421=222%:%
-%:%422=222%:%
-%:%423=223%:%
-%:%424=223%:%
-%:%425=223%:%
-%:%426=224%:%
-%:%427=225%:%
-%:%428=225%:%
-%:%429=226%:%
-%:%430=226%:%
-%:%431=226%:%
-%:%432=227%:%
-%:%433=227%:%
-%:%434=228%:%
-%:%435=228%:%
-%:%436=229%:%
-%:%437=229%:%
-%:%438=230%:%
-%:%444=230%:%
-%:%447=231%:%
-%:%448=232%:%
-%:%449=233%:%
-%:%450=233%:%
-%:%457=234%:%
-%:%458=234%:%
-%:%459=235%:%
-%:%460=235%:%
-%:%461=236%:%
-%:%465=240%:%
-%:%466=241%:%
-%:%467=241%:%
-%:%468=242%:%
-%:%469=242%:%
-%:%470=243%:%
-%:%471=243%:%
-%:%472=244%:%
-%:%473=244%:%
-%:%474=245%:%
-%:%475=245%:%
-%:%476=246%:%
-%:%478=248%:%
-%:%479=249%:%
-%:%480=249%:%
-%:%481=250%:%
-%:%482=250%:%
-%:%483=251%:%
-%:%484=251%:%
-%:%485=252%:%
-%:%486=252%:%
-%:%487=253%:%
-%:%488=253%:%
-%:%489=254%:%
-%:%495=254%:%
-%:%500=255%:%
-%:%505=256%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Properties.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Properties.tex
deleted file mode 100644
index d3eb734..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Properties.tex
+++ /dev/null
@@ -1,108 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Monotonicity{\isacharunderscore}{\kern0pt}Properties}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Monotonicity{\isacharunderscore}{\kern0pt}Properties\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Result{\isacharunderscore}{\kern0pt}Properties\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ elect\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ q\ {\isacharequal}{\kern0pt}\ elect\ m\ A\ p\ {\isasymor}\ elect\ m\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p\ {\isasymor}\ defer\ m\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite\ A\ {\isasymand}\ w\ {\isasymin}\ defer\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ w\ {\isasymin}\ defer\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Properties/Monotonicity_Properties.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%20=5%:%
-%:%23=6%:%
-%:%24=10%:%
-%:%25=11%:%
-%:%26=11%:%
-%:%27=12%:%
-%:%30=15%:%
-%:%31=16%:%
-%:%32=21%:%
-%:%33=22%:%
-%:%34=22%:%
-%:%35=23%:%
-%:%38=26%:%
-%:%39=27%:%
-%:%40=32%:%
-%:%41=33%:%
-%:%42=33%:%
-%:%43=34%:%
-%:%46=37%:%
-%:%47=38%:%
-%:%48=42%:%
-%:%49=43%:%
-%:%50=43%:%
-%:%51=44%:%
-%:%54=47%:%
-%:%55=48%:%
-%:%58=49%:%
-%:%63=50%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Rules.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Rules.tex
deleted file mode 100644
index b5078d4..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Monotonicity_Rules.tex
+++ /dev/null
@@ -1,2710 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Monotonicity{\isacharunderscore}{\kern0pt}Rules}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Monotonicity{\isacharunderscore}{\kern0pt}Rules\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Monotonicity{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Properties{\isacharslash}{\kern0pt}Weak{\isacharunderscore}{\kern0pt}Monotonicity{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Parallel{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Sequential{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Maximum{\isacharunderscore}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Result{\isacharunderscore}{\kern0pt}Rules\isanewline
-\ \ \ \ \ \ \ \ \ \ Monotonicity{\isacharunderscore}{\kern0pt}Facts\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ lifted{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Profile{\isachardot}{\kern0pt}lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}unchanged{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ q\ {\isasymnoteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}single{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ strong{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mon{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ q\ {\isasymlongrightarrow}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\ \isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ a{\isacharunderscore}{\kern0pt}single{\isacharunderscore}{\kern0pt}defer\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ empty{\isacharunderscore}{\kern0pt}iff\ finite{\isachardot}{\kern0pt}emptyI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\ order{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\ \isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}mem{\isacharunderscore}{\kern0pt}eq\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ empty{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ insertCI\ subset{\isacharunderscore}{\kern0pt}singletonD\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ defers{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}subset\ Diff{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}absorb\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ card{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}eq\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}singleton\ finite{\isachardot}{\kern0pt}emptyI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ finite{\isacharunderscore}{\kern0pt}Diff\ insertCI\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\ \isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ calculation\ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}different{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}defer\ m\ A\ q\ {\isasymnoteq}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}elect\ m\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}equal{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ elect\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}different{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}a\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}profile{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ results{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ same{\isacharunderscore}{\kern0pt}alternatives{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ results{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}equal\ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ same{\isacharunderscore}{\kern0pt}profile\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ still{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}lifted\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ equals{\isadigit{0}}D\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ defer{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}p\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}subset\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ still{\isacharunderscore}{\kern0pt}lifted\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}n\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}p\ same{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}deferred{\isacharunderscore}{\kern0pt}q\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ same{\isacharunderscore}{\kern0pt}alternatives\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ q{\isacharparenright}{\kern0pt}\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}q\ def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}p\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}p\ finite{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ compatible{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ S\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Profile{\isachardot}{\kern0pt}lifted\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}profs{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ compatible\ \isacommand{obtain}\isamarkupfalse%
-\ A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\ A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ m\ S\ x\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ x\ {\isasymin}\ reject\ m\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt\ n\ S\ x\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ S\ p\ {\isasymlongrightarrow}\ x\ {\isasymin}\ reject\ n\ S\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ m\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}x\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ n\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}profs\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}n\ lifted{\isacharunderscore}{\kern0pt}x\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ monotone{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{2}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ lifted{\isacharunderscore}{\kern0pt}x\ a{\isadigit{0}}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ f{\isacharunderscore}{\kern0pt}profs\ IntI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ empty{\isacharunderscore}{\kern0pt}iff\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffI\ lifted{\isacharunderscore}{\kern0pt}x\ compatible\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ reject\ n\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}x\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ defer\ m\ S\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ DiffD{\isadigit{2}}\ compatible\ dcompat{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}by{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ maximum{\isacharunderscore}{\kern0pt}parallel{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ n\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{4}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ lifted{\isacharunderscore}{\kern0pt}x\ a{\isadigit{1}}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ indep{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}alt{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{3}}\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ m\ S\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{2}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}profs\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}x\ defer{\isacharunderscore}{\kern0pt}n\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}profs\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ m\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}rej{\isadigit{1}}\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}profs\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S{\isacharminus}{\kern0pt}A{\isachardot}{\kern0pt}\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ mod{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}contains{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ S\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ compatible\ disjoint{\isacharunderscore}{\kern0pt}compatibility{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}profs\isanewline
-\ \ \ \ \ \ \ \ \ \ eq{\isacharunderscore}{\kern0pt}alts{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}profs{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}results\ max{\isacharunderscore}{\kern0pt}par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}help{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq\ q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ modules{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ profile{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ p\ q\ a\ {\isasymlongrightarrow}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ mono{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}m\ A\ p\ {\isacharequal}{\kern0pt}\ m\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ modules\ profile{\isacharunderscore}{\kern0pt}p\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharequal}{\kern0pt}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ mono{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Aq\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}eq\ mono{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ modules\ new{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}eq\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ modules\ profile{\isacharunderscore}{\kern0pt}p\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}subset\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ \isacommand{have}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}lengths{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}length\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{2}}\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{from}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ modules\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isasymle}\ i\ {\isasymand}\ i\ {\isacharless}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}size\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}lifted\ modules\ mono{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isadigit{0}}\ {\isasymle}\ i\ {\isasymand}\ i\ {\isacharless}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}Ap\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}size\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{0}}\ eql{\isacharunderscore}{\kern0pt}lengths\ mono{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ leI\ not{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}zero\ nth{\isacharunderscore}{\kern0pt}equalityI\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ mono{\isacharunderscore}{\kern0pt}m\ mono{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}help\isanewline
-\ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymand}\ n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ n\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}less\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}help\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}unchanged{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}eq\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ less{\isacharunderscore}{\kern0pt}irrefl\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}helper\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}unchanged\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}order\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}bounded\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ order{\isachardot}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}strict\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ t\ {\isacharparenleft}{\kern0pt}acc\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ monotone{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}comp\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isasymnoteq}\ f\ A\ rsa\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isacharequal}{\kern0pt}\ f\ A\ rsa\ {\isasymor}\ {\isasymnot}\ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymnotin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ {\isasymnot}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed\ monotone{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ q\ {\isasymsubset}\ defer\ acc\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}acc\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ monotone{\isacharunderscore}{\kern0pt}m\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ psubsetI\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ subsetCE\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p\ \isacommand{have}\isamarkupfalse%
-\ rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}q{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ {\isasymand}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}acc{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ q\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ monotone{\isacharunderscore}{\kern0pt}m\ subsetCE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ card{\isacharunderscore}{\kern0pt}changed{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ gr{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}not{\isadigit{0}}\ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}changed\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ monotone{\isacharunderscore}{\kern0pt}m\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\ psubsetI\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ t{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p\ less{\isachardot}{\kern0pt}hyps\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}no{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}incr\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ f{\isacharunderscore}{\kern0pt}prof\ rec{\isacharunderscore}{\kern0pt}step{\isacharunderscore}{\kern0pt}q\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\ subsetCE\ subset{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ t{\isacharunderscore}{\kern0pt}satisfied{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ {\isasymnot}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}lifted\ A\ p\ q\ a\ {\isasymand}\ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}helper\isanewline
-\ \ \ \ \ \ \ \ monotone{\isacharunderscore}{\kern0pt}m\ monotone{\isacharunderscore}{\kern0pt}acc\ profile\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ lifted\ A\ p\ q\ a\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ monotone{\isacharunderscore}{\kern0pt}acc\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ Profile{\isachardot}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ acc{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isasymnoteq}\ f\ A\ rsa\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs\ rsa\ a{\isachardot}{\kern0pt}\ f\ A\ rs\ {\isacharequal}{\kern0pt}\ f\ A\ rsa\ {\isasymor}\ {\isasymnot}\ Profile{\isachardot}{\kern0pt}lifted\ A\ rs\ rsa\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ a\ {\isasymnotin}\ defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ {\isasymnot}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}partit\isanewline
-\ \ \ \ \ \ \ \ \ \ monotone{\isacharunderscore}{\kern0pt}acc\ monotone{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ p\ q\ a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ lifted\ A\ p\ q\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}module{\isachardot}{\kern0pt}simps\ monotone{\isacharunderscore}{\kern0pt}m\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\ monotone{\isacharunderscore}{\kern0pt}m\ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}invariant{\isacharunderscore}{\kern0pt}monotonicity\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}w\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymor}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\ invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}w\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymor}\ defer\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}\ A\ q\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ dl{\isacharunderscore}{\kern0pt}inv{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ defer{\isacharunderscore}{\kern0pt}monotonicity{\isacharunderscore}{\kern0pt}def\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}mono{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}monotonicity\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ q\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ elect{\isacharunderscore}{\kern0pt}w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}p{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Profile{\isachardot}{\kern0pt}lifted\ A\ p\ q\ w{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}w\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect\ defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}w{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}p\ lifted{\isacharunderscore}{\kern0pt}w\ def{\isacharunderscore}{\kern0pt}monotone{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m\ electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Composition_Rules/Monotonicity_Rules.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%16=6%:%
-%:%17=7%:%
-%:%18=8%:%
-%:%19=9%:%
-%:%20=10%:%
-%:%21=11%:%
-%:%26=11%:%
-%:%29=12%:%
-%:%30=17%:%
-%:%31=18%:%
-%:%32=18%:%
-%:%33=19%:%
-%:%34=20%:%
-%:%35=21%:%
-%:%36=22%:%
-%:%37=23%:%
-%:%38=24%:%
-%:%41=25%:%
-%:%45=25%:%
-%:%46=25%:%
-%:%47=26%:%
-%:%48=26%:%
-%:%49=27%:%
-%:%50=27%:%
-%:%51=28%:%
-%:%52=28%:%
-%:%53=29%:%
-%:%54=30%:%
-%:%55=30%:%
-%:%56=31%:%
-%:%57=31%:%
-%:%58=32%:%
-%:%59=32%:%
-%:%60=33%:%
-%:%61=33%:%
-%:%62=34%:%
-%:%63=34%:%
-%:%64=35%:%
-%:%65=35%:%
-%:%66=36%:%
-%:%67=36%:%
-%:%68=37%:%
-%:%69=37%:%
-%:%70=38%:%
-%:%71=38%:%
-%:%72=39%:%
-%:%73=40%:%
-%:%74=41%:%
-%:%75=42%:%
-%:%76=43%:%
-%:%77=43%:%
-%:%78=44%:%
-%:%79=45%:%
-%:%80=46%:%
-%:%81=46%:%
-%:%82=47%:%
-%:%83=47%:%
-%:%84=48%:%
-%:%85=48%:%
-%:%86=49%:%
-%:%87=49%:%
-%:%88=50%:%
-%:%89=50%:%
-%:%90=51%:%
-%:%91=51%:%
-%:%92=52%:%
-%:%93=52%:%
-%:%94=53%:%
-%:%95=53%:%
-%:%96=54%:%
-%:%97=54%:%
-%:%98=55%:%
-%:%99=55%:%
-%:%100=56%:%
-%:%101=56%:%
-%:%102=57%:%
-%:%103=57%:%
-%:%104=58%:%
-%:%105=58%:%
-%:%106=59%:%
-%:%107=59%:%
-%:%108=60%:%
-%:%109=60%:%
-%:%110=61%:%
-%:%111=61%:%
-%:%112=62%:%
-%:%113=62%:%
-%:%114=63%:%
-%:%115=63%:%
-%:%116=64%:%
-%:%117=64%:%
-%:%118=65%:%
-%:%119=65%:%
-%:%120=66%:%
-%:%121=67%:%
-%:%122=68%:%
-%:%123=69%:%
-%:%124=69%:%
-%:%125=70%:%
-%:%126=70%:%
-%:%127=70%:%
-%:%128=71%:%
-%:%129=72%:%
-%:%130=72%:%
-%:%131=73%:%
-%:%132=74%:%
-%:%133=74%:%
-%:%134=75%:%
-%:%135=75%:%
-%:%136=75%:%
-%:%137=76%:%
-%:%138=77%:%
-%:%139=77%:%
-%:%140=78%:%
-%:%141=78%:%
-%:%142=78%:%
-%:%143=79%:%
-%:%144=80%:%
-%:%145=80%:%
-%:%146=81%:%
-%:%147=82%:%
-%:%148=83%:%
-%:%149=84%:%
-%:%150=85%:%
-%:%151=85%:%
-%:%152=86%:%
-%:%153=86%:%
-%:%154=86%:%
-%:%155=87%:%
-%:%156=88%:%
-%:%157=88%:%
-%:%158=89%:%
-%:%159=90%:%
-%:%160=91%:%
-%:%161=91%:%
-%:%162=92%:%
-%:%163=92%:%
-%:%164=92%:%
-%:%165=93%:%
-%:%166=94%:%
-%:%167=94%:%
-%:%168=95%:%
-%:%169=96%:%
-%:%170=96%:%
-%:%171=97%:%
-%:%172=97%:%
-%:%173=97%:%
-%:%174=98%:%
-%:%175=99%:%
-%:%176=99%:%
-%:%177=100%:%
-%:%178=101%:%
-%:%179=102%:%
-%:%180=103%:%
-%:%181=104%:%
-%:%182=105%:%
-%:%183=106%:%
-%:%184=106%:%
-%:%185=107%:%
-%:%186=107%:%
-%:%187=107%:%
-%:%188=108%:%
-%:%189=109%:%
-%:%190=109%:%
-%:%191=110%:%
-%:%192=110%:%
-%:%193=110%:%
-%:%194=111%:%
-%:%195=112%:%
-%:%196=112%:%
-%:%197=113%:%
-%:%198=114%:%
-%:%199=115%:%
-%:%200=116%:%
-%:%201=116%:%
-%:%202=117%:%
-%:%203=117%:%
-%:%204=118%:%
-%:%205=118%:%
-%:%206=119%:%
-%:%207=120%:%
-%:%208=121%:%
-%:%209=122%:%
-%:%210=122%:%
-%:%211=123%:%
-%:%212=123%:%
-%:%213=124%:%
-%:%214=124%:%
-%:%215=125%:%
-%:%216=126%:%
-%:%217=126%:%
-%:%218=127%:%
-%:%219=127%:%
-%:%220=128%:%
-%:%221=128%:%
-%:%222=129%:%
-%:%223=129%:%
-%:%224=129%:%
-%:%225=130%:%
-%:%226=130%:%
-%:%227=131%:%
-%:%228=131%:%
-%:%229=132%:%
-%:%230=132%:%
-%:%231=132%:%
-%:%232=133%:%
-%:%233=134%:%
-%:%234=134%:%
-%:%235=135%:%
-%:%236=135%:%
-%:%237=136%:%
-%:%238=136%:%
-%:%239=137%:%
-%:%240=137%:%
-%:%241=138%:%
-%:%242=138%:%
-%:%243=139%:%
-%:%247=143%:%
-%:%248=144%:%
-%:%249=144%:%
-%:%250=145%:%
-%:%251=146%:%
-%:%252=147%:%
-%:%253=147%:%
-%:%254=148%:%
-%:%255=148%:%
-%:%256=149%:%
-%:%257=149%:%
-%:%258=150%:%
-%:%259=150%:%
-%:%260=151%:%
-%:%261=152%:%
-%:%262=153%:%
-%:%263=153%:%
-%:%264=154%:%
-%:%265=155%:%
-%:%266=156%:%
-%:%267=156%:%
-%:%268=157%:%
-%:%269=157%:%
-%:%270=158%:%
-%:%271=158%:%
-%:%272=159%:%
-%:%273=160%:%
-%:%274=161%:%
-%:%275=161%:%
-%:%276=162%:%
-%:%277=162%:%
-%:%278=162%:%
-%:%279=163%:%
-%:%280=163%:%
-%:%281=164%:%
-%:%282=165%:%
-%:%283=165%:%
-%:%284=166%:%
-%:%285=166%:%
-%:%286=167%:%
-%:%287=167%:%
-%:%288=168%:%
-%:%289=168%:%
-%:%290=169%:%
-%:%291=169%:%
-%:%292=170%:%
-%:%293=170%:%
-%:%294=171%:%
-%:%295=171%:%
-%:%296=172%:%
-%:%297=172%:%
-%:%298=173%:%
-%:%299=174%:%
-%:%300=175%:%
-%:%301=175%:%
-%:%302=176%:%
-%:%303=177%:%
-%:%304=178%:%
-%:%305=178%:%
-%:%306=179%:%
-%:%307=180%:%
-%:%308=181%:%
-%:%309=182%:%
-%:%310=182%:%
-%:%311=183%:%
-%:%312=183%:%
-%:%313=184%:%
-%:%314=185%:%
-%:%315=186%:%
-%:%316=186%:%
-%:%317=187%:%
-%:%318=187%:%
-%:%319=188%:%
-%:%320=188%:%
-%:%321=189%:%
-%:%322=190%:%
-%:%323=191%:%
-%:%324=192%:%
-%:%325=193%:%
-%:%326=193%:%
-%:%327=194%:%
-%:%328=194%:%
-%:%329=195%:%
-%:%330=196%:%
-%:%331=197%:%
-%:%332=197%:%
-%:%333=198%:%
-%:%334=199%:%
-%:%335=200%:%
-%:%336=200%:%
-%:%337=201%:%
-%:%338=201%:%
-%:%339=202%:%
-%:%340=203%:%
-%:%341=203%:%
-%:%342=204%:%
-%:%343=205%:%
-%:%344=205%:%
-%:%345=206%:%
-%:%346=206%:%
-%:%347=207%:%
-%:%348=208%:%
-%:%349=209%:%
-%:%350=209%:%
-%:%351=210%:%
-%:%352=211%:%
-%:%353=212%:%
-%:%354=213%:%
-%:%355=214%:%
-%:%356=214%:%
-%:%357=215%:%
-%:%358=215%:%
-%:%359=216%:%
-%:%360=216%:%
-%:%361=217%:%
-%:%362=217%:%
-%:%363=218%:%
-%:%364=218%:%
-%:%365=219%:%
-%:%366=220%:%
-%:%367=221%:%
-%:%368=221%:%
-%:%369=222%:%
-%:%370=223%:%
-%:%371=223%:%
-%:%372=224%:%
-%:%373=224%:%
-%:%374=225%:%
-%:%376=227%:%
-%:%377=228%:%
-%:%378=228%:%
-%:%379=229%:%
-%:%380=230%:%
-%:%381=230%:%
-%:%382=231%:%
-%:%383=231%:%
-%:%384=232%:%
-%:%387=235%:%
-%:%388=236%:%
-%:%389=236%:%
-%:%390=237%:%
-%:%391=237%:%
-%:%392=238%:%
-%:%393=238%:%
-%:%394=239%:%
-%:%395=239%:%
-%:%396=240%:%
-%:%397=241%:%
-%:%398=242%:%
-%:%399=243%:%
-%:%400=244%:%
-%:%401=244%:%
-%:%402=245%:%
-%:%403=245%:%
-%:%404=246%:%
-%:%405=246%:%
-%:%406=247%:%
-%:%412=247%:%
-%:%415=248%:%
-%:%416=252%:%
-%:%417=253%:%
-%:%418=253%:%
-%:%419=254%:%
-%:%420=255%:%
-%:%421=256%:%
-%:%422=257%:%
-%:%423=258%:%
-%:%426=259%:%
-%:%430=259%:%
-%:%431=259%:%
-%:%432=260%:%
-%:%433=260%:%
-%:%434=261%:%
-%:%435=261%:%
-%:%436=262%:%
-%:%437=262%:%
-%:%438=263%:%
-%:%439=263%:%
-%:%440=264%:%
-%:%441=264%:%
-%:%442=265%:%
-%:%443=265%:%
-%:%444=266%:%
-%:%445=266%:%
-%:%446=267%:%
-%:%447=267%:%
-%:%448=268%:%
-%:%449=268%:%
-%:%450=269%:%
-%:%451=269%:%
-%:%452=270%:%
-%:%453=270%:%
-%:%454=271%:%
-%:%455=271%:%
-%:%456=272%:%
-%:%457=273%:%
-%:%458=274%:%
-%:%459=275%:%
-%:%460=276%:%
-%:%461=276%:%
-%:%462=277%:%
-%:%463=278%:%
-%:%464=279%:%
-%:%465=279%:%
-%:%466=280%:%
-%:%467=280%:%
-%:%468=281%:%
-%:%469=281%:%
-%:%470=281%:%
-%:%471=282%:%
-%:%474=285%:%
-%:%475=286%:%
-%:%476=286%:%
-%:%477=287%:%
-%:%478=287%:%
-%:%479=288%:%
-%:%480=288%:%
-%:%481=289%:%
-%:%482=290%:%
-%:%483=290%:%
-%:%484=291%:%
-%:%485=291%:%
-%:%486=292%:%
-%:%487=292%:%
-%:%488=293%:%
-%:%489=293%:%
-%:%490=294%:%
-%:%491=294%:%
-%:%492=295%:%
-%:%493=295%:%
-%:%494=295%:%
-%:%495=296%:%
-%:%496=296%:%
-%:%497=297%:%
-%:%498=298%:%
-%:%499=298%:%
-%:%500=299%:%
-%:%501=299%:%
-%:%502=300%:%
-%:%503=301%:%
-%:%504=301%:%
-%:%505=302%:%
-%:%506=303%:%
-%:%507=303%:%
-%:%508=304%:%
-%:%509=304%:%
-%:%510=304%:%
-%:%511=305%:%
-%:%512=305%:%
-%:%513=306%:%
-%:%514=307%:%
-%:%515=307%:%
-%:%516=308%:%
-%:%517=308%:%
-%:%518=308%:%
-%:%519=309%:%
-%:%520=310%:%
-%:%521=310%:%
-%:%522=311%:%
-%:%523=312%:%
-%:%524=312%:%
-%:%525=313%:%
-%:%526=313%:%
-%:%527=313%:%
-%:%528=314%:%
-%:%529=315%:%
-%:%530=315%:%
-%:%531=316%:%
-%:%532=316%:%
-%:%533=317%:%
-%:%534=318%:%
-%:%535=318%:%
-%:%536=319%:%
-%:%537=320%:%
-%:%538=320%:%
-%:%539=321%:%
-%:%540=321%:%
-%:%541=321%:%
-%:%542=322%:%
-%:%543=322%:%
-%:%544=323%:%
-%:%545=324%:%
-%:%546=325%:%
-%:%547=325%:%
-%:%548=326%:%
-%:%549=326%:%
-%:%550=326%:%
-%:%551=327%:%
-%:%552=328%:%
-%:%553=328%:%
-%:%554=329%:%
-%:%555=330%:%
-%:%556=330%:%
-%:%557=331%:%
-%:%558=331%:%
-%:%559=331%:%
-%:%560=332%:%
-%:%561=333%:%
-%:%562=333%:%
-%:%563=334%:%
-%:%564=334%:%
-%:%565=335%:%
-%:%566=335%:%
-%:%567=336%:%
-%:%568=336%:%
-%:%569=337%:%
-%:%570=337%:%
-%:%571=338%:%
-%:%572=338%:%
-%:%573=339%:%
-%:%574=339%:%
-%:%575=340%:%
-%:%576=340%:%
-%:%577=341%:%
-%:%578=342%:%
-%:%579=342%:%
-%:%580=343%:%
-%:%581=343%:%
-%:%582=344%:%
-%:%583=344%:%
-%:%584=345%:%
-%:%585=345%:%
-%:%586=346%:%
-%:%587=346%:%
-%:%588=346%:%
-%:%589=347%:%
-%:%590=347%:%
-%:%591=348%:%
-%:%592=349%:%
-%:%593=350%:%
-%:%594=351%:%
-%:%595=352%:%
-%:%596=352%:%
-%:%597=353%:%
-%:%598=353%:%
-%:%599=354%:%
-%:%600=355%:%
-%:%601=355%:%
-%:%602=356%:%
-%:%603=357%:%
-%:%604=357%:%
-%:%605=358%:%
-%:%606=358%:%
-%:%607=358%:%
-%:%608=359%:%
-%:%609=359%:%
-%:%610=360%:%
-%:%611=361%:%
-%:%612=362%:%
-%:%613=362%:%
-%:%614=363%:%
-%:%615=363%:%
-%:%616=363%:%
-%:%617=364%:%
-%:%618=365%:%
-%:%619=365%:%
-%:%620=366%:%
-%:%621=367%:%
-%:%622=367%:%
-%:%623=368%:%
-%:%624=368%:%
-%:%625=368%:%
-%:%626=369%:%
-%:%627=370%:%
-%:%628=370%:%
-%:%629=371%:%
-%:%630=371%:%
-%:%631=372%:%
-%:%632=373%:%
-%:%633=373%:%
-%:%634=374%:%
-%:%635=375%:%
-%:%636=375%:%
-%:%637=376%:%
-%:%638=376%:%
-%:%639=376%:%
-%:%640=377%:%
-%:%641=377%:%
-%:%642=378%:%
-%:%643=379%:%
-%:%644=379%:%
-%:%645=380%:%
-%:%646=380%:%
-%:%647=380%:%
-%:%648=381%:%
-%:%649=382%:%
-%:%650=382%:%
-%:%651=383%:%
-%:%652=384%:%
-%:%653=384%:%
-%:%654=385%:%
-%:%655=385%:%
-%:%656=385%:%
-%:%657=386%:%
-%:%658=387%:%
-%:%659=387%:%
-%:%660=388%:%
-%:%661=388%:%
-%:%662=389%:%
-%:%663=389%:%
-%:%664=390%:%
-%:%665=390%:%
-%:%666=391%:%
-%:%667=391%:%
-%:%668=392%:%
-%:%669=392%:%
-%:%670=393%:%
-%:%671=393%:%
-%:%672=394%:%
-%:%673=394%:%
-%:%674=395%:%
-%:%675=396%:%
-%:%676=396%:%
-%:%677=397%:%
-%:%683=397%:%
-%:%686=398%:%
-%:%687=399%:%
-%:%688=399%:%
-%:%689=400%:%
-%:%690=401%:%
-%:%691=402%:%
-%:%692=403%:%
-%:%693=404%:%
-%:%700=405%:%
-%:%701=405%:%
-%:%702=406%:%
-%:%703=406%:%
-%:%704=407%:%
-%:%705=407%:%
-%:%706=408%:%
-%:%707=408%:%
-%:%708=409%:%
-%:%709=409%:%
-%:%710=410%:%
-%:%711=410%:%
-%:%712=410%:%
-%:%713=411%:%
-%:%714=412%:%
-%:%715=412%:%
-%:%716=413%:%
-%:%717=413%:%
-%:%718=414%:%
-%:%719=414%:%
-%:%720=415%:%
-%:%721=415%:%
-%:%722=416%:%
-%:%723=416%:%
-%:%724=416%:%
-%:%725=417%:%
-%:%726=417%:%
-%:%727=418%:%
-%:%728=418%:%
-%:%729=418%:%
-%:%730=419%:%
-%:%731=419%:%
-%:%732=420%:%
-%:%733=420%:%
-%:%734=421%:%
-%:%735=421%:%
-%:%736=422%:%
-%:%737=422%:%
-%:%738=423%:%
-%:%739=424%:%
-%:%740=424%:%
-%:%741=425%:%
-%:%742=425%:%
-%:%743=426%:%
-%:%744=426%:%
-%:%745=427%:%
-%:%746=427%:%
-%:%747=428%:%
-%:%748=429%:%
-%:%749=429%:%
-%:%750=430%:%
-%:%751=430%:%
-%:%752=431%:%
-%:%753=431%:%
-%:%754=432%:%
-%:%755=433%:%
-%:%756=433%:%
-%:%757=434%:%
-%:%758=434%:%
-%:%759=435%:%
-%:%760=435%:%
-%:%761=436%:%
-%:%762=436%:%
-%:%763=437%:%
-%:%764=438%:%
-%:%765=438%:%
-%:%766=439%:%
-%:%767=439%:%
-%:%768=440%:%
-%:%769=440%:%
-%:%770=441%:%
-%:%771=441%:%
-%:%772=441%:%
-%:%773=442%:%
-%:%774=442%:%
-%:%775=443%:%
-%:%776=443%:%
-%:%777=443%:%
-%:%778=444%:%
-%:%779=445%:%
-%:%780=445%:%
-%:%781=446%:%
-%:%782=446%:%
-%:%783=447%:%
-%:%784=447%:%
-%:%785=447%:%
-%:%786=448%:%
-%:%787=448%:%
-%:%788=449%:%
-%:%789=450%:%
-%:%790=450%:%
-%:%791=451%:%
-%:%792=451%:%
-%:%793=452%:%
-%:%794=452%:%
-%:%795=452%:%
-%:%796=453%:%
-%:%797=453%:%
-%:%798=454%:%
-%:%799=454%:%
-%:%800=455%:%
-%:%801=455%:%
-%:%802=455%:%
-%:%803=455%:%
-%:%804=456%:%
-%:%805=457%:%
-%:%806=457%:%
-%:%807=458%:%
-%:%808=458%:%
-%:%809=458%:%
-%:%810=459%:%
-%:%814=463%:%
-%:%815=464%:%
-%:%816=464%:%
-%:%817=465%:%
-%:%818=465%:%
-%:%819=466%:%
-%:%820=466%:%
-%:%821=466%:%
-%:%822=467%:%
-%:%823=468%:%
-%:%824=469%:%
-%:%825=469%:%
-%:%826=470%:%
-%:%827=470%:%
-%:%828=471%:%
-%:%829=471%:%
-%:%830=471%:%
-%:%831=472%:%
-%:%833=474%:%
-%:%834=475%:%
-%:%835=475%:%
-%:%836=476%:%
-%:%837=477%:%
-%:%838=478%:%
-%:%839=478%:%
-%:%840=479%:%
-%:%841=479%:%
-%:%842=480%:%
-%:%843=480%:%
-%:%844=481%:%
-%:%845=481%:%
-%:%846=482%:%
-%:%847=482%:%
-%:%848=483%:%
-%:%849=483%:%
-%:%850=484%:%
-%:%851=484%:%
-%:%852=485%:%
-%:%853=485%:%
-%:%854=486%:%
-%:%855=486%:%
-%:%856=487%:%
-%:%857=487%:%
-%:%858=488%:%
-%:%864=488%:%
-%:%867=489%:%
-%:%868=490%:%
-%:%869=491%:%
-%:%870=491%:%
-%:%871=492%:%
-%:%872=493%:%
-%:%873=494%:%
-%:%874=495%:%
-%:%877=496%:%
-%:%881=496%:%
-%:%882=496%:%
-%:%883=497%:%
-%:%884=498%:%
-%:%885=498%:%
-%:%890=498%:%
-%:%893=499%:%
-%:%894=500%:%
-%:%895=500%:%
-%:%896=501%:%
-%:%897=502%:%
-%:%898=503%:%
-%:%899=504%:%
-%:%900=505%:%
-%:%905=510%:%
-%:%912=511%:%
-%:%913=511%:%
-%:%914=512%:%
-%:%915=512%:%
-%:%916=513%:%
-%:%917=513%:%
-%:%918=514%:%
-%:%920=516%:%
-%:%921=517%:%
-%:%922=517%:%
-%:%923=518%:%
-%:%924=518%:%
-%:%925=519%:%
-%:%926=519%:%
-%:%927=520%:%
-%:%929=522%:%
-%:%930=523%:%
-%:%931=523%:%
-%:%932=524%:%
-%:%933=524%:%
-%:%934=525%:%
-%:%936=527%:%
-%:%937=528%:%
-%:%938=528%:%
-%:%939=529%:%
-%:%940=529%:%
-%:%941=530%:%
-%:%942=530%:%
-%:%943=531%:%
-%:%944=531%:%
-%:%945=532%:%
-%:%946=532%:%
-%:%947=533%:%
-%:%948=533%:%
-%:%949=534%:%
-%:%950=534%:%
-%:%951=535%:%
-%:%953=537%:%
-%:%954=538%:%
-%:%955=538%:%
-%:%956=539%:%
-%:%957=540%:%
-%:%958=541%:%
-%:%959=542%:%
-%:%960=543%:%
-%:%961=543%:%
-%:%962=544%:%
-%:%963=544%:%
-%:%964=544%:%
-%:%965=544%:%
-%:%966=545%:%
-%:%967=546%:%
-%:%968=546%:%
-%:%969=547%:%
-%:%970=548%:%
-%:%971=548%:%
-%:%972=549%:%
-%:%973=549%:%
-%:%974=549%:%
-%:%975=550%:%
-%:%979=554%:%
-%:%980=555%:%
-%:%981=555%:%
-%:%982=556%:%
-%:%983=556%:%
-%:%984=557%:%
-%:%985=557%:%
-%:%986=558%:%
-%:%987=558%:%
-%:%988=559%:%
-%:%989=559%:%
-%:%990=560%:%
-%:%991=560%:%
-%:%992=561%:%
-%:%993=561%:%
-%:%994=562%:%
-%:%995=563%:%
-%:%996=563%:%
-%:%997=563%:%
-%:%998=564%:%
-%:%999=565%:%
-%:%1000=566%:%
-%:%1001=566%:%
-%:%1002=567%:%
-%:%1003=568%:%
-%:%1004=568%:%
-%:%1005=569%:%
-%:%1006=569%:%
-%:%1007=569%:%
-%:%1008=570%:%
-%:%1010=572%:%
-%:%1011=573%:%
-%:%1012=573%:%
-%:%1013=574%:%
-%:%1014=574%:%
-%:%1015=575%:%
-%:%1016=575%:%
-%:%1017=576%:%
-%:%1018=576%:%
-%:%1019=577%:%
-%:%1020=577%:%
-%:%1021=578%:%
-%:%1022=578%:%
-%:%1023=579%:%
-%:%1025=581%:%
-%:%1026=582%:%
-%:%1027=582%:%
-%:%1028=583%:%
-%:%1029=583%:%
-%:%1030=584%:%
-%:%1031=584%:%
-%:%1032=585%:%
-%:%1033=585%:%
-%:%1034=586%:%
-%:%1036=588%:%
-%:%1037=589%:%
-%:%1038=589%:%
-%:%1039=590%:%
-%:%1040=590%:%
-%:%1041=591%:%
-%:%1046=596%:%
-%:%1047=597%:%
-%:%1048=597%:%
-%:%1049=598%:%
-%:%1050=598%:%
-%:%1051=599%:%
-%:%1052=599%:%
-%:%1053=600%:%
-%:%1054=600%:%
-%:%1055=601%:%
-%:%1056=601%:%
-%:%1057=602%:%
-%:%1058=602%:%
-%:%1059=603%:%
-%:%1060=603%:%
-%:%1061=604%:%
-%:%1063=606%:%
-%:%1064=607%:%
-%:%1065=607%:%
-%:%1066=608%:%
-%:%1067=609%:%
-%:%1068=610%:%
-%:%1069=611%:%
-%:%1070=611%:%
-%:%1071=612%:%
-%:%1072=612%:%
-%:%1073=612%:%
-%:%1074=613%:%
-%:%1077=616%:%
-%:%1078=617%:%
-%:%1079=617%:%
-%:%1080=618%:%
-%:%1081=619%:%
-%:%1082=620%:%
-%:%1083=621%:%
-%:%1084=621%:%
-%:%1085=622%:%
-%:%1086=622%:%
-%:%1087=623%:%
-%:%1088=624%:%
-%:%1089=625%:%
-%:%1090=625%:%
-%:%1091=626%:%
-%:%1092=627%:%
-%:%1093=628%:%
-%:%1094=629%:%
-%:%1095=629%:%
-%:%1096=630%:%
-%:%1097=630%:%
-%:%1098=631%:%
-%:%1099=631%:%
-%:%1100=632%:%
-%:%1101=633%:%
-%:%1102=634%:%
-%:%1103=635%:%
-%:%1104=635%:%
-%:%1105=636%:%
-%:%1106=636%:%
-%:%1107=637%:%
-%:%1108=637%:%
-%:%1109=638%:%
-%:%1110=638%:%
-%:%1111=639%:%
-%:%1112=639%:%
-%:%1113=640%:%
-%:%1114=640%:%
-%:%1115=641%:%
-%:%1116=641%:%
-%:%1117=642%:%
-%:%1118=642%:%
-%:%1119=643%:%
-%:%1125=643%:%
-%:%1128=644%:%
-%:%1129=645%:%
-%:%1130=645%:%
-%:%1131=646%:%
-%:%1132=647%:%
-%:%1133=648%:%
-%:%1134=649%:%
-%:%1135=650%:%
-%:%1136=651%:%
-%:%1137=652%:%
-%:%1140=653%:%
-%:%1144=653%:%
-%:%1145=653%:%
-%:%1146=654%:%
-%:%1147=655%:%
-%:%1148=655%:%
-%:%1153=655%:%
-%:%1156=656%:%
-%:%1157=657%:%
-%:%1158=657%:%
-%:%1159=658%:%
-%:%1160=659%:%
-%:%1161=660%:%
-%:%1162=661%:%
-%:%1163=662%:%
-%:%1166=665%:%
-%:%1169=666%:%
-%:%1173=666%:%
-%:%1174=666%:%
-%:%1175=667%:%
-%:%1176=667%:%
-%:%1181=667%:%
-%:%1184=668%:%
-%:%1185=669%:%
-%:%1186=669%:%
-%:%1187=670%:%
-%:%1188=671%:%
-%:%1191=672%:%
-%:%1195=672%:%
-%:%1196=672%:%
-%:%1197=673%:%
-%:%1198=673%:%
-%:%1203=673%:%
-%:%1206=674%:%
-%:%1207=675%:%
-%:%1208=675%:%
-%:%1209=676%:%
-%:%1210=677%:%
-%:%1211=678%:%
-%:%1212=679%:%
-%:%1219=680%:%
-%:%1220=680%:%
-%:%1221=681%:%
-%:%1222=681%:%
-%:%1223=682%:%
-%:%1230=689%:%
-%:%1231=690%:%
-%:%1232=690%:%
-%:%1233=691%:%
-%:%1234=691%:%
-%:%1235=692%:%
-%:%1236=692%:%
-%:%1237=693%:%
-%:%1238=693%:%
-%:%1239=694%:%
-%:%1240=695%:%
-%:%1241=696%:%
-%:%1242=696%:%
-%:%1243=697%:%
-%:%1249=697%:%
-%:%1252=698%:%
-%:%1253=699%:%
-%:%1254=700%:%
-%:%1255=700%:%
-%:%1256=701%:%
-%:%1257=702%:%
-%:%1264=703%:%
-%:%1265=703%:%
-%:%1266=704%:%
-%:%1267=704%:%
-%:%1268=705%:%
-%:%1269=706%:%
-%:%1270=706%:%
-%:%1271=707%:%
-%:%1272=708%:%
-%:%1273=709%:%
-%:%1274=709%:%
-%:%1275=710%:%
-%:%1276=711%:%
-%:%1277=712%:%
-%:%1278=712%:%
-%:%1279=713%:%
-%:%1280=713%:%
-%:%1281=714%:%
-%:%1282=714%:%
-%:%1283=715%:%
-%:%1284=716%:%
-%:%1285=717%:%
-%:%1286=718%:%
-%:%1287=718%:%
-%:%1288=719%:%
-%:%1294=719%:%
-%:%1297=720%:%
-%:%1298=724%:%
-%:%1299=725%:%
-%:%1300=725%:%
-%:%1301=726%:%
-%:%1302=727%:%
-%:%1309=728%:%
-%:%1310=728%:%
-%:%1311=729%:%
-%:%1312=729%:%
-%:%1313=730%:%
-%:%1314=731%:%
-%:%1315=731%:%
-%:%1316=732%:%
-%:%1317=732%:%
-%:%1318=733%:%
-%:%1319=733%:%
-%:%1320=733%:%
-%:%1321=734%:%
-%:%1322=734%:%
-%:%1323=735%:%
-%:%1324=735%:%
-%:%1325=736%:%
-%:%1326=736%:%
-%:%1327=736%:%
-%:%1328=737%:%
-%:%1329=737%:%
-%:%1330=738%:%
-%:%1331=738%:%
-%:%1332=739%:%
-%:%1333=739%:%
-%:%1334=739%:%
-%:%1336=741%:%
-%:%1337=742%:%
-%:%1338=742%:%
-%:%1339=743%:%
-%:%1340=743%:%
-%:%1341=744%:%
-%:%1342=744%:%
-%:%1343=745%:%
-%:%1344=745%:%
-%:%1345=746%:%
-%:%1351=746%:%
-%:%1354=747%:%
-%:%1355=751%:%
-%:%1356=752%:%
-%:%1357=752%:%
-%:%1358=753%:%
-%:%1359=754%:%
-%:%1362=755%:%
-%:%1366=755%:%
-%:%1367=755%:%
-%:%1368=756%:%
-%:%1369=756%:%
-%:%1374=756%:%
-%:%1377=757%:%
-%:%1378=763%:%
-%:%1379=764%:%
-%:%1380=764%:%
-%:%1381=765%:%
-%:%1382=766%:%
-%:%1383=767%:%
-%:%1384=768%:%
-%:%1385=769%:%
-%:%1386=770%:%
-%:%1389=771%:%
-%:%1393=771%:%
-%:%1394=771%:%
-%:%1395=772%:%
-%:%1396=772%:%
-%:%1397=773%:%
-%:%1398=773%:%
-%:%1399=774%:%
-%:%1400=774%:%
-%:%1401=775%:%
-%:%1402=775%:%
-%:%1403=776%:%
-%:%1404=776%:%
-%:%1405=777%:%
-%:%1406=777%:%
-%:%1407=778%:%
-%:%1408=778%:%
-%:%1409=779%:%
-%:%1410=779%:%
-%:%1411=780%:%
-%:%1412=780%:%
-%:%1413=781%:%
-%:%1414=781%:%
-%:%1415=782%:%
-%:%1416=782%:%
-%:%1417=783%:%
-%:%1418=783%:%
-%:%1419=784%:%
-%:%1420=785%:%
-%:%1421=786%:%
-%:%1422=787%:%
-%:%1423=788%:%
-%:%1424=788%:%
-%:%1425=789%:%
-%:%1426=790%:%
-%:%1427=791%:%
-%:%1428=792%:%
-%:%1429=792%:%
-%:%1430=793%:%
-%:%1431=794%:%
-%:%1432=794%:%
-%:%1433=795%:%
-%:%1434=795%:%
-%:%1435=796%:%
-%:%1436=796%:%
-%:%1437=797%:%
-%:%1438=797%:%
-%:%1439=798%:%
-%:%1440=799%:%
-%:%1441=800%:%
-%:%1442=800%:%
-%:%1443=801%:%
-%:%1449=801%:%
-%:%1454=802%:%
-%:%1459=803%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Nanson_Baldwin_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Nanson_Baldwin_Rule.tex
deleted file mode 100644
index f13ef1c..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Nanson_Baldwin_Rule.tex
+++ /dev/null
@@ -1,103 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Nanson{\isacharunderscore}{\kern0pt}Baldwin{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Nanson-Baldwin Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Nanson{\isacharunderscore}{\kern0pt}Baldwin{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Nanson-Baldwin voting rule. It excludes alternatives with the
-lowest Borda score from the set of possible winners and then adjusts the
-Borda score to the new (remaining) set of still eligible alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ nanson{\isacharunderscore}{\kern0pt}baldwin{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}nanson{\isacharunderscore}{\kern0pt}baldwin{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Voting_Rules/Nanson_Baldwin_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%53=19%:%
-%:%63=21%:%
-%:%64=21%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%69=24%:%
-%:%74=25%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Pairwise_Majority_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Pairwise_Majority_Rule.tex
deleted file mode 100644
index d697f79..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Pairwise_Majority_Rule.tex
+++ /dev/null
@@ -1,195 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Pairwise{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Pairwise Majority Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Pairwise{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Rules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Condorcet{\isacharunderscore}{\kern0pt}Facts{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the pairwise majority rule, a voting rule that implements the
-Condorcet criterion, i.e., it elects the Condorcet winner if it exists,
-otherwise a tie remains between all alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\ elector\ condorcet\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-{\isachardoublequoteopen}condorcet{\isacharprime}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ condorcet{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isacharprime}{\kern0pt}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-{\isachardoublequoteopen}pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isacharprime}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ iterelect\ condorcet{\isacharprime}{\kern0pt}\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Condorcet Consistency Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}condorcet{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}consistency\ {\isacharparenleft}{\kern0pt}elector\ condorcet{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}dcc\ dcc{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}cc{\isacharunderscore}{\kern0pt}elector\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}consistency{\isadigit{2}}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ pairwise{\isacharunderscore}{\kern0pt}majority{\isacharunderscore}{\kern0pt}rule{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Voting_Rules/Pairwise_Majority_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%35=14%:%
-%:%36=15%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%56=22%:%
-%:%66=24%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%73=29%:%
-%:%74=30%:%
-%:%75=31%:%
-%:%76=31%:%
-%:%77=32%:%
-%:%84=34%:%
-%:%94=36%:%
-%:%95=36%:%
-%:%102=37%:%
-%:%103=37%:%
-%:%104=38%:%
-%:%105=38%:%
-%:%106=39%:%
-%:%107=40%:%
-%:%108=40%:%
-%:%109=41%:%
-%:%110=41%:%
-%:%111=42%:%
-%:%112=42%:%
-%:%113=43%:%
-%:%114=43%:%
-%:%115=44%:%
-%:%116=45%:%
-%:%117=45%:%
-%:%118=46%:%
-%:%124=46%:%
-%:%129=47%:%
-%:%134=48%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Parallel_Composition.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Parallel_Composition.tex
deleted file mode 100644
index 3941087..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Parallel_Composition.tex
+++ /dev/null
@@ -1,214 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Parallel{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Parallel Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Parallel{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The parallel composition composes a new electoral module from
-two electoral modules combined with an aggregator.
-Therein, the two modules each make a decision and the aggregator combines
-them to a single (aggregated) result.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ parallel{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}parallel{\isacharunderscore}{\kern0pt}composition\ m\ n\ agg\ A\ p\ {\isacharequal}{\kern0pt}\ agg\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ parallel\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Aggregator\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{5}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{5}}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymparallel}\isactrlsub a\ n\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ parallel{\isacharunderscore}{\kern0pt}composition\ m\ n\ a{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ agg{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}aggregator\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ aggregator{\isacharunderscore}{\kern0pt}def\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ mod{\isacharunderscore}{\kern0pt}m\ mod{\isacharunderscore}{\kern0pt}n\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ agg{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Compositional_Structures/Parallel_Composition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%55=21%:%
-%:%65=23%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=27%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%73=29%:%
-%:%74=30%:%
-%:%81=32%:%
-%:%91=34%:%
-%:%92=34%:%
-%:%93=35%:%
-%:%94=36%:%
-%:%95=37%:%
-%:%96=38%:%
-%:%97=39%:%
-%:%100=40%:%
-%:%104=40%:%
-%:%105=40%:%
-%:%106=41%:%
-%:%107=41%:%
-%:%108=42%:%
-%:%109=42%:%
-%:%110=43%:%
-%:%111=44%:%
-%:%112=45%:%
-%:%113=45%:%
-%:%114=46%:%
-%:%115=47%:%
-%:%116=48%:%
-%:%117=48%:%
-%:%118=49%:%
-%:%119=49%:%
-%:%120=50%:%
-%:%121=51%:%
-%:%122=51%:%
-%:%123=52%:%
-%:%124=52%:%
-%:%125=53%:%
-%:%126=53%:%
-%:%127=54%:%
-%:%133=54%:%
-%:%138=55%:%
-%:%143=56%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Pass_Module.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Pass_Module.tex
deleted file mode 100644
index 8ff0149..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Pass_Module.tex
+++ /dev/null
@@ -1,270 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Pass{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Pass Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Pass{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is a family of electoral modules. For a natural number n and a
-lexicon (linear order) r of all alternatives, the according pass module
-defers the lexicographically first n alternatives (from A) and rejects
-the rest. It is primarily used as counterpart to the drop module in a
-parallel composition in order to segment the alternatives into two groups.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}mod\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectI\ not{\isacharunderscore}{\kern0pt}less\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ n{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}mod\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}modI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Pass_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%55=21%:%
-%:%65=23%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%77=29%:%
-%:%87=31%:%
-%:%88=31%:%
-%:%89=32%:%
-%:%90=33%:%
-%:%97=34%:%
-%:%98=34%:%
-%:%99=35%:%
-%:%100=35%:%
-%:%101=36%:%
-%:%102=36%:%
-%:%103=37%:%
-%:%105=39%:%
-%:%106=40%:%
-%:%107=40%:%
-%:%108=41%:%
-%:%109=41%:%
-%:%110=42%:%
-%:%111=42%:%
-%:%112=43%:%
-%:%114=45%:%
-%:%115=46%:%
-%:%116=46%:%
-%:%117=47%:%
-%:%118=47%:%
-%:%119=48%:%
-%:%120=49%:%
-%:%121=49%:%
-%:%122=50%:%
-%:%123=50%:%
-%:%124=51%:%
-%:%126=53%:%
-%:%127=54%:%
-%:%128=54%:%
-%:%129=55%:%
-%:%130=55%:%
-%:%131=56%:%
-%:%133=58%:%
-%:%134=59%:%
-%:%135=59%:%
-%:%136=60%:%
-%:%137=60%:%
-%:%138=61%:%
-%:%139=62%:%
-%:%140=62%:%
-%:%141=63%:%
-%:%142=63%:%
-%:%143=64%:%
-%:%144=64%:%
-%:%145=65%:%
-%:%146=66%:%
-%:%147=66%:%
-%:%148=67%:%
-%:%149=67%:%
-%:%150=68%:%
-%:%151=69%:%
-%:%152=69%:%
-%:%153=70%:%
-%:%154=70%:%
-%:%155=71%:%
-%:%156=71%:%
-%:%157=72%:%
-%:%158=72%:%
-%:%159=73%:%
-%:%165=73%:%
-%:%170=74%:%
-%:%175=75%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Plurality_Module.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Plurality_Module.tex
deleted file mode 100644
index 62fbfcf..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Plurality_Module.tex
+++ /dev/null
@@ -1,257 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Plurality{\isacharunderscore}{\kern0pt}Module}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Plurality Module%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Plurality{\isacharunderscore}{\kern0pt}Module\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The plurality module implements the plurality voting rule.
-The plurality rule elects all modules with the maximum amount of top
-preferences among all alternatives, and rejects all the other alternatives.
-It is electing and induces the classical plurality (voting) rule
-from social-choice theory.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ plurality\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}plurality\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ elect\ {\isasyminter}\ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ disjoint{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}elect{\isacharcomma}{\kern0pt}\ reject{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ elect\ {\isasymunion}\ reject\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ unity{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\ elect\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymexists}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isachargreater}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}\ in\isanewline
-\ \ \ \ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}elect{\isacharcomma}{\kern0pt}\ reject{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint\ unity\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}modI{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimML
-\isanewline
-%
-\endisadelimML
-%
-\isatagML
-\isacommand{ML}\isamarkupfalse%
-\ {\isacartoucheopen}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharasterisk}{\kern0pt}\ val\ w\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}all{\isacharunderscore}{\kern0pt}specifications{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Theory{\isachardot}{\kern0pt}defs{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ x\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ String{\isachardot}{\kern0pt}isPrefix\ {\isachardoublequote}{\kern0pt}Seq{\isachardoublequote}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ snd\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}all{\isacharunderscore}{\kern0pt}specifications{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Theory{\isachardot}{\kern0pt}defs{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ y\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}Plurality{\isacharunderscore}{\kern0pt}Module{\isachardot}{\kern0pt}plurality{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}all{\isacharunderscore}{\kern0pt}specifications{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Theory{\isachardot}{\kern0pt}defs{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ z\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}Plurality{\isacharunderscore}{\kern0pt}Module{\isachardot}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}def{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharhash}{\kern0pt}description\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}hd\ {\isacharparenleft}{\kern0pt}snd\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ val\ def\ {\isacharequal}{\kern0pt}\ hd\ z\isanewline
-\ \ \ \ val\ def{\isacharunderscore}{\kern0pt}rhs\ {\isacharequal}{\kern0pt}\ {\isacharhash}{\kern0pt}rhs\ def\isanewline
-\isanewline
-\ \ \ \ val\ def{\isacharunderscore}{\kern0pt}funs\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}fun{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}def{\isacharunderscore}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ def{\isacharunderscore}{\kern0pt}consts\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ not\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}fun{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}def{\isacharunderscore}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ val\ fun{\isacharunderscore}{\kern0pt}type\ {\isacharequal}{\kern0pt}\ Defs{\isachardot}{\kern0pt}pretty{\isacharunderscore}{\kern0pt}entry\ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}global{\isacharunderscore}{\kern0pt}context\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}List{\isachardot}{\kern0pt}last\ def{\isacharunderscore}{\kern0pt}funs{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ {\isacharparenleft}{\kern0pt}{\isacharasterisk}{\kern0pt}\ val\ a\ {\isacharequal}{\kern0pt}\ Defs{\isachardot}{\kern0pt}pretty{\isacharunderscore}{\kern0pt}entry\ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}global{\isacharunderscore}{\kern0pt}context\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}nth\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}{\isacharhash}{\kern0pt}lhs\ {\isacharparenleft}{\kern0pt}hd\ z{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharhash}{\kern0pt}rhs\ {\isacharparenleft}{\kern0pt}hd\ z{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ {\isacartoucheclose}%
-\endisatagML
-{\isafoldML}%
-%
-\isadelimML
-\isanewline
-%
-\endisadelimML
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Basic_Modules/Plurality_Module.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%55=21%:%
-%:%65=23%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%77=29%:%
-%:%87=31%:%
-%:%88=31%:%
-%:%95=32%:%
-%:%96=32%:%
-%:%97=33%:%
-%:%98=33%:%
-%:%99=34%:%
-%:%102=37%:%
-%:%103=38%:%
-%:%104=38%:%
-%:%105=39%:%
-%:%106=39%:%
-%:%107=40%:%
-%:%110=43%:%
-%:%111=44%:%
-%:%112=44%:%
-%:%113=45%:%
-%:%114=45%:%
-%:%115=46%:%
-%:%118=49%:%
-%:%119=50%:%
-%:%120=50%:%
-%:%121=51%:%
-%:%122=51%:%
-%:%123=52%:%
-%:%124=52%:%
-%:%125=53%:%
-%:%128=56%:%
-%:%129=57%:%
-%:%130=57%:%
-%:%131=58%:%
-%:%132=58%:%
-%:%133=58%:%
-%:%134=59%:%
-%:%135=59%:%
-%:%136=60%:%
-%:%142=60%:%
-%:%147=61%:%
-%:%152=62%:%
-%:%153=62%:%
-%:%175=79%:%
-%:%180=80%:%
-%:%185=81%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Preference_Relation.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Preference_Relation.tex
deleted file mode 100644
index a6f4bdc..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Preference_Relation.tex
+++ /dev/null
@@ -1,4290 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Preference{\isacharunderscore}{\kern0pt}Relation}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupchapter{Social-Choice Types%
-}
-\isamarkuptrue%
-%
-\isamarkupsection{Preference Relation%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Preference{\isacharunderscore}{\kern0pt}Relation\isanewline
-\ \ \isakeyword{imports}\ Main\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The very core of the composable modules voting framework: types and
-functions, derivations, lemmata, operations on preference relations, etc.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ rel{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}preferred{\isacharunderscore}{\kern0pt}than\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}\ {\isasympreceq}\isactrlsub {\isacharunderscore}{\kern0pt}\ {\isacharunderscore}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbrackleft}{\kern0pt}{\isadigit{5}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcomma}{\kern0pt}\ {\isadigit{5}}{\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Ranking%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ rank\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}rank\ r\ x\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ r\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ rank{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ refl{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}rank\ r\ x\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ FieldI{\isadigit{2}}\ refl\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ fin\ finite{\isacharunderscore}{\kern0pt}Field{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card{\isacharbraceleft}{\kern0pt}y\ {\isasymin}\ Field\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharbraceright}{\kern0pt}\ {\isasymge}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}cong\ FieldI{\isadigit{2}}\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ less{\isacharunderscore}{\kern0pt}one\ not{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}less\ rank{\isachardot}{\kern0pt}elims\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Limited Preference%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ limited\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}limited\ A\ r\ {\isasymequiv}\ r\ {\isasymsubseteq}\ A\ {\isasymtimes}\ A{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limitedI{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}x\ y{\isachardot}{\kern0pt}\ {\isasymlbrakk}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ \ x\ {\isasymin}\ A\ {\isasymand}\ y\ {\isasymin}\ A{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ limited\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}dest{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymAnd}x\ y{\isachardot}{\kern0pt}\ {\isasymlbrakk}\ x\ {\isasympreceq}\isactrlsub r\ y{\isacharsemicolon}{\kern0pt}\ limited\ A\ r\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ \ x\ {\isasymin}\ A\ {\isasymand}\ y\ {\isasymin}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ limit\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}limit\ A\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ {\isasymand}\ b\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ connex\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}connex\ A\ r\ {\isasymequiv}\ limited\ A\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymor}\ y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}r\ {\isasymsubseteq}\ A\ {\isasymtimes}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}def\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ assms\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}onD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ assms\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}onD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ y\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ assms\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}onD{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ refl{\isacharunderscore}{\kern0pt}onD\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}antsym{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}trans{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ connex{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ antisym{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ trans{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ preorder{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}r\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}r\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}r\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}r\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}r\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ y{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymor}\ y\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ connex{\isacharunderscore}{\kern0pt}r\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ connex{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}connex\ S\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}s\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\ b\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ b\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasympreceq}\isactrlsub r\ x\ {\isasymor}\ x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\ connex\ connex{\isacharunderscore}{\kern0pt}def\ in{\isacharunderscore}{\kern0pt}mono\ subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub {\isacharquery}{\kern0pt}s\ y\ {\isasymor}\ y\ {\isasympreceq}\isactrlsub {\isacharquery}{\kern0pt}s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub {\isacharquery}{\kern0pt}s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}antisym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ antisymmetric{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}antisym\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}def\ antisymmetric\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ transitive{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ \ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}trans\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ z\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ asm{\isadigit{5}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}z\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ asm{\isadigit{4}}\ transE\ transitive\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ S\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}antsym{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}trans{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}antisym\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}trans\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}let\ s\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ y{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ x{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}y\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}C\ {\isasymsubseteq}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit\ C\ r\ {\isacharequal}{\kern0pt}\ limit\ C\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}r\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnot}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ subrelI\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}singleton{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymforall}r{\isachardot}{\kern0pt}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ r\ {\isasymlongrightarrow}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ r\ {\isasymlongrightarrow}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ singletonI\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardot}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ a\ {\isasymand}\ y\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ split{\isacharunderscore}{\kern0pt}beta\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ b\ {\isasymsubseteq}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Collect{\isacharunderscore}{\kern0pt}mono\ above{\isacharunderscore}{\kern0pt}def\ assms\ transE\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}refl{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}a\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ assms\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ fastforce%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ s{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}above\ r\ a\ {\isasymsubseteq}\ above\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ above{\isacharunderscore}{\kern0pt}refl\ insert{\isacharunderscore}{\kern0pt}absorb\isanewline
-\ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ singletonI\ subset{\isacharunderscore}{\kern0pt}singletonD\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}a\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ above{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ b\ {\isasymlongleftrightarrow}\ b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ B\ {\isasymand}\ b\ {\isasymin}\ B{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ assms\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}b\ {\isasymin}\ B{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ n{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ \isakeyword{where}\ n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}plus{\isadigit{1}}\ antisym{\isacharunderscore}{\kern0pt}conv{\isadigit{2}}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ gr{\isadigit{0}}{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}Suc\ le{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymexists}a{\isachardot}{\kern0pt}\ a{\isasymin}A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ n\ arbitrary{\isacharcolon}{\kern0pt}\ A\ r{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ {\isadigit{0}}{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ add{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ above{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a{\isasymin}A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}Suc\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ Suc\ n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isacharequal}{\kern0pt}\ card\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ B\ \isakeyword{where}\ B{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ B\ {\isacharequal}{\kern0pt}\ n{\isacharplus}{\kern0pt}{\isadigit{1}}\ {\isasymand}\ B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharunderscore}{\kern0pt}inject\ add{\isacharunderscore}{\kern0pt}Suc\ card{\isachardot}{\kern0pt}insert{\isacharunderscore}{\kern0pt}remove\ finite{\isachardot}{\kern0pt}cases\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}single\ subset{\isacharunderscore}{\kern0pt}insertI\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ B{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Suc{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}plus{\isadigit{1}}\ add{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}left{\isacharprime}{\kern0pt}\ asm\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset\ finite{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}b{\isasymin}B{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ B\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isachardot}{\kern0pt}IH\ add{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}cancel{\isacharunderscore}{\kern0pt}left{\isacharprime}{\kern0pt}\ asm\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ diff{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}self\ finite{\isacharunderscore}{\kern0pt}subset\ leD\ lessI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ b\ \isakeyword{where}\ b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r\ a\ aa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ refl{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ r\ {\isasymor}\ a\ {\isasymin}\ A\ {\isasymand}\ aa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\ {\isasymnot}\ connex\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isasymor}\ refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\ {\isasymnot}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isasymor}\ connex\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}refl{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ asm\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ A\ {\isasymand}\ b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ asm{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\ b\ {\isacharequal}{\kern0pt}\ a\ {\isasymor}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{5}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ singletonI\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{6}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ A\ Aa{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ A\ {\isacharminus}{\kern0pt}\ Aa\ {\isasymor}\ a\ {\isasymin}\ A\ {\isasymand}\ a\ {\isasymnotin}\ Aa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ ff{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ ff{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ aa\ {\isasymin}\ B\ {\isasymand}\ a\ {\isasymin}\ B{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{5}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isacharunderscore}{\kern0pt}B{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ B\ above{\isacharunderscore}{\kern0pt}def\ f{\isadigit{4}}\ ff{\isadigit{1}}\ ff{\isadigit{2}}\ CollectI\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Product{\isacharunderscore}{\kern0pt}Type{\isachardot}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}case{\isacharunderscore}{\kern0pt}prodD\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{1}}\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ a\ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasymin}\ above\ r\ b\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectD\ above{\isacharunderscore}{\kern0pt}def\ antisym{\isacharunderscore}{\kern0pt}def\ asm\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasymin}\ above\ r\ b\ {\isasymlongleftrightarrow}\ x\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ b\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ subsetI\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ r\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}a\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ B\ DiffE\ a\ asm\ b\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ limited{\isacharunderscore}{\kern0pt}dest\ singletonI\ subset{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ g{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r\ Aa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ Aa\ {\isasymsubseteq}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ Aa\ {\isacharparenleft}{\kern0pt}limit\ Aa\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ g{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ B{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ g{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ B\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ total{\isacharunderscore}{\kern0pt}on\ B\ {\isacharparenleft}{\kern0pt}limit\ B\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ g{\isadigit{1}}\ B\ asm\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ total{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}aa\ {\isasymnotin}\ A\ {\isasymor}\ a\ {\isacharequal}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ B\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymnotin}\ B\ {\isasymor}\ a\ {\isacharequal}{\kern0pt}\ aa\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r\ {\isasymor}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ B\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ g{\isadigit{3}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isadigit{2}}\ B\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ asm\ b\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ singletonI\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ g{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}wins{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}trans\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ transE\ b{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ b{\isacharunderscore}{\kern0pt}wins{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ B{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ nothing{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a\ asm\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ above{\isacharunderscore}{\kern0pt}connex\ Diff{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasymin}\ above\ r\ a\ {\isasymlongleftrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}def\ asm\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ nothing{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}a\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ CollectD\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ a\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ asm\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ a\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a{\isasymin}A\ {\isasymand}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}iff\ above{\isacharunderscore}{\kern0pt}trans\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ empty{\isacharunderscore}{\kern0pt}Diff\ insertE\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}if\ insert{\isacharunderscore}{\kern0pt}absorb\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}defs{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ total{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}emp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A\ {\isasymand}\ A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ above{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ above\ r\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}a\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ b\ {\isasympreceq}\isactrlsub r\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isadigit{1}}\ singletonI\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord\ fin{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}emp\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}one{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}connex\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isacharequal}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isadigit{1}}\ connex{\isacharunderscore}{\kern0pt}def\ limited{\isacharunderscore}{\kern0pt}dest\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lifting Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a\ {\isasymequiv}\isanewline
-\ \ \ \ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ s\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ lifted\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}lifted\ A\ r\ s\ a\ {\isasymequiv}\isanewline
-\ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ trivial{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ p\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ order{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ lifted\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ r\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ s\ {\isasymand}\ a\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}mono{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub s\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}exist{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasympreceq}\isactrlsub s\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}antisym\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}x\ y{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisym{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ imp{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymlbrakk}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isacharsemicolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isasymrbrakk}\ {\isasymLongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ lift{\isacharunderscore}{\kern0pt}ex{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ lift{\isacharunderscore}{\kern0pt}ex\ \isacommand{obtain}\isamarkupfalse%
-\ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ \isakeyword{where}\isanewline
-\ \ \ \ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub r\ y\ {\isasymand}\ y\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ trans{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x\ y\ z{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ partial{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ preorder{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}def\ trans{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ f{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ f{\isadigit{2}}\ f{\isadigit{2}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ insertE\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}Diff\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}a\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ trans\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ imp{\isacharunderscore}{\kern0pt}x{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ x{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}a\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ f{\isadigit{2}}\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ transE\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}y{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ antisymD\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ DiffD{\isadigit{2}}\ f{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ f{\isadigit{2}}\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\ singletonI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ refl{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}domain\ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ rest{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ ex{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}cases\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ True\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{case}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ insertE\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ lifted\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ x{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}{\isadigit{0}}\ ex{\isacharunderscore}{\kern0pt}lifted\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\ rest{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ trans{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ s\ a\ {\isasymsubseteq}\ above\ r\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ rest{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ y\ z{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}s{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ y\ z{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ s\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ s\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ trans{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}trans\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ assms\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ refl{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x\ assms\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ refl{\isacharunderscore}{\kern0pt}onD{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ a{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}x\ lifted{\isacharunderscore}{\kern0pt}r\ rest{\isacharunderscore}{\kern0pt}eq\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \ \ trans{\isacharunderscore}{\kern0pt}r\ trans{\isacharunderscore}{\kern0pt}s\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ refl{\isacharunderscore}{\kern0pt}r\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ x\ {\isasymsubseteq}\ above\ s\ x\ {\isasymunion}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ y\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}r{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ above\ r\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ y{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}s{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymnotin}\ above\ s\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}z\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ z\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ z{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}z\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ z{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}z\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ z\ {\isasymin}\ above\ r\ x\ {\isasymlongleftrightarrow}\ z\ {\isasymin}\ above\ s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasymin}\ above\ r\ x\ {\isasymlongleftrightarrow}\ y\ {\isasymin}\ above\ s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ y{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}s\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}mono{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ limited{\isacharunderscore}{\kern0pt}dest\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ member{\isacharunderscore}{\kern0pt}remove\isanewline
-\ \ \ \ \ \ \ \ \ \ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ remove{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}r\ y{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}s\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ S\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}limit\ A\ r\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ {\isasymor}\isanewline
-\ \ \ \ \ \ lifted\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ lifted\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ S\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ S\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ subset\ \isacommand{have}\isamarkupfalse%
-\ temp{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}rs{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{from}\isamarkupfalse%
-\ lifted\ subset\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ a{\isadigit{1}}\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ keep{\isacharunderscore}{\kern0pt}lift{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ a\ {\isasympreceq}\isactrlsub q\ x{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ u\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ in\ x\ {\isasympreceq}\isactrlsub u\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{1}}\ temp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymexists}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}worse{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isasymnot}{\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub s\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ subset\ lifted{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ subset\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ connex{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\ connex\ A\ r\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limited\ A\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymin}\ A\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymin}\ A\ {\isasymlongrightarrow}\ a\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub r\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Ball{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}raw\ connex{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ limit{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ aa\ {\isasymnotin}\ A\ {\isasymor}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r\ {\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex\ connex{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ limit{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ aa\ A\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ limit\ A\ r\ {\isasymor}\ a\ {\isasympreceq}\isactrlsub r\ aa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ in\ a\ {\isasympreceq}\isactrlsub q\ aa\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub q\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ connex{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ aa{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ s\ {\isasymor}\ {\isacharparenleft}{\kern0pt}aa{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ {\isasympreceq}\isactrlsub r\ x\ {\isasymand}\ a\ {\isasympreceq}\isactrlsub s\ x{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ x\ {\isasympreceq}\isactrlsub s\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\ a{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ r{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}s{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isacharparenright}{\kern0pt}\ {\isasymor}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ limit{\isadigit{2}}\ limit{\isadigit{1}}\ connex{\isadigit{2}}\ a{\isadigit{1}}\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ not{\isacharunderscore}{\kern0pt}worse\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ a\ {\isasympreceq}\isactrlsub q\ x{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ s\ in\ a\ {\isasympreceq}\isactrlsub q\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{1}}\ a{\isadigit{1}}{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ not{\isacharunderscore}{\kern0pt}worse\ DiffD{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}prefs{\isadigit{2}}\ connex{\isadigit{2}}\ limit{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isadigit{1}}\ connex\ connex{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}refl\ refl{\isacharunderscore}{\kern0pt}onD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\ limited\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ y{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ s{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}rs\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ a{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ eql{\isacharunderscore}{\kern0pt}rs\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}limit\ A\ s{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}limits\ limited{\isacharunderscore}{\kern0pt}dest\ subrelI\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ change{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ notInA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit\ A\ r\ {\isacharequal}{\kern0pt}\ limit\ A\ s{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ notInA\ subset\ subset{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}insert{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ x\ {\isasympreceq}\isactrlsub r\ y\ {\isasymlongleftrightarrow}\ x\ {\isasympreceq}\isactrlsub s\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}meson\ change\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ in{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ above{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ s\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}one\ lifted{\isacharunderscore}{\kern0pt}a\ above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}above\ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ s\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ s\ x\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}y\ {\isasymin}\ A{\isachardot}{\kern0pt}\ y\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ aa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ imp{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymlongrightarrow}\ aa\ {\isasymnotin}\ A\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ singletonD\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ r{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}connex\ A\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymexists}aa{\isachardot}{\kern0pt}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}aa{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasympreceq}\isactrlsub r\ a\ {\isasymand}\ {\isasymnot}\ a\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymand}\ aa\ {\isasymin}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\ a\ {\isasymin}\ A{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ limited\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymforall}aa{\isachardot}{\kern0pt}\ aa\ {\isasympreceq}\isactrlsub r\ a\ {\isasymor}\ a\ {\isasympreceq}\isactrlsub r\ aa\ {\isasymor}\ aa\ {\isasymnotin}\ A{\isacharparenright}{\kern0pt}\ {\isasymor}\ a\ {\isasymnotin}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\ limited\ A\ r\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ connex\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ eq{\isacharunderscore}{\kern0pt}exc{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}aa\ {\isasymnotin}\ A\ {\isasymor}\ aa\ {\isasympreceq}\isactrlsub r\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ above{\isacharunderscore}{\kern0pt}x\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ limited{\isacharunderscore}{\kern0pt}dest\ insertCI\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ eq{\isacharunderscore}{\kern0pt}exc{\isacharunderscore}{\kern0pt}a\ above{\isacharunderscore}{\kern0pt}one\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\ above{\isacharunderscore}{\kern0pt}x\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ mk{\isacharunderscore}{\kern0pt}disjoint{\isacharunderscore}{\kern0pt}insert\ insertE\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ r\ s\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\ asm{\isadigit{1}}\ assms\ calculation\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ member{\isacharunderscore}{\kern0pt}remove\ remove{\isacharunderscore}{\kern0pt}def\ insert{\isacharunderscore}{\kern0pt}absorb\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ y\ {\isasympreceq}\isactrlsub s\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffD{\isadigit{1}}\ lifted{\isacharunderscore}{\kern0pt}a\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}others{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ above\ s\ y\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm{\isadigit{2}}\ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ all{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}conv\ lifted{\isacharunderscore}{\kern0pt}a\ fin{\isacharunderscore}{\kern0pt}A\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one\ singleton{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}above\ r\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ r\ s\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ above{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ s\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ asm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ x\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ y\ \isakeyword{where}\ y{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ r\ y\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\ fin{\isacharunderscore}{\kern0pt}A\ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}empty\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}def\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ s\ y\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}y{\isacharbraceright}{\kern0pt}\ {\isasymor}\ above\ s\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}a\ fin{\isacharunderscore}{\kern0pt}A\ lifted{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}winner\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}b{\isachardot}{\kern0pt}\ above\ s\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ b\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ all{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}conv\ lifted{\isacharunderscore}{\kern0pt}a\ above{\isacharunderscore}{\kern0pt}x\ lifted{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A\ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\ above{\isacharunderscore}{\kern0pt}one{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}y\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ asm\ y\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/Preference_Relation.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%18=10%:%
-%:%34=12%:%
-%:%35=12%:%
-%:%36=13%:%
-%:%37=14%:%
-%:%46=17%:%
-%:%47=18%:%
-%:%56=20%:%
-%:%66=26%:%
-%:%67=26%:%
-%:%68=27%:%
-%:%69=28%:%
-%:%70=28%:%
-%:%71=29%:%
-%:%72=30%:%
-%:%73=31%:%
-%:%74=32%:%
-%:%75=32%:%
-%:%76=33%:%
-%:%77=34%:%
-%:%80=35%:%
-%:%84=35%:%
-%:%85=35%:%
-%:%86=36%:%
-%:%87=36%:%
-%:%92=36%:%
-%:%95=37%:%
-%:%96=38%:%
-%:%97=38%:%
-%:%98=39%:%
-%:%99=40%:%
-%:%102=41%:%
-%:%106=41%:%
-%:%107=41%:%
-%:%108=42%:%
-%:%109=42%:%
-%:%123=44%:%
-%:%133=46%:%
-%:%134=46%:%
-%:%135=47%:%
-%:%136=48%:%
-%:%137=49%:%
-%:%138=49%:%
-%:%139=50%:%
-%:%140=51%:%
-%:%141=52%:%
-%:%142=53%:%
-%:%149=54%:%
-%:%150=54%:%
-%:%151=55%:%
-%:%152=55%:%
-%:%153=56%:%
-%:%154=56%:%
-%:%155=57%:%
-%:%156=57%:%
-%:%157=58%:%
-%:%158=58%:%
-%:%159=59%:%
-%:%160=59%:%
-%:%161=60%:%
-%:%162=60%:%
-%:%163=61%:%
-%:%164=61%:%
-%:%165=62%:%
-%:%166=62%:%
-%:%167=62%:%
-%:%168=63%:%
-%:%169=63%:%
-%:%170=64%:%
-%:%171=64%:%
-%:%172=65%:%
-%:%173=65%:%
-%:%174=65%:%
-%:%175=66%:%
-%:%176=66%:%
-%:%177=67%:%
-%:%178=68%:%
-%:%179=68%:%
-%:%180=69%:%
-%:%195=71%:%
-%:%205=73%:%
-%:%206=73%:%
-%:%207=74%:%
-%:%208=75%:%
-%:%209=76%:%
-%:%210=76%:%
-%:%211=77%:%
-%:%214=78%:%
-%:%218=78%:%
-%:%219=78%:%
-%:%220=79%:%
-%:%221=79%:%
-%:%226=79%:%
-%:%229=80%:%
-%:%230=81%:%
-%:%231=81%:%
-%:%232=82%:%
-%:%235=83%:%
-%:%239=83%:%
-%:%240=83%:%
-%:%241=84%:%
-%:%242=84%:%
-%:%247=84%:%
-%:%250=85%:%
-%:%251=86%:%
-%:%252=86%:%
-%:%253=87%:%
-%:%254=88%:%
-%:%255=89%:%
-%:%256=89%:%
-%:%257=90%:%
-%:%258=91%:%
-%:%259=92%:%
-%:%260=92%:%
-%:%261=93%:%
-%:%262=94%:%
-%:%269=95%:%
-%:%270=95%:%
-%:%271=96%:%
-%:%272=96%:%
-%:%273=97%:%
-%:%274=97%:%
-%:%275=98%:%
-%:%276=98%:%
-%:%277=99%:%
-%:%278=99%:%
-%:%279=100%:%
-%:%280=100%:%
-%:%281=101%:%
-%:%282=102%:%
-%:%283=102%:%
-%:%284=103%:%
-%:%285=104%:%
-%:%286=104%:%
-%:%287=105%:%
-%:%288=105%:%
-%:%289=106%:%
-%:%290=106%:%
-%:%291=107%:%
-%:%292=107%:%
-%:%293=108%:%
-%:%294=108%:%
-%:%295=109%:%
-%:%301=109%:%
-%:%304=110%:%
-%:%305=111%:%
-%:%306=111%:%
-%:%307=112%:%
-%:%308=113%:%
-%:%311=114%:%
-%:%315=114%:%
-%:%316=114%:%
-%:%317=115%:%
-%:%318=115%:%
-%:%319=116%:%
-%:%320=116%:%
-%:%321=117%:%
-%:%322=118%:%
-%:%323=119%:%
-%:%324=119%:%
-%:%325=120%:%
-%:%326=121%:%
-%:%327=121%:%
-%:%328=122%:%
-%:%329=122%:%
-%:%330=123%:%
-%:%331=124%:%
-%:%332=124%:%
-%:%333=125%:%
-%:%334=125%:%
-%:%335=126%:%
-%:%336=126%:%
-%:%337=127%:%
-%:%338=128%:%
-%:%339=129%:%
-%:%340=129%:%
-%:%341=130%:%
-%:%342=131%:%
-%:%343=131%:%
-%:%344=132%:%
-%:%345=132%:%
-%:%346=133%:%
-%:%347=134%:%
-%:%348=134%:%
-%:%349=135%:%
-%:%350=135%:%
-%:%351=136%:%
-%:%352=136%:%
-%:%353=137%:%
-%:%354=138%:%
-%:%355=139%:%
-%:%356=139%:%
-%:%357=140%:%
-%:%358=141%:%
-%:%359=142%:%
-%:%360=143%:%
-%:%361=143%:%
-%:%362=144%:%
-%:%363=144%:%
-%:%364=145%:%
-%:%365=145%:%
-%:%366=146%:%
-%:%367=146%:%
-%:%368=147%:%
-%:%369=147%:%
-%:%370=148%:%
-%:%371=149%:%
-%:%372=149%:%
-%:%373=150%:%
-%:%374=150%:%
-%:%375=151%:%
-%:%376=151%:%
-%:%377=152%:%
-%:%383=152%:%
-%:%386=153%:%
-%:%387=154%:%
-%:%388=154%:%
-%:%389=155%:%
-%:%390=156%:%
-%:%391=157%:%
-%:%392=158%:%
-%:%393=159%:%
-%:%396=160%:%
-%:%400=160%:%
-%:%401=160%:%
-%:%402=161%:%
-%:%403=162%:%
-%:%404=162%:%
-%:%405=163%:%
-%:%406=163%:%
-%:%407=164%:%
-%:%408=165%:%
-%:%409=166%:%
-%:%410=166%:%
-%:%411=167%:%
-%:%412=168%:%
-%:%413=168%:%
-%:%414=169%:%
-%:%415=169%:%
-%:%416=170%:%
-%:%417=170%:%
-%:%418=171%:%
-%:%419=171%:%
-%:%420=172%:%
-%:%421=172%:%
-%:%422=173%:%
-%:%423=174%:%
-%:%424=175%:%
-%:%425=175%:%
-%:%426=176%:%
-%:%427=177%:%
-%:%428=177%:%
-%:%429=178%:%
-%:%430=178%:%
-%:%431=179%:%
-%:%432=179%:%
-%:%433=180%:%
-%:%434=180%:%
-%:%435=181%:%
-%:%436=181%:%
-%:%437=182%:%
-%:%438=183%:%
-%:%439=183%:%
-%:%440=184%:%
-%:%441=185%:%
-%:%442=185%:%
-%:%443=186%:%
-%:%444=186%:%
-%:%445=187%:%
-%:%446=187%:%
-%:%447=188%:%
-%:%448=188%:%
-%:%449=189%:%
-%:%450=189%:%
-%:%451=190%:%
-%:%452=190%:%
-%:%453=191%:%
-%:%454=191%:%
-%:%455=192%:%
-%:%456=192%:%
-%:%457=193%:%
-%:%458=193%:%
-%:%459=194%:%
-%:%460=194%:%
-%:%461=195%:%
-%:%462=195%:%
-%:%463=196%:%
-%:%464=196%:%
-%:%465=197%:%
-%:%466=197%:%
-%:%467=198%:%
-%:%468=199%:%
-%:%469=200%:%
-%:%470=200%:%
-%:%471=201%:%
-%:%472=202%:%
-%:%473=203%:%
-%:%474=204%:%
-%:%475=205%:%
-%:%476=205%:%
-%:%477=206%:%
-%:%478=206%:%
-%:%479=207%:%
-%:%480=207%:%
-%:%481=208%:%
-%:%482=208%:%
-%:%483=209%:%
-%:%484=209%:%
-%:%485=210%:%
-%:%486=210%:%
-%:%487=211%:%
-%:%488=211%:%
-%:%489=212%:%
-%:%490=212%:%
-%:%491=213%:%
-%:%497=213%:%
-%:%500=214%:%
-%:%501=215%:%
-%:%502=215%:%
-%:%505=216%:%
-%:%509=216%:%
-%:%510=216%:%
-%:%511=217%:%
-%:%512=217%:%
-%:%517=217%:%
-%:%520=218%:%
-%:%521=219%:%
-%:%522=219%:%
-%:%523=220%:%
-%:%524=221%:%
-%:%525=222%:%
-%:%526=223%:%
-%:%529=224%:%
-%:%533=224%:%
-%:%534=224%:%
-%:%535=225%:%
-%:%536=225%:%
-%:%537=226%:%
-%:%538=226%:%
-%:%539=227%:%
-%:%540=227%:%
-%:%541=228%:%
-%:%542=229%:%
-%:%543=230%:%
-%:%544=231%:%
-%:%545=232%:%
-%:%546=232%:%
-%:%547=233%:%
-%:%548=234%:%
-%:%549=235%:%
-%:%550=236%:%
-%:%551=236%:%
-%:%552=237%:%
-%:%553=237%:%
-%:%554=238%:%
-%:%555=238%:%
-%:%556=239%:%
-%:%557=239%:%
-%:%558=240%:%
-%:%559=241%:%
-%:%560=241%:%
-%:%561=242%:%
-%:%562=242%:%
-%:%563=243%:%
-%:%564=243%:%
-%:%565=244%:%
-%:%566=244%:%
-%:%567=245%:%
-%:%568=245%:%
-%:%569=246%:%
-%:%570=246%:%
-%:%571=247%:%
-%:%572=247%:%
-%:%573=248%:%
-%:%579=248%:%
-%:%582=249%:%
-%:%583=250%:%
-%:%584=250%:%
-%:%585=251%:%
-%:%586=252%:%
-%:%587=253%:%
-%:%588=254%:%
-%:%591=255%:%
-%:%595=255%:%
-%:%596=255%:%
-%:%597=256%:%
-%:%598=256%:%
-%:%603=256%:%
-%:%606=257%:%
-%:%607=258%:%
-%:%608=258%:%
-%:%609=259%:%
-%:%610=260%:%
-%:%611=261%:%
-%:%612=262%:%
-%:%615=263%:%
-%:%619=263%:%
-%:%620=263%:%
-%:%621=264%:%
-%:%622=264%:%
-%:%623=265%:%
-%:%624=265%:%
-%:%625=266%:%
-%:%626=267%:%
-%:%627=268%:%
-%:%628=269%:%
-%:%629=269%:%
-%:%630=270%:%
-%:%631=271%:%
-%:%632=272%:%
-%:%633=273%:%
-%:%634=274%:%
-%:%635=275%:%
-%:%636=275%:%
-%:%637=276%:%
-%:%638=276%:%
-%:%639=277%:%
-%:%640=277%:%
-%:%641=278%:%
-%:%647=278%:%
-%:%650=279%:%
-%:%651=280%:%
-%:%652=280%:%
-%:%653=281%:%
-%:%654=282%:%
-%:%655=283%:%
-%:%656=284%:%
-%:%659=285%:%
-%:%663=285%:%
-%:%664=285%:%
-%:%665=286%:%
-%:%666=287%:%
-%:%667=288%:%
-%:%668=289%:%
-%:%669=290%:%
-%:%670=290%:%
-%:%675=290%:%
-%:%678=291%:%
-%:%679=292%:%
-%:%680=292%:%
-%:%681=293%:%
-%:%682=294%:%
-%:%683=295%:%
-%:%684=296%:%
-%:%685=297%:%
-%:%688=298%:%
-%:%692=298%:%
-%:%693=298%:%
-%:%694=299%:%
-%:%695=299%:%
-%:%700=299%:%
-%:%703=300%:%
-%:%704=301%:%
-%:%705=301%:%
-%:%706=302%:%
-%:%707=303%:%
-%:%710=304%:%
-%:%714=304%:%
-%:%715=304%:%
-%:%716=305%:%
-%:%717=305%:%
-%:%722=305%:%
-%:%725=306%:%
-%:%726=307%:%
-%:%727=307%:%
-%:%728=308%:%
-%:%729=309%:%
-%:%730=310%:%
-%:%731=311%:%
-%:%732=312%:%
-%:%735=313%:%
-%:%739=313%:%
-%:%740=313%:%
-%:%741=314%:%
-%:%742=314%:%
-%:%747=314%:%
-%:%750=315%:%
-%:%751=316%:%
-%:%752=316%:%
-%:%753=317%:%
-%:%754=318%:%
-%:%757=319%:%
-%:%761=319%:%
-%:%762=319%:%
-%:%763=320%:%
-%:%764=321%:%
-%:%765=321%:%
-%:%770=321%:%
-%:%773=322%:%
-%:%774=323%:%
-%:%775=323%:%
-%:%776=324%:%
-%:%783=325%:%
-%:%784=325%:%
-%:%785=326%:%
-%:%786=326%:%
-%:%787=327%:%
-%:%788=327%:%
-%:%789=328%:%
-%:%790=328%:%
-%:%791=329%:%
-%:%792=329%:%
-%:%793=330%:%
-%:%794=330%:%
-%:%795=331%:%
-%:%796=331%:%
-%:%797=332%:%
-%:%798=332%:%
-%:%799=333%:%
-%:%800=333%:%
-%:%801=333%:%
-%:%802=334%:%
-%:%803=334%:%
-%:%804=335%:%
-%:%805=336%:%
-%:%806=336%:%
-%:%807=337%:%
-%:%808=337%:%
-%:%809=337%:%
-%:%810=338%:%
-%:%811=338%:%
-%:%812=339%:%
-%:%813=339%:%
-%:%814=340%:%
-%:%829=342%:%
-%:%839=344%:%
-%:%840=344%:%
-%:%841=345%:%
-%:%842=346%:%
-%:%843=347%:%
-%:%844=348%:%
-%:%847=349%:%
-%:%851=349%:%
-%:%852=349%:%
-%:%853=350%:%
-%:%854=350%:%
-%:%859=350%:%
-%:%862=351%:%
-%:%863=352%:%
-%:%864=352%:%
-%:%865=353%:%
-%:%866=354%:%
-%:%867=355%:%
-%:%868=356%:%
-%:%871=357%:%
-%:%875=357%:%
-%:%876=357%:%
-%:%877=358%:%
-%:%878=358%:%
-%:%883=358%:%
-%:%886=359%:%
-%:%887=360%:%
-%:%888=360%:%
-%:%889=361%:%
-%:%890=362%:%
-%:%891=363%:%
-%:%892=364%:%
-%:%893=365%:%
-%:%896=366%:%
-%:%900=366%:%
-%:%901=366%:%
-%:%902=367%:%
-%:%903=368%:%
-%:%904=369%:%
-%:%905=369%:%
-%:%910=369%:%
-%:%913=370%:%
-%:%914=371%:%
-%:%915=371%:%
-%:%916=372%:%
-%:%917=373%:%
-%:%918=374%:%
-%:%919=375%:%
-%:%922=376%:%
-%:%926=376%:%
-%:%927=376%:%
-%:%928=377%:%
-%:%929=377%:%
-%:%934=377%:%
-%:%937=378%:%
-%:%938=379%:%
-%:%939=379%:%
-%:%942=380%:%
-%:%946=380%:%
-%:%947=380%:%
-%:%952=380%:%
-%:%955=381%:%
-%:%956=382%:%
-%:%957=382%:%
-%:%958=383%:%
-%:%959=384%:%
-%:%960=385%:%
-%:%960=386%:%
-%:%961=387%:%
-%:%962=388%:%
-%:%965=389%:%
-%:%969=389%:%
-%:%970=389%:%
-%:%971=390%:%
-%:%972=390%:%
-%:%977=390%:%
-%:%980=391%:%
-%:%981=392%:%
-%:%982=392%:%
-%:%983=393%:%
-%:%984=394%:%
-%:%985=395%:%
-%:%986=396%:%
-%:%987=397%:%
-%:%988=398%:%
-%:%989=399%:%
-%:%992=400%:%
-%:%996=400%:%
-%:%997=400%:%
-%:%998=401%:%
-%:%999=401%:%
-%:%1000=402%:%
-%:%1001=403%:%
-%:%1002=403%:%
-%:%1007=403%:%
-%:%1010=404%:%
-%:%1011=405%:%
-%:%1012=405%:%
-%:%1013=406%:%
-%:%1014=407%:%
-%:%1015=408%:%
-%:%1016=409%:%
-%:%1023=410%:%
-%:%1024=410%:%
-%:%1025=411%:%
-%:%1026=411%:%
-%:%1027=412%:%
-%:%1028=412%:%
-%:%1029=413%:%
-%:%1030=414%:%
-%:%1031=414%:%
-%:%1032=415%:%
-%:%1033=415%:%
-%:%1034=416%:%
-%:%1035=417%:%
-%:%1036=418%:%
-%:%1037=418%:%
-%:%1038=419%:%
-%:%1039=419%:%
-%:%1040=420%:%
-%:%1041=420%:%
-%:%1042=421%:%
-%:%1043=421%:%
-%:%1044=422%:%
-%:%1045=422%:%
-%:%1046=423%:%
-%:%1047=423%:%
-%:%1048=423%:%
-%:%1049=424%:%
-%:%1050=424%:%
-%:%1051=425%:%
-%:%1052=425%:%
-%:%1053=426%:%
-%:%1054=426%:%
-%:%1055=427%:%
-%:%1056=427%:%
-%:%1057=428%:%
-%:%1058=429%:%
-%:%1059=429%:%
-%:%1060=430%:%
-%:%1061=430%:%
-%:%1062=431%:%
-%:%1063=431%:%
-%:%1064=432%:%
-%:%1065=432%:%
-%:%1066=433%:%
-%:%1067=433%:%
-%:%1068=434%:%
-%:%1069=434%:%
-%:%1070=435%:%
-%:%1071=435%:%
-%:%1072=436%:%
-%:%1073=436%:%
-%:%1074=437%:%
-%:%1075=437%:%
-%:%1076=438%:%
-%:%1077=439%:%
-%:%1078=439%:%
-%:%1079=439%:%
-%:%1080=440%:%
-%:%1081=440%:%
-%:%1082=441%:%
-%:%1083=442%:%
-%:%1084=442%:%
-%:%1085=443%:%
-%:%1086=443%:%
-%:%1087=443%:%
-%:%1088=444%:%
-%:%1089=444%:%
-%:%1090=445%:%
-%:%1091=446%:%
-%:%1092=446%:%
-%:%1093=447%:%
-%:%1094=447%:%
-%:%1095=448%:%
-%:%1096=448%:%
-%:%1097=449%:%
-%:%1098=450%:%
-%:%1099=451%:%
-%:%1100=451%:%
-%:%1101=452%:%
-%:%1102=452%:%
-%:%1103=452%:%
-%:%1104=453%:%
-%:%1105=453%:%
-%:%1106=454%:%
-%:%1107=454%:%
-%:%1108=455%:%
-%:%1109=455%:%
-%:%1110=456%:%
-%:%1111=456%:%
-%:%1112=457%:%
-%:%1113=458%:%
-%:%1114=458%:%
-%:%1115=459%:%
-%:%1116=460%:%
-%:%1117=461%:%
-%:%1118=461%:%
-%:%1119=462%:%
-%:%1120=462%:%
-%:%1121=463%:%
-%:%1122=463%:%
-%:%1123=464%:%
-%:%1124=465%:%
-%:%1125=465%:%
-%:%1126=466%:%
-%:%1127=466%:%
-%:%1128=467%:%
-%:%1129=467%:%
-%:%1130=468%:%
-%:%1131=469%:%
-%:%1132=469%:%
-%:%1133=470%:%
-%:%1134=470%:%
-%:%1135=471%:%
-%:%1136=471%:%
-%:%1137=472%:%
-%:%1138=472%:%
-%:%1139=473%:%
-%:%1140=473%:%
-%:%1141=474%:%
-%:%1142=474%:%
-%:%1143=475%:%
-%:%1144=475%:%
-%:%1145=476%:%
-%:%1146=476%:%
-%:%1147=477%:%
-%:%1148=478%:%
-%:%1149=478%:%
-%:%1150=479%:%
-%:%1151=479%:%
-%:%1152=480%:%
-%:%1153=480%:%
-%:%1154=481%:%
-%:%1155=482%:%
-%:%1156=482%:%
-%:%1157=483%:%
-%:%1158=483%:%
-%:%1159=484%:%
-%:%1160=484%:%
-%:%1161=485%:%
-%:%1162=486%:%
-%:%1163=486%:%
-%:%1164=487%:%
-%:%1165=487%:%
-%:%1166=488%:%
-%:%1167=489%:%
-%:%1168=489%:%
-%:%1169=490%:%
-%:%1170=490%:%
-%:%1171=491%:%
-%:%1172=491%:%
-%:%1173=492%:%
-%:%1174=493%:%
-%:%1175=493%:%
-%:%1176=494%:%
-%:%1177=494%:%
-%:%1178=495%:%
-%:%1179=495%:%
-%:%1180=495%:%
-%:%1181=496%:%
-%:%1182=497%:%
-%:%1183=497%:%
-%:%1184=498%:%
-%:%1185=499%:%
-%:%1186=499%:%
-%:%1187=500%:%
-%:%1188=500%:%
-%:%1189=500%:%
-%:%1190=501%:%
-%:%1191=501%:%
-%:%1192=502%:%
-%:%1193=502%:%
-%:%1194=503%:%
-%:%1195=503%:%
-%:%1196=503%:%
-%:%1197=504%:%
-%:%1198=505%:%
-%:%1199=505%:%
-%:%1200=506%:%
-%:%1201=506%:%
-%:%1202=507%:%
-%:%1203=507%:%
-%:%1204=508%:%
-%:%1205=508%:%
-%:%1206=509%:%
-%:%1207=509%:%
-%:%1208=510%:%
-%:%1209=510%:%
-%:%1210=511%:%
-%:%1211=511%:%
-%:%1212=512%:%
-%:%1213=512%:%
-%:%1214=513%:%
-%:%1215=513%:%
-%:%1216=513%:%
-%:%1217=514%:%
-%:%1218=514%:%
-%:%1219=515%:%
-%:%1220=516%:%
-%:%1221=516%:%
-%:%1222=517%:%
-%:%1223=517%:%
-%:%1224=517%:%
-%:%1225=518%:%
-%:%1226=518%:%
-%:%1227=519%:%
-%:%1228=519%:%
-%:%1229=520%:%
-%:%1230=520%:%
-%:%1231=521%:%
-%:%1232=521%:%
-%:%1233=522%:%
-%:%1234=522%:%
-%:%1235=523%:%
-%:%1236=523%:%
-%:%1237=524%:%
-%:%1238=524%:%
-%:%1239=525%:%
-%:%1240=525%:%
-%:%1241=526%:%
-%:%1242=526%:%
-%:%1243=527%:%
-%:%1244=528%:%
-%:%1245=529%:%
-%:%1246=529%:%
-%:%1247=530%:%
-%:%1248=530%:%
-%:%1249=531%:%
-%:%1250=531%:%
-%:%1251=532%:%
-%:%1252=532%:%
-%:%1253=533%:%
-%:%1256=536%:%
-%:%1257=537%:%
-%:%1258=537%:%
-%:%1259=538%:%
-%:%1260=538%:%
-%:%1261=539%:%
-%:%1262=539%:%
-%:%1263=540%:%
-%:%1264=541%:%
-%:%1265=541%:%
-%:%1266=542%:%
-%:%1267=542%:%
-%:%1268=543%:%
-%:%1269=543%:%
-%:%1270=544%:%
-%:%1271=544%:%
-%:%1272=545%:%
-%:%1273=545%:%
-%:%1274=546%:%
-%:%1275=547%:%
-%:%1276=547%:%
-%:%1277=548%:%
-%:%1278=548%:%
-%:%1279=549%:%
-%:%1280=549%:%
-%:%1281=550%:%
-%:%1283=552%:%
-%:%1284=553%:%
-%:%1285=553%:%
-%:%1286=554%:%
-%:%1287=554%:%
-%:%1288=555%:%
-%:%1289=555%:%
-%:%1290=556%:%
-%:%1292=558%:%
-%:%1293=559%:%
-%:%1294=559%:%
-%:%1295=560%:%
-%:%1296=560%:%
-%:%1297=561%:%
-%:%1298=561%:%
-%:%1299=562%:%
-%:%1300=562%:%
-%:%1301=563%:%
-%:%1302=564%:%
-%:%1303=565%:%
-%:%1304=565%:%
-%:%1305=566%:%
-%:%1306=566%:%
-%:%1307=567%:%
-%:%1308=568%:%
-%:%1309=568%:%
-%:%1310=569%:%
-%:%1311=569%:%
-%:%1312=570%:%
-%:%1313=571%:%
-%:%1314=571%:%
-%:%1315=572%:%
-%:%1316=572%:%
-%:%1317=573%:%
-%:%1318=573%:%
-%:%1319=574%:%
-%:%1320=574%:%
-%:%1321=575%:%
-%:%1322=575%:%
-%:%1323=576%:%
-%:%1324=576%:%
-%:%1325=577%:%
-%:%1326=577%:%
-%:%1327=578%:%
-%:%1328=578%:%
-%:%1329=579%:%
-%:%1330=579%:%
-%:%1331=580%:%
-%:%1332=580%:%
-%:%1333=581%:%
-%:%1334=581%:%
-%:%1335=582%:%
-%:%1336=583%:%
-%:%1337=583%:%
-%:%1338=584%:%
-%:%1339=584%:%
-%:%1340=585%:%
-%:%1341=585%:%
-%:%1342=586%:%
-%:%1343=587%:%
-%:%1344=588%:%
-%:%1345=588%:%
-%:%1346=589%:%
-%:%1347=589%:%
-%:%1348=589%:%
-%:%1349=590%:%
-%:%1350=590%:%
-%:%1351=591%:%
-%:%1352=592%:%
-%:%1353=592%:%
-%:%1354=593%:%
-%:%1355=593%:%
-%:%1356=593%:%
-%:%1357=594%:%
-%:%1358=594%:%
-%:%1359=595%:%
-%:%1360=595%:%
-%:%1361=596%:%
-%:%1362=596%:%
-%:%1363=597%:%
-%:%1364=597%:%
-%:%1365=598%:%
-%:%1366=598%:%
-%:%1367=599%:%
-%:%1368=599%:%
-%:%1369=600%:%
-%:%1370=600%:%
-%:%1371=601%:%
-%:%1372=601%:%
-%:%1373=602%:%
-%:%1374=602%:%
-%:%1375=603%:%
-%:%1376=603%:%
-%:%1377=604%:%
-%:%1378=604%:%
-%:%1379=605%:%
-%:%1380=605%:%
-%:%1381=606%:%
-%:%1382=606%:%
-%:%1383=607%:%
-%:%1384=608%:%
-%:%1385=609%:%
-%:%1386=609%:%
-%:%1387=610%:%
-%:%1393=610%:%
-%:%1396=611%:%
-%:%1397=612%:%
-%:%1398=612%:%
-%:%1399=613%:%
-%:%1400=614%:%
-%:%1401=615%:%
-%:%1402=616%:%
-%:%1403=617%:%
-%:%1410=618%:%
-%:%1411=618%:%
-%:%1412=619%:%
-%:%1413=619%:%
-%:%1414=620%:%
-%:%1415=620%:%
-%:%1416=621%:%
-%:%1417=621%:%
-%:%1418=622%:%
-%:%1419=622%:%
-%:%1420=622%:%
-%:%1421=623%:%
-%:%1422=624%:%
-%:%1423=625%:%
-%:%1424=625%:%
-%:%1425=626%:%
-%:%1426=626%:%
-%:%1427=627%:%
-%:%1428=627%:%
-%:%1429=627%:%
-%:%1430=628%:%
-%:%1431=628%:%
-%:%1432=629%:%
-%:%1433=629%:%
-%:%1434=630%:%
-%:%1435=630%:%
-%:%1436=630%:%
-%:%1437=631%:%
-%:%1438=631%:%
-%:%1439=632%:%
-%:%1440=632%:%
-%:%1441=633%:%
-%:%1447=633%:%
-%:%1450=634%:%
-%:%1451=635%:%
-%:%1452=635%:%
-%:%1453=636%:%
-%:%1454=637%:%
-%:%1457=638%:%
-%:%1461=638%:%
-%:%1462=638%:%
-%:%1463=639%:%
-%:%1464=639%:%
-%:%1478=641%:%
-%:%1488=643%:%
-%:%1489=643%:%
-%:%1490=644%:%
-%:%1491=645%:%
-%:%1493=647%:%
-%:%1494=648%:%
-%:%1495=649%:%
-%:%1496=649%:%
-%:%1497=650%:%
-%:%1498=651%:%
-%:%1499=652%:%
-%:%1500=653%:%
-%:%1501=654%:%
-%:%1502=654%:%
-%:%1503=655%:%
-%:%1504=656%:%
-%:%1507=657%:%
-%:%1511=657%:%
-%:%1512=657%:%
-%:%1517=657%:%
-%:%1520=658%:%
-%:%1521=659%:%
-%:%1522=659%:%
-%:%1523=660%:%
-%:%1524=661%:%
-%:%1531=662%:%
-%:%1532=662%:%
-%:%1533=663%:%
-%:%1534=663%:%
-%:%1535=663%:%
-%:%1536=664%:%
-%:%1537=665%:%
-%:%1538=666%:%
-%:%1539=666%:%
-%:%1540=667%:%
-%:%1541=667%:%
-%:%1542=668%:%
-%:%1543=668%:%
-%:%1544=669%:%
-%:%1550=669%:%
-%:%1553=670%:%
-%:%1554=671%:%
-%:%1555=671%:%
-%:%1556=672%:%
-%:%1557=673%:%
-%:%1564=674%:%
-%:%1565=674%:%
-%:%1566=675%:%
-%:%1567=675%:%
-%:%1568=676%:%
-%:%1569=677%:%
-%:%1570=677%:%
-%:%1571=678%:%
-%:%1572=679%:%
-%:%1573=680%:%
-%:%1574=681%:%
-%:%1575=682%:%
-%:%1576=683%:%
-%:%1577=683%:%
-%:%1578=684%:%
-%:%1579=684%:%
-%:%1580=685%:%
-%:%1581=685%:%
-%:%1582=686%:%
-%:%1583=686%:%
-%:%1584=687%:%
-%:%1585=687%:%
-%:%1586=688%:%
-%:%1587=688%:%
-%:%1588=689%:%
-%:%1589=689%:%
-%:%1590=690%:%
-%:%1591=690%:%
-%:%1592=691%:%
-%:%1593=692%:%
-%:%1594=693%:%
-%:%1595=693%:%
-%:%1596=694%:%
-%:%1597=694%:%
-%:%1598=695%:%
-%:%1599=696%:%
-%:%1600=696%:%
-%:%1601=697%:%
-%:%1602=697%:%
-%:%1603=698%:%
-%:%1604=698%:%
-%:%1605=699%:%
-%:%1606=700%:%
-%:%1607=700%:%
-%:%1608=701%:%
-%:%1609=701%:%
-%:%1610=702%:%
-%:%1611=702%:%
-%:%1612=703%:%
-%:%1613=703%:%
-%:%1614=704%:%
-%:%1615=704%:%
-%:%1616=704%:%
-%:%1617=705%:%
-%:%1618=706%:%
-%:%1619=706%:%
-%:%1620=707%:%
-%:%1621=707%:%
-%:%1622=708%:%
-%:%1623=709%:%
-%:%1624=709%:%
-%:%1625=710%:%
-%:%1626=710%:%
-%:%1627=711%:%
-%:%1628=712%:%
-%:%1629=712%:%
-%:%1630=713%:%
-%:%1631=713%:%
-%:%1632=714%:%
-%:%1633=714%:%
-%:%1634=715%:%
-%:%1635=716%:%
-%:%1636=716%:%
-%:%1637=717%:%
-%:%1638=717%:%
-%:%1639=718%:%
-%:%1640=718%:%
-%:%1641=719%:%
-%:%1642=720%:%
-%:%1643=720%:%
-%:%1644=721%:%
-%:%1645=721%:%
-%:%1646=722%:%
-%:%1647=722%:%
-%:%1648=723%:%
-%:%1649=724%:%
-%:%1650=724%:%
-%:%1651=725%:%
-%:%1652=725%:%
-%:%1653=726%:%
-%:%1654=726%:%
-%:%1655=727%:%
-%:%1656=728%:%
-%:%1657=728%:%
-%:%1658=729%:%
-%:%1659=729%:%
-%:%1660=730%:%
-%:%1661=730%:%
-%:%1662=731%:%
-%:%1663=732%:%
-%:%1664=733%:%
-%:%1665=733%:%
-%:%1666=734%:%
-%:%1667=734%:%
-%:%1668=735%:%
-%:%1669=735%:%
-%:%1670=736%:%
-%:%1671=737%:%
-%:%1672=737%:%
-%:%1673=738%:%
-%:%1679=738%:%
-%:%1682=739%:%
-%:%1683=740%:%
-%:%1684=740%:%
-%:%1685=741%:%
-%:%1686=742%:%
-%:%1687=743%:%
-%:%1688=744%:%
-%:%1695=745%:%
-%:%1696=745%:%
-%:%1697=746%:%
-%:%1698=746%:%
-%:%1699=747%:%
-%:%1700=747%:%
-%:%1701=748%:%
-%:%1702=748%:%
-%:%1703=749%:%
-%:%1704=749%:%
-%:%1705=750%:%
-%:%1706=750%:%
-%:%1707=751%:%
-%:%1708=752%:%
-%:%1709=753%:%
-%:%1710=753%:%
-%:%1711=754%:%
-%:%1712=754%:%
-%:%1713=755%:%
-%:%1714=755%:%
-%:%1715=756%:%
-%:%1716=756%:%
-%:%1717=757%:%
-%:%1718=757%:%
-%:%1719=758%:%
-%:%1720=759%:%
-%:%1721=759%:%
-%:%1722=760%:%
-%:%1723=760%:%
-%:%1724=761%:%
-%:%1725=761%:%
-%:%1726=762%:%
-%:%1727=762%:%
-%:%1728=763%:%
-%:%1729=763%:%
-%:%1730=764%:%
-%:%1731=765%:%
-%:%1732=765%:%
-%:%1733=766%:%
-%:%1734=766%:%
-%:%1735=767%:%
-%:%1736=767%:%
-%:%1737=768%:%
-%:%1738=768%:%
-%:%1739=769%:%
-%:%1740=769%:%
-%:%1741=770%:%
-%:%1742=770%:%
-%:%1743=771%:%
-%:%1744=772%:%
-%:%1745=772%:%
-%:%1746=773%:%
-%:%1747=773%:%
-%:%1748=774%:%
-%:%1749=774%:%
-%:%1750=775%:%
-%:%1751=775%:%
-%:%1752=776%:%
-%:%1753=776%:%
-%:%1754=777%:%
-%:%1755=778%:%
-%:%1756=779%:%
-%:%1757=780%:%
-%:%1758=780%:%
-%:%1759=781%:%
-%:%1760=781%:%
-%:%1761=782%:%
-%:%1767=782%:%
-%:%1770=783%:%
-%:%1771=784%:%
-%:%1772=784%:%
-%:%1773=785%:%
-%:%1774=786%:%
-%:%1777=787%:%
-%:%1781=787%:%
-%:%1782=787%:%
-%:%1783=788%:%
-%:%1784=788%:%
-%:%1785=789%:%
-%:%1786=789%:%
-%:%1787=790%:%
-%:%1788=791%:%
-%:%1789=791%:%
-%:%1790=792%:%
-%:%1791=793%:%
-%:%1792=793%:%
-%:%1793=794%:%
-%:%1794=794%:%
-%:%1795=795%:%
-%:%1796=795%:%
-%:%1797=796%:%
-%:%1798=796%:%
-%:%1799=797%:%
-%:%1800=798%:%
-%:%1801=798%:%
-%:%1802=799%:%
-%:%1803=799%:%
-%:%1804=800%:%
-%:%1805=800%:%
-%:%1806=801%:%
-%:%1807=801%:%
-%:%1808=802%:%
-%:%1809=802%:%
-%:%1810=803%:%
-%:%1811=804%:%
-%:%1812=804%:%
-%:%1813=805%:%
-%:%1814=805%:%
-%:%1815=806%:%
-%:%1816=807%:%
-%:%1817=807%:%
-%:%1818=808%:%
-%:%1819=809%:%
-%:%1820=809%:%
-%:%1821=810%:%
-%:%1822=810%:%
-%:%1823=811%:%
-%:%1824=812%:%
-%:%1825=812%:%
-%:%1826=813%:%
-%:%1827=814%:%
-%:%1828=814%:%
-%:%1829=815%:%
-%:%1830=815%:%
-%:%1831=816%:%
-%:%1832=817%:%
-%:%1833=817%:%
-%:%1834=818%:%
-%:%1835=819%:%
-%:%1836=819%:%
-%:%1837=820%:%
-%:%1838=820%:%
-%:%1839=821%:%
-%:%1840=821%:%
-%:%1841=822%:%
-%:%1842=823%:%
-%:%1843=823%:%
-%:%1844=824%:%
-%:%1845=824%:%
-%:%1846=825%:%
-%:%1847=825%:%
-%:%1848=826%:%
-%:%1849=827%:%
-%:%1850=827%:%
-%:%1851=828%:%
-%:%1857=828%:%
-%:%1860=829%:%
-%:%1861=830%:%
-%:%1862=830%:%
-%:%1863=831%:%
-%:%1864=832%:%
-%:%1865=833%:%
-%:%1866=834%:%
-%:%1873=835%:%
-%:%1874=835%:%
-%:%1875=836%:%
-%:%1876=836%:%
-%:%1877=837%:%
-%:%1878=837%:%
-%:%1879=838%:%
-%:%1880=839%:%
-%:%1881=840%:%
-%:%1882=840%:%
-%:%1883=841%:%
-%:%1884=841%:%
-%:%1885=842%:%
-%:%1886=842%:%
-%:%1887=843%:%
-%:%1888=843%:%
-%:%1889=844%:%
-%:%1890=844%:%
-%:%1891=845%:%
-%:%1892=845%:%
-%:%1893=846%:%
-%:%1894=846%:%
-%:%1895=847%:%
-%:%1896=847%:%
-%:%1897=848%:%
-%:%1898=848%:%
-%:%1899=849%:%
-%:%1900=850%:%
-%:%1901=851%:%
-%:%1902=852%:%
-%:%1903=852%:%
-%:%1904=853%:%
-%:%1905=853%:%
-%:%1906=854%:%
-%:%1907=854%:%
-%:%1908=855%:%
-%:%1909=855%:%
-%:%1910=856%:%
-%:%1916=856%:%
-%:%1919=857%:%
-%:%1920=858%:%
-%:%1921=858%:%
-%:%1922=859%:%
-%:%1923=860%:%
-%:%1924=861%:%
-%:%1925=862%:%
-%:%1926=863%:%
-%:%1927=864%:%
-%:%1934=865%:%
-%:%1935=865%:%
-%:%1936=866%:%
-%:%1937=866%:%
-%:%1938=866%:%
-%:%1939=867%:%
-%:%1940=868%:%
-%:%1941=868%:%
-%:%1942=869%:%
-%:%1943=869%:%
-%:%1944=869%:%
-%:%1945=870%:%
-%:%1946=871%:%
-%:%1947=871%:%
-%:%1948=872%:%
-%:%1949=872%:%
-%:%1950=873%:%
-%:%1951=874%:%
-%:%1952=875%:%
-%:%1953=875%:%
-%:%1954=876%:%
-%:%1955=876%:%
-%:%1956=877%:%
-%:%1957=877%:%
-%:%1958=878%:%
-%:%1959=878%:%
-%:%1960=879%:%
-%:%1961=879%:%
-%:%1962=880%:%
-%:%1963=880%:%
-%:%1964=881%:%
-%:%1965=881%:%
-%:%1966=882%:%
-%:%1967=883%:%
-%:%1968=883%:%
-%:%1969=884%:%
-%:%1970=884%:%
-%:%1971=884%:%
-%:%1972=885%:%
-%:%1973=886%:%
-%:%1974=886%:%
-%:%1975=887%:%
-%:%1976=887%:%
-%:%1977=888%:%
-%:%1978=888%:%
-%:%1979=888%:%
-%:%1980=888%:%
-%:%1981=889%:%
-%:%1982=890%:%
-%:%1983=891%:%
-%:%1984=891%:%
-%:%1985=892%:%
-%:%1986=892%:%
-%:%1987=893%:%
-%:%1988=893%:%
-%:%1989=893%:%
-%:%1990=894%:%
-%:%1991=894%:%
-%:%1992=895%:%
-%:%1993=895%:%
-%:%1994=896%:%
-%:%1995=896%:%
-%:%1996=897%:%
-%:%1997=897%:%
-%:%1998=898%:%
-%:%1999=899%:%
-%:%2000=899%:%
-%:%2001=900%:%
-%:%2002=901%:%
-%:%2003=901%:%
-%:%2004=902%:%
-%:%2005=902%:%
-%:%2006=902%:%
-%:%2007=903%:%
-%:%2008=904%:%
-%:%2009=904%:%
-%:%2010=905%:%
-%:%2011=905%:%
-%:%2012=906%:%
-%:%2013=906%:%
-%:%2014=906%:%
-%:%2015=907%:%
-%:%2016=908%:%
-%:%2017=908%:%
-%:%2018=909%:%
-%:%2019=910%:%
-%:%2020=910%:%
-%:%2021=911%:%
-%:%2022=911%:%
-%:%2023=911%:%
-%:%2024=912%:%
-%:%2026=914%:%
-%:%2027=915%:%
-%:%2028=915%:%
-%:%2029=916%:%
-%:%2030=916%:%
-%:%2031=917%:%
-%:%2035=921%:%
-%:%2036=922%:%
-%:%2037=922%:%
-%:%2038=923%:%
-%:%2039=923%:%
-%:%2040=924%:%
-%:%2041=924%:%
-%:%2042=925%:%
-%:%2043=926%:%
-%:%2044=926%:%
-%:%2045=927%:%
-%:%2046=927%:%
-%:%2047=928%:%
-%:%2048=928%:%
-%:%2049=929%:%
-%:%2052=932%:%
-%:%2053=933%:%
-%:%2054=933%:%
-%:%2055=934%:%
-%:%2056=934%:%
-%:%2057=935%:%
-%:%2058=935%:%
-%:%2059=936%:%
-%:%2062=939%:%
-%:%2063=940%:%
-%:%2064=940%:%
-%:%2065=941%:%
-%:%2066=941%:%
-%:%2067=941%:%
-%:%2068=942%:%
-%:%2069=943%:%
-%:%2070=943%:%
-%:%2071=944%:%
-%:%2072=944%:%
-%:%2073=945%:%
-%:%2074=945%:%
-%:%2075=946%:%
-%:%2076=947%:%
-%:%2077=947%:%
-%:%2078=948%:%
-%:%2079=948%:%
-%:%2080=949%:%
-%:%2081=950%:%
-%:%2082=950%:%
-%:%2083=951%:%
-%:%2084=951%:%
-%:%2085=952%:%
-%:%2086=952%:%
-%:%2087=953%:%
-%:%2088=954%:%
-%:%2089=955%:%
-%:%2090=955%:%
-%:%2091=956%:%
-%:%2092=956%:%
-%:%2093=956%:%
-%:%2094=957%:%
-%:%2095=958%:%
-%:%2096=958%:%
-%:%2097=959%:%
-%:%2098=959%:%
-%:%2099=960%:%
-%:%2100=960%:%
-%:%2101=960%:%
-%:%2102=961%:%
-%:%2103=962%:%
-%:%2104=962%:%
-%:%2105=963%:%
-%:%2106=963%:%
-%:%2107=964%:%
-%:%2108=964%:%
-%:%2109=964%:%
-%:%2110=965%:%
-%:%2111=966%:%
-%:%2112=966%:%
-%:%2113=967%:%
-%:%2114=967%:%
-%:%2115=968%:%
-%:%2116=968%:%
-%:%2117=968%:%
-%:%2118=969%:%
-%:%2119=970%:%
-%:%2120=970%:%
-%:%2121=971%:%
-%:%2122=971%:%
-%:%2123=972%:%
-%:%2124=972%:%
-%:%2125=973%:%
-%:%2126=973%:%
-%:%2127=974%:%
-%:%2128=974%:%
-%:%2129=975%:%
-%:%2130=975%:%
-%:%2131=976%:%
-%:%2132=976%:%
-%:%2133=977%:%
-%:%2134=977%:%
-%:%2135=977%:%
-%:%2136=978%:%
-%:%2137=979%:%
-%:%2138=979%:%
-%:%2139=980%:%
-%:%2140=980%:%
-%:%2141=981%:%
-%:%2142=981%:%
-%:%2143=982%:%
-%:%2144=982%:%
-%:%2145=983%:%
-%:%2146=983%:%
-%:%2147=984%:%
-%:%2153=984%:%
-%:%2156=985%:%
-%:%2157=986%:%
-%:%2158=986%:%
-%:%2159=987%:%
-%:%2160=988%:%
-%:%2161=989%:%
-%:%2162=990%:%
-%:%2163=991%:%
-%:%2170=992%:%
-%:%2171=992%:%
-%:%2172=993%:%
-%:%2173=993%:%
-%:%2174=994%:%
-%:%2175=994%:%
-%:%2176=995%:%
-%:%2177=995%:%
-%:%2178=996%:%
-%:%2179=996%:%
-%:%2180=997%:%
-%:%2181=997%:%
-%:%2182=998%:%
-%:%2183=998%:%
-%:%2184=999%:%
-%:%2190=999%:%
-%:%2193=1000%:%
-%:%2194=1001%:%
-%:%2195=1001%:%
-%:%2196=1002%:%
-%:%2197=1003%:%
-%:%2198=1004%:%
-%:%2199=1005%:%
-%:%2200=1006%:%
-%:%2207=1007%:%
-%:%2208=1007%:%
-%:%2209=1008%:%
-%:%2210=1008%:%
-%:%2211=1009%:%
-%:%2212=1009%:%
-%:%2213=1010%:%
-%:%2214=1010%:%
-%:%2215=1011%:%
-%:%2216=1012%:%
-%:%2217=1012%:%
-%:%2218=1013%:%
-%:%2219=1013%:%
-%:%2220=1014%:%
-%:%2221=1014%:%
-%:%2222=1015%:%
-%:%2223=1015%:%
-%:%2224=1016%:%
-%:%2225=1016%:%
-%:%2226=1017%:%
-%:%2227=1017%:%
-%:%2228=1018%:%
-%:%2229=1018%:%
-%:%2230=1019%:%
-%:%2231=1019%:%
-%:%2232=1020%:%
-%:%2233=1020%:%
-%:%2234=1021%:%
-%:%2235=1021%:%
-%:%2236=1022%:%
-%:%2237=1022%:%
-%:%2238=1023%:%
-%:%2239=1023%:%
-%:%2240=1024%:%
-%:%2241=1024%:%
-%:%2242=1025%:%
-%:%2243=1025%:%
-%:%2244=1026%:%
-%:%2245=1026%:%
-%:%2246=1027%:%
-%:%2247=1027%:%
-%:%2248=1028%:%
-%:%2249=1028%:%
-%:%2250=1028%:%
-%:%2251=1029%:%
-%:%2256=1034%:%
-%:%2257=1035%:%
-%:%2258=1035%:%
-%:%2259=1036%:%
-%:%2260=1036%:%
-%:%2261=1037%:%
-%:%2262=1037%:%
-%:%2263=1037%:%
-%:%2264=1038%:%
-%:%2265=1039%:%
-%:%2266=1039%:%
-%:%2267=1040%:%
-%:%2268=1040%:%
-%:%2269=1041%:%
-%:%2270=1041%:%
-%:%2271=1041%:%
-%:%2272=1042%:%
-%:%2273=1042%:%
-%:%2274=1043%:%
-%:%2275=1044%:%
-%:%2276=1044%:%
-%:%2277=1045%:%
-%:%2278=1045%:%
-%:%2279=1046%:%
-%:%2280=1046%:%
-%:%2281=1047%:%
-%:%2282=1048%:%
-%:%2283=1049%:%
-%:%2284=1049%:%
-%:%2285=1050%:%
-%:%2286=1050%:%
-%:%2287=1051%:%
-%:%2288=1051%:%
-%:%2289=1051%:%
-%:%2290=1052%:%
-%:%2291=1052%:%
-%:%2292=1053%:%
-%:%2293=1053%:%
-%:%2294=1054%:%
-%:%2295=1054%:%
-%:%2296=1054%:%
-%:%2297=1055%:%
-%:%2298=1055%:%
-%:%2299=1056%:%
-%:%2300=1057%:%
-%:%2301=1058%:%
-%:%2302=1058%:%
-%:%2303=1059%:%
-%:%2304=1059%:%
-%:%2305=1059%:%
-%:%2306=1060%:%
-%:%2307=1060%:%
-%:%2308=1061%:%
-%:%2309=1061%:%
-%:%2310=1062%:%
-%:%2311=1062%:%
-%:%2312=1063%:%
-%:%2313=1063%:%
-%:%2314=1064%:%
-%:%2315=1064%:%
-%:%2316=1065%:%
-%:%2317=1065%:%
-%:%2318=1066%:%
-%:%2319=1066%:%
-%:%2320=1067%:%
-%:%2321=1068%:%
-%:%2322=1068%:%
-%:%2323=1069%:%
-%:%2324=1069%:%
-%:%2325=1070%:%
-%:%2326=1070%:%
-%:%2327=1071%:%
-%:%2328=1071%:%
-%:%2329=1072%:%
-%:%2335=1072%:%
-%:%2338=1073%:%
-%:%2339=1074%:%
-%:%2340=1074%:%
-%:%2341=1075%:%
-%:%2342=1076%:%
-%:%2343=1077%:%
-%:%2344=1078%:%
-%:%2345=1079%:%
-%:%2348=1080%:%
-%:%2352=1080%:%
-%:%2353=1080%:%
-%:%2354=1081%:%
-%:%2355=1081%:%
-%:%2360=1081%:%
-%:%2363=1082%:%
-%:%2364=1083%:%
-%:%2365=1083%:%
-%:%2366=1084%:%
-%:%2367=1085%:%
-%:%2368=1086%:%
-%:%2369=1087%:%
-%:%2370=1088%:%
-%:%2371=1089%:%
-%:%2378=1090%:%
-%:%2379=1090%:%
-%:%2380=1091%:%
-%:%2381=1091%:%
-%:%2382=1092%:%
-%:%2383=1092%:%
-%:%2384=1092%:%
-%:%2385=1093%:%
-%:%2386=1093%:%
-%:%2387=1094%:%
-%:%2388=1095%:%
-%:%2389=1095%:%
-%:%2390=1096%:%
-%:%2391=1096%:%
-%:%2392=1097%:%
-%:%2393=1097%:%
-%:%2394=1098%:%
-%:%2395=1098%:%
-%:%2396=1099%:%
-%:%2397=1099%:%
-%:%2398=1099%:%
-%:%2399=1100%:%
-%:%2400=1100%:%
-%:%2401=1101%:%
-%:%2402=1102%:%
-%:%2403=1102%:%
-%:%2404=1103%:%
-%:%2405=1103%:%
-%:%2406=1103%:%
-%:%2407=1104%:%
-%:%2408=1104%:%
-%:%2409=1105%:%
-%:%2410=1105%:%
-%:%2411=1106%:%
-%:%2412=1106%:%
-%:%2413=1106%:%
-%:%2414=1107%:%
-%:%2415=1107%:%
-%:%2416=1108%:%
-%:%2417=1108%:%
-%:%2418=1109%:%
-%:%2419=1109%:%
-%:%2420=1109%:%
-%:%2421=1110%:%
-%:%2422=1110%:%
-%:%2423=1111%:%
-%:%2429=1111%:%
-%:%2434=1112%:%
-%:%2439=1113%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Profile.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Profile.tex
deleted file mode 100644
index b370d59..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Profile.tex
+++ /dev/null
@@ -1,2959 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Profile}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-\isanewline
-%
-\isamarkupsection{Preference Profile%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Profile\isanewline
-\ \ \isakeyword{imports}\ Preference{\isacharunderscore}{\kern0pt}Relation\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-Preference profiles denote the decisions made by the individual voters on
-the eligible alternatives. They are represented in the form of one preference
-relation (e.g., selected on a ballot) per voter, collectively captured in a
-list of such preference relations.
-Unlike a the common preference profiles in the social-choice sense, the
-profiles described here considers only the (sub-)set of alternatives that are
-received.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Profile\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isacharparenright}{\kern0pt}\ list{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ profile\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}profile\ A\ p\ {\isasymequiv}\ {\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ profile{\isacharunderscore}{\kern0pt}set\ {\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p\ {\isasymequiv}\ {\isacharparenleft}{\kern0pt}{\isasymforall}b\ {\isasymin}\ {\isacharparenleft}{\kern0pt}set\ p{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ b{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ all{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}conv{\isacharunderscore}{\kern0pt}all{\isacharunderscore}{\kern0pt}nth\ profile{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymequiv}\ finite\ A\ {\isasymand}\ profile\ A\ p{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Preference Counts and Comparisons%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ Nil\ a\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}above\ p\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ a{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}equiv{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ p\ rule{\isacharcolon}{\kern0pt}\ rev{\isacharunderscore}{\kern0pt}induct{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}snoc\ a\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ base{\isacharunderscore}{\kern0pt}case{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ size{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}length\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharless}{\kern0pt}length\ p{\isachardot}{\kern0pt}\ p{\isacharbang}{\kern0pt}i\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nth{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ in\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ q\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}above\ r\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ one{\isacharunderscore}{\kern0pt}ballot{\isacharunderscore}{\kern0pt}equiv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ size{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nth{\isacharunderscore}{\kern0pt}Cons{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}induct{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isacharplus}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}above\ a\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ then\ {\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}\ else\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Collect{\isacharunderscore}{\kern0pt}conv{\isacharunderscore}{\kern0pt}if{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ shift{\isacharunderscore}{\kern0pt}idx{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ xaa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymnoteq}\ length\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xaa\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymnoteq}\ length\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ insertI{\isadigit{1}}\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ xaa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymin}\ above\ a\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ length\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ xaa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ xb\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymin}\ above\ a\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymnoteq}\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xb\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xb\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ xaa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymin}\ above\ a\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ insertI{\isadigit{1}}\ less{\isacharunderscore}{\kern0pt}antisym\ nth{\isacharunderscore}{\kern0pt}append\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ above\ a\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ length\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ insertI{\isadigit{1}}\ less{\isacharunderscore}{\kern0pt}antisym\ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ xb\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ above\ a\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xb\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xb\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ insertI{\isadigit{1}}\ less{\isacharunderscore}{\kern0pt}antisym\ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\ singletonD\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ above\ a\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ insertI{\isadigit{1}}\ less{\isacharunderscore}{\kern0pt}antisym\ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ xaa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ length\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xaa\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xaa\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nth{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ length\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nth{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}n{\isachardot}{\kern0pt}\ n\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}n{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}n{\isachardot}{\kern0pt}\ n\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ f{\isadigit{2}}\ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}disjoint\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq\ shift{\isacharunderscore}{\kern0pt}idx{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}induct{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ {\isacharplus}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ p{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ aa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ {\isacharplus}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}aa{\isacharhash}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}aa{\isacharhash}{\kern0pt}p{\isacharparenright}{\kern0pt}\ x\ {\isacharplus}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}induct\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}induct\isanewline
-\ \ \ \ \ \ \ \ \ \ base{\isacharunderscore}{\kern0pt}case\ one{\isacharunderscore}{\kern0pt}ballot{\isacharunderscore}{\kern0pt}equiv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ prefer{\isacharunderscore}{\kern0pt}count\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ Nil\ x\ y\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ y\ {\isasympreceq}\isactrlsub p\ x\ then\ {\isadigit{1}}\ else\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ x\ y{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}equiv{\isacharbrackleft}{\kern0pt}code{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ p\ rule{\isacharcolon}{\kern0pt}\ rev{\isacharunderscore}{\kern0pt}induct{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}snoc\ a\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ base{\isacharunderscore}{\kern0pt}case{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ size{\isacharunderscore}{\kern0pt}one{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}length\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharless}{\kern0pt}length\ p{\isachardot}{\kern0pt}\ p{\isacharbang}{\kern0pt}i\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nth{\isacharunderscore}{\kern0pt}append{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ q\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ in\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ q\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ one{\isacharunderscore}{\kern0pt}ballot{\isacharunderscore}{\kern0pt}equiv{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ size{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ nth{\isacharunderscore}{\kern0pt}Cons{\isacharprime}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}induct{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharplus}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x\ y{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}if\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ a{\isacharparenright}{\kern0pt}\ then\ {\isacharbraceleft}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}\ else\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Collect{\isacharunderscore}{\kern0pt}conv{\isacharunderscore}{\kern0pt}if{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ shift{\isacharunderscore}{\kern0pt}idx{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymnoteq}\ length\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}xa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ length\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}xa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ nth{\isacharunderscore}{\kern0pt}append{\isacharunderscore}{\kern0pt}length\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isacharless}{\kern0pt}\ length\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}xa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}xa{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}antisym\ p{\isacharunderscore}{\kern0pt}pos{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}ps\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}n{\isachardot}{\kern0pt}\ n\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}n{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharbraceleft}{\kern0pt}n{\isachardot}{\kern0pt}\ n\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}{\isadigit{0}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ f{\isadigit{2}}\ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}disjoint\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq\ shift{\isacharunderscore}{\kern0pt}idx{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}induct{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y\ {\isacharplus}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}\ x\ y{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ assm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ Suc\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ p{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ assm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induction\ p{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharat}{\kern0pt}{\isacharbrackleft}{\kern0pt}a{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ x\ y{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}induct\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}induct\isanewline
-\ \ \ \ \ \ \ \ \ \ base{\isacharunderscore}{\kern0pt}case\ one{\isacharunderscore}{\kern0pt}ballot{\isacharunderscore}{\kern0pt}equiv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}compr{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}\ f\ x\ {\isacharbar}{\kern0pt}\ x\ {\isachardot}{\kern0pt}\ x\ {\isasymin}\ S\ {\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ f\ {\isacharbackquote}{\kern0pt}\ S{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}compr{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharbar}{\kern0pt}\ y\ {\isachardot}{\kern0pt}\ y\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x{\isacharparenright}{\kern0pt}\ {\isacharbackquote}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}y\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{assumes}\ neq{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ y{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ y\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ length\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prof\ profile{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ connex\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}def\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ y{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ antisym\ \ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}y{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ antisymD\ neq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat\ {\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Collect{\isacharunderscore}{\kern0pt}mono{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{3}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card{\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub r\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{2}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card{\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}let\ r\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ in\ {\isacharparenleft}{\kern0pt}x\ {\isasympreceq}\isactrlsub r\ y{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{3}}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharequal}{\kern0pt}\ length\ p\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ y\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{finally}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{2}}{\isadigit{0}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{3}}{\isadigit{0}}{\isachardoublequoteclose}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}sym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \isakeyword{assumes}\ p{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ a\ x\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ neq{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{assumes}\ neq{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymnoteq}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x\ {\isasymge}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ prof\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ neq{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ a\ x\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ prof\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ neq{\isadigit{2}}\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ b\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}\ {\isasymge}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ calculation\ p{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}length\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ diff{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}eq\ diff{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}self\ neq{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ pref{\isacharunderscore}{\kern0pt}count\ prof\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ b\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{3}}{\isachardoublequoteclose}\ calculation\ p{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ empty{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}p\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}\ x\ y{\isachardot}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ empty{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}p\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymforall}\ x\ y{\isachardot}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ p\ x\ y\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}incr{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}y\ {\isasympreceq}\isactrlsub p\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isacharplus}{\kern0pt}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ pref{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}smaller{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}constant{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ ps\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}y\ {\isasympreceq}\isactrlsub p\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}prefer{\isacharunderscore}{\kern0pt}count{\isacharunderscore}{\kern0pt}code\ {\isacharparenleft}{\kern0pt}p{\isacharhash}{\kern0pt}ps{\isacharparenright}{\kern0pt}\ x\ y\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ wins\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}wins\ x\ p\ y\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}prefer{\isacharunderscore}{\kern0pt}count\ p\ x\ y\ {\isachargreater}{\kern0pt}\ prefer{\isacharunderscore}{\kern0pt}count\ p\ y\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ wins{\isacharunderscore}{\kern0pt}antisym{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}wins\ a\ p\ b{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ b\ p\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ wins{\isacharunderscore}{\kern0pt}irreflex{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ wins\ w\ p\ w{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ wins{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Condorcet Winner%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ condorcet{\isacharunderscore}{\kern0pt}winner\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\ wins\ w\ p\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharunderscore}{\kern0pt}c{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ c{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ winner{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}w\ {\isacharequal}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}rule\ ccontr{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ assumption{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}w\ {\isasymnoteq}\ c{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}wins\ w\ p\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assumption\ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}iff\ winner{\isacharunderscore}{\kern0pt}c\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}\ wins\ c\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ wins{\isacharunderscore}{\kern0pt}antisym{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}c\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ c{\isacharunderscore}{\kern0pt}wins{\isacharunderscore}{\kern0pt}against{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}wins\ c\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}iff\ assumption\isanewline
-\ \ \ \ \ \ \ \ \ \ singletonD\ winner{\isacharunderscore}{\kern0pt}w\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ False\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ winner{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ w{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isasymnot}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}w\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique\ winner\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ cond{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}unique{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ w{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ condorcet{\isacharunderscore}{\kern0pt}winner\ A\ p\ a{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharcomma}{\kern0pt}\ safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}xa\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ xa{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}xa{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\ assm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ assm\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ assm\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}w{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isasymnoteq}\ w\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}wins\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}x\ x{\isacharunderscore}{\kern0pt}beats{\isacharunderscore}{\kern0pt}w\ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ w{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ xa{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}xa{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ \ w\ {\isasymin}\ A\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x\ {\isasymin}\ A\ {\isacharminus}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}w{\isacharcomma}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharbraceleft}{\kern0pt}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymin}\ p{\isacharbang}{\kern0pt}i{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ w{\isacharunderscore}{\kern0pt}wins\ insert{\isacharunderscore}{\kern0pt}Diff\ insert{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Limited Profile%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}profile\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ p{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}B\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}C\ {\isasymsubseteq}\ B{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ C\ p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ C\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ B\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ assms\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ profile\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ length{\isacharunderscore}{\kern0pt}map\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \ \ \ \ \ \ profile{\isacharunderscore}{\kern0pt}def\ subset\ infinite{\isacharunderscore}{\kern0pt}super\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}size{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ S\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ subset{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}length\ p\ {\isacharequal}{\kern0pt}\ length\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lifting Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{definition}\isamarkupfalse%
-\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\isanewline
-\ \ \ \ \ \ a\ {\isasymin}\ A\ {\isasymand}\ length\ p\ {\isacharequal}{\kern0pt}\ length\ q\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ lifted\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}lifted\ A\ p\ q\ a\ {\isasymequiv}\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymand}\isanewline
-\ \ \ \ \ \ a\ {\isasymin}\ A\ {\isasymand}\ length\ p\ {\isacharequal}{\kern0pt}\ length\ q\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymand}\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ p\ q\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ i\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ nat\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}ok{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}i\ {\isacharless}{\kern0pt}\ length\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ i{\isacharunderscore}{\kern0pt}ok\ lifted\ profile{\isacharunderscore}{\kern0pt}def\ trivial{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ change{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ p\ q\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ notInA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ equiv{\isacharunderscore}{\kern0pt}rel{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\ S\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ change\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ notInA\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit\ subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ p\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}limit\ A{\isacharparenright}{\kern0pt}\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ change\ equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ length{\isacharunderscore}{\kern0pt}map\ nth{\isacharunderscore}{\kern0pt}equalityI\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ lifted{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}lifted\ S\ p\ q\ a{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymsubseteq}\ S{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ A\ q\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ lifted\ A\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ A\ q{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ inA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ S\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ one{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ a\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}lifted\ subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ length{\isacharunderscore}{\kern0pt}map\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}equalityI\ nth{\isacharunderscore}{\kern0pt}map\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}mono{\isacharunderscore}{\kern0pt}tags{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ assm{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ A\ {\isacharparenleft}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}q\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ A\ q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}profile\ A\ {\isacharquery}{\kern0pt}p\ {\isasymand}\ profile\ A\ {\isacharquery}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound\ subset\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}length\ {\isacharquery}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}q{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ lifted\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\ i\ {\isacharless}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}p\ {\isasymand}\ Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assm\ lifted{\isacharunderscore}{\kern0pt}def\ length{\isacharunderscore}{\kern0pt}map\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}map\ one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}i{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}i\ {\isacharless}{\kern0pt}\ length\ {\isacharquery}{\kern0pt}p\ {\isasymand}\ {\isasymnot}Preference{\isacharunderscore}{\kern0pt}Relation{\isachardot}{\kern0pt}lifted\ A\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}p{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}q{\isacharbang}{\kern0pt}i{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ length{\isacharunderscore}{\kern0pt}map\ lifted\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ limit{\isacharunderscore}{\kern0pt}profile{\isachardot}{\kern0pt}simps\ nth{\isacharunderscore}{\kern0pt}map\ one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}lifted\ A\ {\isacharquery}{\kern0pt}p\ {\isacharquery}{\kern0pt}q\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted{\isacharunderscore}{\kern0pt}def\ inA\ lifted\ rev{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}subset\ subset\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymnotin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lifted\ negl{\isacharunderscore}{\kern0pt}diff{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}limit{\isacharunderscore}{\kern0pt}prof\ subset\isanewline
-\ \ \ \ \ \ \ \ \ \ lifted{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}equiv{\isacharunderscore}{\kern0pt}prof{\isacharunderscore}{\kern0pt}except{\isacharunderscore}{\kern0pt}a\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/Profile.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%13=6%:%
-%:%15=9%:%
-%:%31=11%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%48=21%:%
-%:%49=22%:%
-%:%58=24%:%
-%:%68=27%:%
-%:%69=27%:%
-%:%70=28%:%
-%:%71=32%:%
-%:%72=33%:%
-%:%73=33%:%
-%:%74=34%:%
-%:%75=35%:%
-%:%76=36%:%
-%:%77=36%:%
-%:%80=37%:%
-%:%84=37%:%
-%:%85=37%:%
-%:%90=37%:%
-%:%93=38%:%
-%:%94=39%:%
-%:%95=39%:%
-%:%96=40%:%
-%:%103=42%:%
-%:%113=48%:%
-%:%114=48%:%
-%:%115=49%:%
-%:%116=50%:%
-%:%117=51%:%
-%:%118=52%:%
-%:%119=52%:%
-%:%120=53%:%
-%:%121=54%:%
-%:%122=55%:%
-%:%123=56%:%
-%:%124=57%:%
-%:%125=57%:%
-%:%132=58%:%
-%:%133=58%:%
-%:%134=59%:%
-%:%135=59%:%
-%:%136=60%:%
-%:%137=60%:%
-%:%138=61%:%
-%:%139=62%:%
-%:%140=63%:%
-%:%141=63%:%
-%:%142=64%:%
-%:%143=65%:%
-%:%144=66%:%
-%:%145=66%:%
-%:%146=67%:%
-%:%147=67%:%
-%:%148=68%:%
-%:%149=68%:%
-%:%150=69%:%
-%:%151=70%:%
-%:%152=70%:%
-%:%153=71%:%
-%:%154=71%:%
-%:%155=72%:%
-%:%158=75%:%
-%:%159=76%:%
-%:%160=76%:%
-%:%161=77%:%
-%:%162=77%:%
-%:%163=78%:%
-%:%164=79%:%
-%:%165=79%:%
-%:%166=80%:%
-%:%167=80%:%
-%:%168=81%:%
-%:%169=81%:%
-%:%170=82%:%
-%:%171=83%:%
-%:%172=84%:%
-%:%173=84%:%
-%:%174=85%:%
-%:%175=85%:%
-%:%176=86%:%
-%:%177=87%:%
-%:%178=88%:%
-%:%179=88%:%
-%:%180=89%:%
-%:%181=89%:%
-%:%182=90%:%
-%:%183=91%:%
-%:%184=92%:%
-%:%185=92%:%
-%:%186=93%:%
-%:%187=93%:%
-%:%188=94%:%
-%:%190=96%:%
-%:%191=97%:%
-%:%192=97%:%
-%:%193=98%:%
-%:%194=98%:%
-%:%195=99%:%
-%:%196=100%:%
-%:%197=101%:%
-%:%198=101%:%
-%:%199=102%:%
-%:%200=103%:%
-%:%201=104%:%
-%:%202=105%:%
-%:%203=106%:%
-%:%204=106%:%
-%:%205=107%:%
-%:%206=107%:%
-%:%207=108%:%
-%:%208=108%:%
-%:%209=109%:%
-%:%210=109%:%
-%:%211=110%:%
-%:%212=110%:%
-%:%213=111%:%
-%:%214=112%:%
-%:%215=112%:%
-%:%216=113%:%
-%:%217=114%:%
-%:%218=115%:%
-%:%219=116%:%
-%:%220=116%:%
-%:%221=117%:%
-%:%222=117%:%
-%:%223=118%:%
-%:%224=118%:%
-%:%225=119%:%
-%:%226=119%:%
-%:%227=120%:%
-%:%228=120%:%
-%:%229=121%:%
-%:%230=122%:%
-%:%231=123%:%
-%:%232=123%:%
-%:%233=124%:%
-%:%234=125%:%
-%:%235=126%:%
-%:%236=127%:%
-%:%237=128%:%
-%:%238=128%:%
-%:%239=129%:%
-%:%240=129%:%
-%:%241=130%:%
-%:%242=131%:%
-%:%243=131%:%
-%:%244=132%:%
-%:%245=132%:%
-%:%246=133%:%
-%:%247=133%:%
-%:%248=134%:%
-%:%249=135%:%
-%:%250=136%:%
-%:%251=137%:%
-%:%252=137%:%
-%:%253=138%:%
-%:%254=139%:%
-%:%255=140%:%
-%:%256=141%:%
-%:%257=142%:%
-%:%258=143%:%
-%:%259=143%:%
-%:%260=144%:%
-%:%261=144%:%
-%:%262=145%:%
-%:%263=146%:%
-%:%264=146%:%
-%:%265=147%:%
-%:%266=147%:%
-%:%267=148%:%
-%:%268=148%:%
-%:%269=149%:%
-%:%270=150%:%
-%:%271=151%:%
-%:%272=151%:%
-%:%273=152%:%
-%:%274=153%:%
-%:%275=154%:%
-%:%276=155%:%
-%:%277=156%:%
-%:%278=156%:%
-%:%279=157%:%
-%:%280=157%:%
-%:%281=158%:%
-%:%282=159%:%
-%:%283=159%:%
-%:%284=160%:%
-%:%285=160%:%
-%:%286=161%:%
-%:%287=161%:%
-%:%288=162%:%
-%:%289=163%:%
-%:%290=163%:%
-%:%291=164%:%
-%:%292=165%:%
-%:%293=166%:%
-%:%294=167%:%
-%:%295=167%:%
-%:%296=168%:%
-%:%297=168%:%
-%:%298=169%:%
-%:%299=169%:%
-%:%300=170%:%
-%:%301=170%:%
-%:%302=171%:%
-%:%303=171%:%
-%:%304=172%:%
-%:%305=173%:%
-%:%306=174%:%
-%:%307=174%:%
-%:%308=175%:%
-%:%309=176%:%
-%:%310=177%:%
-%:%311=178%:%
-%:%312=179%:%
-%:%313=179%:%
-%:%314=180%:%
-%:%315=180%:%
-%:%316=181%:%
-%:%317=182%:%
-%:%318=182%:%
-%:%319=183%:%
-%:%320=183%:%
-%:%321=184%:%
-%:%322=184%:%
-%:%323=185%:%
-%:%324=186%:%
-%:%325=186%:%
-%:%326=187%:%
-%:%327=188%:%
-%:%328=189%:%
-%:%329=190%:%
-%:%330=190%:%
-%:%331=191%:%
-%:%332=191%:%
-%:%333=192%:%
-%:%334=192%:%
-%:%335=193%:%
-%:%336=193%:%
-%:%337=194%:%
-%:%338=194%:%
-%:%339=195%:%
-%:%340=196%:%
-%:%341=197%:%
-%:%342=197%:%
-%:%343=198%:%
-%:%344=199%:%
-%:%345=200%:%
-%:%346=201%:%
-%:%347=201%:%
-%:%348=202%:%
-%:%349=202%:%
-%:%350=203%:%
-%:%351=203%:%
-%:%352=204%:%
-%:%353=204%:%
-%:%354=205%:%
-%:%355=206%:%
-%:%356=206%:%
-%:%357=207%:%
-%:%358=208%:%
-%:%359=209%:%
-%:%360=209%:%
-%:%361=210%:%
-%:%362=210%:%
-%:%363=211%:%
-%:%364=211%:%
-%:%365=212%:%
-%:%366=212%:%
-%:%367=213%:%
-%:%368=214%:%
-%:%369=214%:%
-%:%370=215%:%
-%:%371=215%:%
-%:%372=216%:%
-%:%373=217%:%
-%:%374=217%:%
-%:%375=218%:%
-%:%376=218%:%
-%:%377=219%:%
-%:%380=222%:%
-%:%381=223%:%
-%:%382=223%:%
-%:%383=224%:%
-%:%384=224%:%
-%:%385=225%:%
-%:%386=225%:%
-%:%387=226%:%
-%:%389=228%:%
-%:%390=229%:%
-%:%391=229%:%
-%:%392=230%:%
-%:%393=230%:%
-%:%394=231%:%
-%:%395=231%:%
-%:%396=232%:%
-%:%397=232%:%
-%:%398=233%:%
-%:%399=234%:%
-%:%400=235%:%
-%:%401=235%:%
-%:%402=236%:%
-%:%403=236%:%
-%:%404=237%:%
-%:%405=238%:%
-%:%406=239%:%
-%:%407=239%:%
-%:%408=240%:%
-%:%409=241%:%
-%:%410=242%:%
-%:%411=242%:%
-%:%412=243%:%
-%:%413=244%:%
-%:%414=245%:%
-%:%415=245%:%
-%:%416=246%:%
-%:%417=246%:%
-%:%418=247%:%
-%:%419=247%:%
-%:%420=248%:%
-%:%421=248%:%
-%:%422=249%:%
-%:%423=250%:%
-%:%424=250%:%
-%:%425=251%:%
-%:%431=251%:%
-%:%434=252%:%
-%:%435=253%:%
-%:%436=253%:%
-%:%437=254%:%
-%:%438=255%:%
-%:%439=256%:%
-%:%440=257%:%
-%:%441=257%:%
-%:%442=258%:%
-%:%443=259%:%
-%:%444=260%:%
-%:%445=261%:%
-%:%446=262%:%
-%:%447=262%:%
-%:%454=263%:%
-%:%455=263%:%
-%:%456=264%:%
-%:%457=264%:%
-%:%458=265%:%
-%:%459=265%:%
-%:%460=266%:%
-%:%461=267%:%
-%:%462=268%:%
-%:%463=268%:%
-%:%464=269%:%
-%:%465=270%:%
-%:%466=271%:%
-%:%467=271%:%
-%:%468=272%:%
-%:%469=272%:%
-%:%470=273%:%
-%:%471=273%:%
-%:%472=274%:%
-%:%473=275%:%
-%:%474=275%:%
-%:%475=276%:%
-%:%476=276%:%
-%:%477=277%:%
-%:%480=280%:%
-%:%481=281%:%
-%:%482=281%:%
-%:%483=282%:%
-%:%484=282%:%
-%:%485=283%:%
-%:%486=284%:%
-%:%487=284%:%
-%:%488=285%:%
-%:%489=285%:%
-%:%490=286%:%
-%:%491=286%:%
-%:%492=287%:%
-%:%493=288%:%
-%:%494=289%:%
-%:%495=289%:%
-%:%496=290%:%
-%:%497=290%:%
-%:%498=291%:%
-%:%499=292%:%
-%:%500=293%:%
-%:%501=293%:%
-%:%502=294%:%
-%:%503=294%:%
-%:%504=295%:%
-%:%505=296%:%
-%:%506=297%:%
-%:%507=297%:%
-%:%508=298%:%
-%:%509=298%:%
-%:%510=299%:%
-%:%512=301%:%
-%:%513=302%:%
-%:%514=302%:%
-%:%515=303%:%
-%:%516=303%:%
-%:%517=304%:%
-%:%518=305%:%
-%:%519=305%:%
-%:%520=306%:%
-%:%521=307%:%
-%:%522=308%:%
-%:%523=309%:%
-%:%524=309%:%
-%:%525=310%:%
-%:%526=310%:%
-%:%527=311%:%
-%:%528=311%:%
-%:%529=312%:%
-%:%530=312%:%
-%:%531=313%:%
-%:%532=313%:%
-%:%533=314%:%
-%:%534=315%:%
-%:%535=315%:%
-%:%536=316%:%
-%:%537=317%:%
-%:%538=318%:%
-%:%539=319%:%
-%:%540=319%:%
-%:%541=320%:%
-%:%542=320%:%
-%:%543=321%:%
-%:%544=321%:%
-%:%545=322%:%
-%:%546=322%:%
-%:%547=323%:%
-%:%548=323%:%
-%:%549=324%:%
-%:%550=325%:%
-%:%551=325%:%
-%:%552=326%:%
-%:%553=327%:%
-%:%554=328%:%
-%:%555=329%:%
-%:%556=329%:%
-%:%557=330%:%
-%:%558=330%:%
-%:%559=331%:%
-%:%560=331%:%
-%:%561=332%:%
-%:%562=332%:%
-%:%563=333%:%
-%:%564=333%:%
-%:%565=334%:%
-%:%566=335%:%
-%:%567=335%:%
-%:%568=336%:%
-%:%569=337%:%
-%:%570=338%:%
-%:%571=338%:%
-%:%572=339%:%
-%:%573=339%:%
-%:%574=340%:%
-%:%575=340%:%
-%:%576=341%:%
-%:%577=341%:%
-%:%578=342%:%
-%:%579=342%:%
-%:%580=343%:%
-%:%581=344%:%
-%:%582=344%:%
-%:%583=345%:%
-%:%584=345%:%
-%:%585=346%:%
-%:%586=347%:%
-%:%587=347%:%
-%:%588=348%:%
-%:%589=348%:%
-%:%590=349%:%
-%:%593=352%:%
-%:%594=353%:%
-%:%595=353%:%
-%:%596=354%:%
-%:%597=354%:%
-%:%598=355%:%
-%:%599=355%:%
-%:%600=356%:%
-%:%602=358%:%
-%:%603=359%:%
-%:%604=359%:%
-%:%605=360%:%
-%:%606=360%:%
-%:%607=361%:%
-%:%608=361%:%
-%:%609=362%:%
-%:%610=362%:%
-%:%611=363%:%
-%:%612=364%:%
-%:%613=365%:%
-%:%614=365%:%
-%:%615=366%:%
-%:%616=366%:%
-%:%617=367%:%
-%:%618=368%:%
-%:%619=368%:%
-%:%620=369%:%
-%:%621=370%:%
-%:%622=370%:%
-%:%623=371%:%
-%:%624=371%:%
-%:%625=372%:%
-%:%626=372%:%
-%:%627=373%:%
-%:%628=373%:%
-%:%629=374%:%
-%:%630=374%:%
-%:%631=375%:%
-%:%632=375%:%
-%:%633=376%:%
-%:%634=376%:%
-%:%635=377%:%
-%:%636=378%:%
-%:%637=378%:%
-%:%638=379%:%
-%:%639=380%:%
-%:%640=380%:%
-%:%641=381%:%
-%:%642=381%:%
-%:%643=382%:%
-%:%644=383%:%
-%:%645=383%:%
-%:%646=384%:%
-%:%647=384%:%
-%:%648=385%:%
-%:%649=385%:%
-%:%650=386%:%
-%:%651=386%:%
-%:%652=387%:%
-%:%653=387%:%
-%:%654=388%:%
-%:%655=388%:%
-%:%656=389%:%
-%:%657=389%:%
-%:%658=390%:%
-%:%659=391%:%
-%:%660=391%:%
-%:%661=392%:%
-%:%667=392%:%
-%:%670=393%:%
-%:%671=394%:%
-%:%672=394%:%
-%:%675=395%:%
-%:%679=395%:%
-%:%680=395%:%
-%:%685=395%:%
-%:%688=396%:%
-%:%689=397%:%
-%:%690=397%:%
-%:%691=398%:%
-%:%694=399%:%
-%:%698=399%:%
-%:%699=399%:%
-%:%704=399%:%
-%:%707=400%:%
-%:%708=401%:%
-%:%709=401%:%
-%:%710=402%:%
-%:%711=403%:%
-%:%712=404%:%
-%:%713=405%:%
-%:%714=406%:%
-%:%721=407%:%
-%:%722=407%:%
-%:%723=408%:%
-%:%724=408%:%
-%:%725=409%:%
-%:%726=409%:%
-%:%727=410%:%
-%:%728=410%:%
-%:%729=411%:%
-%:%731=413%:%
-%:%732=414%:%
-%:%733=414%:%
-%:%734=415%:%
-%:%735=415%:%
-%:%736=416%:%
-%:%737=416%:%
-%:%738=417%:%
-%:%739=417%:%
-%:%740=418%:%
-%:%741=418%:%
-%:%742=419%:%
-%:%743=419%:%
-%:%744=420%:%
-%:%745=420%:%
-%:%746=421%:%
-%:%747=422%:%
-%:%748=422%:%
-%:%749=423%:%
-%:%750=423%:%
-%:%751=424%:%
-%:%752=424%:%
-%:%753=424%:%
-%:%754=425%:%
-%:%755=426%:%
-%:%756=426%:%
-%:%757=427%:%
-%:%758=427%:%
-%:%759=428%:%
-%:%760=428%:%
-%:%761=429%:%
-%:%762=429%:%
-%:%763=430%:%
-%:%764=430%:%
-%:%765=431%:%
-%:%766=431%:%
-%:%767=432%:%
-%:%768=433%:%
-%:%769=433%:%
-%:%770=434%:%
-%:%771=434%:%
-%:%772=434%:%
-%:%773=435%:%
-%:%774=436%:%
-%:%775=437%:%
-%:%776=437%:%
-%:%777=438%:%
-%:%778=438%:%
-%:%779=439%:%
-%:%780=440%:%
-%:%781=441%:%
-%:%782=441%:%
-%:%783=442%:%
-%:%784=442%:%
-%:%785=443%:%
-%:%787=445%:%
-%:%788=446%:%
-%:%789=446%:%
-%:%790=447%:%
-%:%791=447%:%
-%:%792=448%:%
-%:%793=448%:%
-%:%794=449%:%
-%:%795=450%:%
-%:%796=451%:%
-%:%797=451%:%
-%:%798=452%:%
-%:%799=452%:%
-%:%800=453%:%
-%:%803=456%:%
-%:%804=457%:%
-%:%805=457%:%
-%:%806=458%:%
-%:%807=458%:%
-%:%808=459%:%
-%:%809=460%:%
-%:%810=460%:%
-%:%811=461%:%
-%:%812=461%:%
-%:%813=461%:%
-%:%814=462%:%
-%:%815=463%:%
-%:%816=464%:%
-%:%817=464%:%
-%:%818=465%:%
-%:%819=465%:%
-%:%820=466%:%
-%:%821=466%:%
-%:%822=466%:%
-%:%823=467%:%
-%:%824=468%:%
-%:%825=469%:%
-%:%826=469%:%
-%:%827=470%:%
-%:%828=470%:%
-%:%829=471%:%
-%:%830=471%:%
-%:%831=471%:%
-%:%832=472%:%
-%:%833=473%:%
-%:%834=473%:%
-%:%835=474%:%
-%:%836=474%:%
-%:%837=474%:%
-%:%838=475%:%
-%:%839=475%:%
-%:%840=476%:%
-%:%846=476%:%
-%:%849=477%:%
-%:%850=478%:%
-%:%851=478%:%
-%:%852=479%:%
-%:%853=480%:%
-%:%854=481%:%
-%:%855=482%:%
-%:%856=483%:%
-%:%857=484%:%
-%:%858=485%:%
-%:%859=486%:%
-%:%866=487%:%
-%:%867=487%:%
-%:%868=488%:%
-%:%869=488%:%
-%:%870=488%:%
-%:%871=489%:%
-%:%872=490%:%
-%:%873=490%:%
-%:%874=491%:%
-%:%875=491%:%
-%:%876=492%:%
-%:%877=492%:%
-%:%878=492%:%
-%:%879=492%:%
-%:%880=493%:%
-%:%881=494%:%
-%:%882=494%:%
-%:%883=495%:%
-%:%884=495%:%
-%:%885=496%:%
-%:%886=496%:%
-%:%887=497%:%
-%:%888=498%:%
-%:%889=498%:%
-%:%890=499%:%
-%:%891=499%:%
-%:%892=500%:%
-%:%893=500%:%
-%:%894=501%:%
-%:%895=502%:%
-%:%896=502%:%
-%:%897=503%:%
-%:%898=504%:%
-%:%899=504%:%
-%:%900=505%:%
-%:%901=505%:%
-%:%902=506%:%
-%:%903=506%:%
-%:%904=507%:%
-%:%905=507%:%
-%:%906=508%:%
-%:%907=508%:%
-%:%908=509%:%
-%:%909=509%:%
-%:%910=510%:%
-%:%916=510%:%
-%:%919=511%:%
-%:%920=512%:%
-%:%921=512%:%
-%:%922=513%:%
-%:%923=514%:%
-%:%926=515%:%
-%:%930=515%:%
-%:%931=515%:%
-%:%932=516%:%
-%:%933=516%:%
-%:%938=516%:%
-%:%941=517%:%
-%:%942=518%:%
-%:%943=518%:%
-%:%944=519%:%
-%:%945=520%:%
-%:%948=521%:%
-%:%952=521%:%
-%:%953=521%:%
-%:%954=522%:%
-%:%955=522%:%
-%:%960=522%:%
-%:%963=523%:%
-%:%964=524%:%
-%:%965=524%:%
-%:%966=525%:%
-%:%967=526%:%
-%:%968=527%:%
-%:%969=528%:%
-%:%972=529%:%
-%:%976=529%:%
-%:%977=529%:%
-%:%978=530%:%
-%:%979=530%:%
-%:%984=530%:%
-%:%987=531%:%
-%:%988=532%:%
-%:%989=532%:%
-%:%990=533%:%
-%:%991=534%:%
-%:%992=535%:%
-%:%993=536%:%
-%:%996=537%:%
-%:%1000=537%:%
-%:%1001=537%:%
-%:%1002=538%:%
-%:%1003=538%:%
-%:%1008=538%:%
-%:%1011=539%:%
-%:%1012=540%:%
-%:%1013=540%:%
-%:%1014=541%:%
-%:%1015=542%:%
-%:%1016=543%:%
-%:%1017=544%:%
-%:%1018=545%:%
-%:%1019=545%:%
-%:%1020=546%:%
-%:%1021=547%:%
-%:%1024=548%:%
-%:%1028=548%:%
-%:%1029=548%:%
-%:%1030=549%:%
-%:%1031=549%:%
-%:%1036=549%:%
-%:%1039=550%:%
-%:%1040=551%:%
-%:%1041=551%:%
-%:%1044=552%:%
-%:%1048=552%:%
-%:%1049=552%:%
-%:%1050=553%:%
-%:%1051=553%:%
-%:%1065=555%:%
-%:%1075=557%:%
-%:%1076=557%:%
-%:%1077=558%:%
-%:%1078=559%:%
-%:%1079=560%:%
-%:%1080=561%:%
-%:%1081=561%:%
-%:%1082=562%:%
-%:%1083=563%:%
-%:%1084=564%:%
-%:%1091=565%:%
-%:%1092=565%:%
-%:%1093=566%:%
-%:%1094=566%:%
-%:%1095=567%:%
-%:%1096=568%:%
-%:%1097=568%:%
-%:%1098=569%:%
-%:%1099=569%:%
-%:%1100=570%:%
-%:%1101=570%:%
-%:%1102=571%:%
-%:%1103=571%:%
-%:%1104=572%:%
-%:%1105=572%:%
-%:%1106=573%:%
-%:%1107=573%:%
-%:%1108=574%:%
-%:%1109=574%:%
-%:%1110=574%:%
-%:%1111=575%:%
-%:%1112=575%:%
-%:%1113=576%:%
-%:%1114=577%:%
-%:%1115=577%:%
-%:%1116=578%:%
-%:%1117=579%:%
-%:%1118=579%:%
-%:%1119=580%:%
-%:%1120=580%:%
-%:%1121=580%:%
-%:%1122=581%:%
-%:%1123=581%:%
-%:%1124=582%:%
-%:%1130=582%:%
-%:%1133=583%:%
-%:%1134=584%:%
-%:%1135=584%:%
-%:%1136=585%:%
-%:%1137=586%:%
-%:%1138=587%:%
-%:%1139=588%:%
-%:%1142=589%:%
-%:%1146=589%:%
-%:%1147=589%:%
-%:%1148=590%:%
-%:%1149=590%:%
-%:%1154=590%:%
-%:%1157=591%:%
-%:%1158=592%:%
-%:%1159=592%:%
-%:%1160=593%:%
-%:%1161=594%:%
-%:%1168=595%:%
-%:%1169=595%:%
-%:%1170=596%:%
-%:%1171=596%:%
-%:%1172=597%:%
-%:%1173=598%:%
-%:%1174=598%:%
-%:%1175=599%:%
-%:%1176=600%:%
-%:%1177=601%:%
-%:%1178=602%:%
-%:%1179=603%:%
-%:%1181=605%:%
-%:%1182=606%:%
-%:%1183=606%:%
-%:%1184=606%:%
-%:%1185=607%:%
-%:%1188=610%:%
-%:%1189=611%:%
-%:%1190=611%:%
-%:%1191=612%:%
-%:%1192=612%:%
-%:%1193=613%:%
-%:%1195=615%:%
-%:%1196=616%:%
-%:%1197=616%:%
-%:%1198=617%:%
-%:%1199=617%:%
-%:%1200=618%:%
-%:%1202=620%:%
-%:%1203=621%:%
-%:%1204=621%:%
-%:%1205=622%:%
-%:%1206=622%:%
-%:%1207=623%:%
-%:%1208=623%:%
-%:%1209=623%:%
-%:%1210=623%:%
-%:%1211=624%:%
-%:%1212=625%:%
-%:%1213=625%:%
-%:%1214=626%:%
-%:%1215=626%:%
-%:%1216=626%:%
-%:%1217=626%:%
-%:%1218=627%:%
-%:%1219=628%:%
-%:%1220=628%:%
-%:%1221=629%:%
-%:%1222=629%:%
-%:%1223=630%:%
-%:%1225=632%:%
-%:%1226=633%:%
-%:%1227=633%:%
-%:%1228=634%:%
-%:%1229=634%:%
-%:%1230=635%:%
-%:%1231=635%:%
-%:%1232=635%:%
-%:%1233=636%:%
-%:%1234=637%:%
-%:%1235=637%:%
-%:%1236=638%:%
-%:%1237=638%:%
-%:%1238=639%:%
-%:%1239=639%:%
-%:%1240=640%:%
-%:%1241=641%:%
-%:%1242=641%:%
-%:%1243=641%:%
-%:%1244=642%:%
-%:%1245=642%:%
-%:%1246=643%:%
-%:%1247=643%:%
-%:%1248=644%:%
-%:%1249=644%:%
-%:%1250=645%:%
-%:%1251=646%:%
-%:%1252=646%:%
-%:%1253=646%:%
-%:%1254=647%:%
-%:%1255=647%:%
-%:%1256=648%:%
-%:%1257=648%:%
-%:%1258=649%:%
-%:%1259=649%:%
-%:%1260=650%:%
-%:%1261=651%:%
-%:%1262=651%:%
-%:%1263=651%:%
-%:%1264=652%:%
-%:%1265=652%:%
-%:%1266=653%:%
-%:%1267=653%:%
-%:%1268=654%:%
-%:%1269=654%:%
-%:%1270=655%:%
-%:%1271=656%:%
-%:%1272=656%:%
-%:%1273=656%:%
-%:%1274=657%:%
-%:%1275=657%:%
-%:%1276=658%:%
-%:%1277=658%:%
-%:%1278=659%:%
-%:%1279=659%:%
-%:%1280=660%:%
-%:%1281=661%:%
-%:%1282=662%:%
-%:%1283=662%:%
-%:%1284=663%:%
-%:%1285=664%:%
-%:%1286=665%:%
-%:%1287=666%:%
-%:%1288=667%:%
-%:%1289=667%:%
-%:%1290=667%:%
-%:%1291=668%:%
-%:%1294=671%:%
-%:%1295=672%:%
-%:%1296=672%:%
-%:%1297=673%:%
-%:%1298=673%:%
-%:%1299=674%:%
-%:%1300=674%:%
-%:%1301=675%:%
-%:%1302=675%:%
-%:%1303=676%:%
-%:%1318=678%:%
-%:%1328=684%:%
-%:%1329=684%:%
-%:%1330=685%:%
-%:%1331=686%:%
-%:%1332=687%:%
-%:%1333=687%:%
-%:%1334=688%:%
-%:%1335=689%:%
-%:%1336=690%:%
-%:%1337=691%:%
-%:%1338=692%:%
-%:%1341=693%:%
-%:%1345=693%:%
-%:%1346=693%:%
-%:%1347=694%:%
-%:%1348=694%:%
-%:%1353=694%:%
-%:%1356=695%:%
-%:%1357=696%:%
-%:%1358=696%:%
-%:%1359=697%:%
-%:%1360=698%:%
-%:%1361=699%:%
-%:%1362=700%:%
-%:%1369=701%:%
-%:%1370=701%:%
-%:%1371=702%:%
-%:%1372=702%:%
-%:%1373=703%:%
-%:%1374=703%:%
-%:%1375=704%:%
-%:%1376=704%:%
-%:%1377=705%:%
-%:%1378=706%:%
-%:%1379=706%:%
-%:%1380=707%:%
-%:%1386=707%:%
-%:%1389=708%:%
-%:%1390=709%:%
-%:%1391=709%:%
-%:%1392=710%:%
-%:%1393=711%:%
-%:%1394=712%:%
-%:%1397=713%:%
-%:%1401=713%:%
-%:%1402=713%:%
-%:%1416=715%:%
-%:%1426=717%:%
-%:%1427=717%:%
-%:%1428=718%:%
-%:%1429=719%:%
-%:%1434=724%:%
-%:%1435=725%:%
-%:%1436=729%:%
-%:%1437=730%:%
-%:%1438=730%:%
-%:%1439=731%:%
-%:%1445=737%:%
-%:%1446=738%:%
-%:%1447=739%:%
-%:%1448=739%:%
-%:%1449=740%:%
-%:%1450=741%:%
-%:%1457=742%:%
-%:%1458=742%:%
-%:%1459=743%:%
-%:%1460=743%:%
-%:%1461=744%:%
-%:%1462=745%:%
-%:%1463=746%:%
-%:%1464=746%:%
-%:%1465=747%:%
-%:%1466=747%:%
-%:%1467=748%:%
-%:%1468=748%:%
-%:%1469=749%:%
-%:%1470=750%:%
-%:%1471=751%:%
-%:%1472=751%:%
-%:%1473=752%:%
-%:%1474=752%:%
-%:%1475=753%:%
-%:%1476=753%:%
-%:%1477=754%:%
-%:%1478=754%:%
-%:%1479=755%:%
-%:%1480=756%:%
-%:%1481=756%:%
-%:%1482=757%:%
-%:%1483=757%:%
-%:%1484=758%:%
-%:%1485=758%:%
-%:%1486=759%:%
-%:%1487=759%:%
-%:%1488=760%:%
-%:%1489=760%:%
-%:%1490=761%:%
-%:%1491=761%:%
-%:%1492=762%:%
-%:%1498=762%:%
-%:%1501=763%:%
-%:%1502=764%:%
-%:%1503=764%:%
-%:%1504=765%:%
-%:%1505=766%:%
-%:%1506=767%:%
-%:%1507=768%:%
-%:%1508=769%:%
-%:%1515=770%:%
-%:%1516=770%:%
-%:%1517=771%:%
-%:%1518=771%:%
-%:%1519=772%:%
-%:%1520=773%:%
-%:%1521=774%:%
-%:%1522=774%:%
-%:%1523=775%:%
-%:%1524=775%:%
-%:%1525=776%:%
-%:%1526=776%:%
-%:%1527=777%:%
-%:%1528=777%:%
-%:%1529=778%:%
-%:%1530=778%:%
-%:%1531=779%:%
-%:%1532=779%:%
-%:%1533=780%:%
-%:%1534=780%:%
-%:%1535=781%:%
-%:%1536=782%:%
-%:%1537=782%:%
-%:%1538=783%:%
-%:%1539=783%:%
-%:%1540=784%:%
-%:%1541=784%:%
-%:%1542=785%:%
-%:%1548=785%:%
-%:%1551=786%:%
-%:%1552=787%:%
-%:%1553=787%:%
-%:%1554=788%:%
-%:%1555=789%:%
-%:%1556=790%:%
-%:%1557=791%:%
-%:%1558=792%:%
-%:%1559=793%:%
-%:%1566=794%:%
-%:%1567=794%:%
-%:%1568=795%:%
-%:%1569=795%:%
-%:%1570=796%:%
-%:%1571=796%:%
-%:%1572=797%:%
-%:%1573=798%:%
-%:%1574=799%:%
-%:%1575=799%:%
-%:%1576=800%:%
-%:%1577=800%:%
-%:%1578=801%:%
-%:%1579=801%:%
-%:%1580=802%:%
-%:%1582=804%:%
-%:%1583=805%:%
-%:%1584=805%:%
-%:%1585=806%:%
-%:%1586=806%:%
-%:%1587=807%:%
-%:%1588=807%:%
-%:%1589=808%:%
-%:%1590=808%:%
-%:%1591=809%:%
-%:%1592=809%:%
-%:%1593=810%:%
-%:%1594=810%:%
-%:%1595=811%:%
-%:%1596=811%:%
-%:%1597=812%:%
-%:%1598=813%:%
-%:%1599=813%:%
-%:%1600=814%:%
-%:%1601=814%:%
-%:%1602=815%:%
-%:%1603=815%:%
-%:%1604=816%:%
-%:%1605=817%:%
-%:%1606=817%:%
-%:%1607=818%:%
-%:%1608=818%:%
-%:%1609=819%:%
-%:%1610=819%:%
-%:%1611=820%:%
-%:%1612=820%:%
-%:%1613=821%:%
-%:%1614=821%:%
-%:%1615=822%:%
-%:%1616=822%:%
-%:%1617=822%:%
-%:%1618=823%:%
-%:%1619=823%:%
-%:%1620=824%:%
-%:%1621=824%:%
-%:%1622=825%:%
-%:%1623=825%:%
-%:%1624=825%:%
-%:%1625=826%:%
-%:%1626=827%:%
-%:%1627=827%:%
-%:%1628=828%:%
-%:%1629=829%:%
-%:%1630=829%:%
-%:%1631=830%:%
-%:%1632=830%:%
-%:%1633=830%:%
-%:%1634=831%:%
-%:%1636=833%:%
-%:%1637=834%:%
-%:%1638=834%:%
-%:%1639=835%:%
-%:%1640=836%:%
-%:%1641=836%:%
-%:%1642=837%:%
-%:%1643=837%:%
-%:%1644=837%:%
-%:%1645=838%:%
-%:%1646=838%:%
-%:%1647=839%:%
-%:%1648=839%:%
-%:%1649=840%:%
-%:%1650=840%:%
-%:%1651=841%:%
-%:%1652=841%:%
-%:%1653=842%:%
-%:%1654=842%:%
-%:%1655=843%:%
-%:%1656=843%:%
-%:%1657=844%:%
-%:%1658=844%:%
-%:%1659=845%:%
-%:%1660=845%:%
-%:%1661=846%:%
-%:%1662=846%:%
-%:%1663=847%:%
-%:%1664=848%:%
-%:%1665=848%:%
-%:%1666=849%:%
-%:%1672=849%:%
-%:%1677=850%:%
-%:%1682=851%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result.tex
deleted file mode 100644
index 77ece34..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result.tex
+++ /dev/null
@@ -1,469 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Result}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Electoral Result%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Result\isanewline
-\ \ \isakeyword{imports}\ Main\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-An electoral result is the principal result type of the composable modules
-voting framework, as it is a generalization of the set of winning alternatives
-from social choice functions. Electoral results are selections of the received
-(possibly empty) set of alternatives into the three disjoint groups of elected,
-rejected and deferred alternatives.
-Any of those sets, e.g., the set of winning (elected) alternatives, may also
-be left empty, as long as they collectively still hold all the received
-alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Result\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isacharasterisk}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ set\ {\isacharasterisk}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Functions%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ disjoint{\isadigit{3}}\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ well{\isacharunderscore}{\kern0pt}formed\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ result\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}disjoint{\isadigit{3}}\ result\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ result{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elect{\isacharunderscore}{\kern0pt}r\ r\ {\isasymequiv}\ fst\ r{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}reject{\isacharunderscore}{\kern0pt}r\ r\ {\isasymequiv}\ fst\ {\isacharparenleft}{\kern0pt}snd\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}r\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}r\ r\ {\isasymequiv}\ snd\ {\isacharparenleft}{\kern0pt}snd\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Auxiliary Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ d{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymnotin}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej\ x{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ r{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}x\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}elec{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ e{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}rej\ x{\isacharunderscore}{\kern0pt}elec\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}rej{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ d{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}rej\ x{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}count{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}e{\isacharcomma}{\kern0pt}\ r{\isacharcomma}{\kern0pt}\ d{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\ card\ e\ {\isacharplus}{\kern0pt}\ card\ r\ {\isacharplus}{\kern0pt}\ card\ d{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{have}\isamarkupfalse%
-\ disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ r\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}e\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}r\ {\isasyminter}\ d\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ \isacommand{have}\isamarkupfalse%
-\ set{\isacharunderscore}{\kern0pt}partit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}e\ {\isasymunion}\ r\ {\isasymunion}\ d\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ disj\ set{\isacharunderscore}{\kern0pt}partit\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ finite{\isacharunderscore}{\kern0pt}Un\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}disjoint\ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}right{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Types/Result.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%48=21%:%
-%:%49=22%:%
-%:%58=24%:%
-%:%68=30%:%
-%:%69=30%:%
-%:%76=32%:%
-%:%86=38%:%
-%:%87=38%:%
-%:%88=39%:%
-%:%91=42%:%
-%:%92=43%:%
-%:%93=44%:%
-%:%94=44%:%
-%:%95=45%:%
-%:%96=46%:%
-%:%97=47%:%
-%:%98=47%:%
-%:%99=48%:%
-%:%100=49%:%
-%:%101=50%:%
-%:%102=51%:%
-%:%103=51%:%
-%:%104=52%:%
-%:%105=53%:%
-%:%106=54%:%
-%:%107=54%:%
-%:%108=55%:%
-%:%109=56%:%
-%:%110=57%:%
-%:%111=57%:%
-%:%112=58%:%
-%:%119=60%:%
-%:%129=62%:%
-%:%130=62%:%
-%:%131=63%:%
-%:%132=64%:%
-%:%139=65%:%
-%:%140=65%:%
-%:%141=66%:%
-%:%142=66%:%
-%:%143=67%:%
-%:%144=68%:%
-%:%145=68%:%
-%:%146=69%:%
-%:%147=70%:%
-%:%148=71%:%
-%:%149=72%:%
-%:%150=72%:%
-%:%151=72%:%
-%:%152=73%:%
-%:%153=74%:%
-%:%154=75%:%
-%:%155=75%:%
-%:%156=76%:%
-%:%157=76%:%
-%:%158=77%:%
-%:%159=77%:%
-%:%160=78%:%
-%:%161=78%:%
-%:%162=79%:%
-%:%163=79%:%
-%:%164=80%:%
-%:%165=80%:%
-%:%166=81%:%
-%:%167=82%:%
-%:%168=82%:%
-%:%169=83%:%
-%:%170=84%:%
-%:%171=84%:%
-%:%172=84%:%
-%:%173=85%:%
-%:%174=86%:%
-%:%175=87%:%
-%:%176=87%:%
-%:%177=88%:%
-%:%178=88%:%
-%:%179=89%:%
-%:%180=89%:%
-%:%181=90%:%
-%:%182=90%:%
-%:%183=91%:%
-%:%184=91%:%
-%:%185=92%:%
-%:%186=92%:%
-%:%187=93%:%
-%:%188=94%:%
-%:%189=94%:%
-%:%190=95%:%
-%:%191=96%:%
-%:%192=97%:%
-%:%193=97%:%
-%:%194=97%:%
-%:%195=98%:%
-%:%196=99%:%
-%:%197=100%:%
-%:%198=100%:%
-%:%199=101%:%
-%:%200=101%:%
-%:%201=102%:%
-%:%202=102%:%
-%:%203=103%:%
-%:%204=103%:%
-%:%205=104%:%
-%:%206=104%:%
-%:%207=105%:%
-%:%208=105%:%
-%:%209=106%:%
-%:%210=107%:%
-%:%211=107%:%
-%:%212=108%:%
-%:%213=109%:%
-%:%214=110%:%
-%:%215=110%:%
-%:%216=110%:%
-%:%217=111%:%
-%:%218=112%:%
-%:%219=113%:%
-%:%220=113%:%
-%:%221=114%:%
-%:%222=114%:%
-%:%223=115%:%
-%:%224=115%:%
-%:%225=116%:%
-%:%226=116%:%
-%:%227=117%:%
-%:%233=117%:%
-%:%236=118%:%
-%:%237=119%:%
-%:%238=119%:%
-%:%239=120%:%
-%:%240=121%:%
-%:%241=122%:%
-%:%242=123%:%
-%:%249=124%:%
-%:%250=124%:%
-%:%251=125%:%
-%:%252=125%:%
-%:%253=125%:%
-%:%254=126%:%
-%:%255=127%:%
-%:%256=127%:%
-%:%257=128%:%
-%:%258=128%:%
-%:%259=128%:%
-%:%260=129%:%
-%:%261=130%:%
-%:%262=130%:%
-%:%263=131%:%
-%:%264=131%:%
-%:%265=132%:%
-%:%266=132%:%
-%:%267=133%:%
-%:%268=134%:%
-%:%269=134%:%
-%:%270=135%:%
-%:%276=135%:%
-%:%281=136%:%
-%:%286=137%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Facts.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Facts.tex
deleted file mode 100644
index 9b427d4..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Facts.tex
+++ /dev/null
@@ -1,2293 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Result{\isacharunderscore}{\kern0pt}Facts}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Result{\isacharunderscore}{\kern0pt}Facts\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Result{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Plurality{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Defer{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Drop{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Revision{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}electing{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}intro\ allI\ impI\ conjI{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ assm{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ max\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ max{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}max\ {\isacharequal}{\kern0pt}\ Max{\isacharparenleft}{\kern0pt}win{\isacharunderscore}{\kern0pt}count\ p\ {\isacharbackquote}{\kern0pt}\ A{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}win{\isacharunderscore}{\kern0pt}count\ p\ a\ {\isacharequal}{\kern0pt}\ max\ {\isasymand}\ a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Max{\isacharunderscore}{\kern0pt}in\ assm{\isadigit{0}}\ empty{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}image\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}imageI\ imageE\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ max\ assm{\isadigit{0}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isasymforall}x\ {\isasymin}\ A{\isachardot}{\kern0pt}\ win{\isacharunderscore}{\kern0pt}count\ p\ x\ {\isasymle}\ win{\isacharunderscore}{\kern0pt}count\ p\ a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}elem{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ elect\ plurality\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ assm{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}elect\ plurality\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}elem\ assm{\isadigit{1}}\ all{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ plurality{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ plurality{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ elect\ plurality\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}electing{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ defer{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ order{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ elim{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ r\ {\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ elim{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\ profile\ less{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}eliminator\ e\ t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ max{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}max{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}min{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isacharless}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ leq{\isacharunderscore}{\kern0pt}avg{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ profile{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}leq{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ e{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}elimination{\isacharunderscore}{\kern0pt}module\ e\ t\ {\isacharparenleft}{\kern0pt}{\isasymle}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ order{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ assms\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ g{\isadigit{0}}{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}n\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ card{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ n\ {\isacharless}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}above\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ r\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ a\ {\isasymin}\ A\ {\isasymand}\ b\ {\isasymin}\ A{\isacharbraceright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order\ top{\isacharunderscore}{\kern0pt}greatest\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\ fin{\isacharunderscore}{\kern0pt}A\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}all{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ n{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ is{\isacharunderscore}{\kern0pt}singletonI\isanewline
-\ \ \ \ \ \ \ \ \ \ is{\isacharunderscore}{\kern0pt}singleton{\isacharunderscore}{\kern0pt}altdef\ leD\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ n\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ card{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}zero{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ {\isadigit{0}}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ card{\isacharunderscore}{\kern0pt}pos{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ aa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Collect\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isasymnot}\ p\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Collect\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ p\ {\isacharparenleft}{\kern0pt}aa\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ moura\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}n{\isachardot}{\kern0pt}\ {\isasymnot}\ {\isacharparenleft}{\kern0pt}n{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}nat{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{0}}\ {\isasymor}\ n\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ Aa{\isachardot}{\kern0pt}\ {\isasymnot}\ connex\ Aa\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymor}\ a\ {\isasymnotin}\ Aa\ {\isasymor}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}connex\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ equals{\isadigit{0}}D\ finite{\isacharunderscore}{\kern0pt}A\ order\ rev{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{0}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{0}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isachardot}{\kern0pt}empty\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ card{\isacharunderscore}{\kern0pt}pos{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isasymle}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}pos\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}exists{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ finite{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}one{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{then}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ w\ \isakeyword{where}\ w{\isacharunderscore}{\kern0pt}unique{\isacharunderscore}{\kern0pt}top{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ w\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ w{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ w{\isacharunderscore}{\kern0pt}top{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ w\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ w{\isacharunderscore}{\kern0pt}unique{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}x{\isasymin}A{\isachardot}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}top\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ winner{\isacharunderscore}{\kern0pt}exists\ w{\isacharunderscore}{\kern0pt}unique{\isacharunderscore}{\kern0pt}top\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}on{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ x\ {\isasympreceq}\isactrlsub q\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex{\isacharunderscore}{\kern0pt}limit\ above{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}x{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}finite{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}A\ finite{\isacharunderscore}{\kern0pt}subset{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}winner{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ above{\isacharunderscore}{\kern0pt}finite\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ equals{\isadigit{0}}D\ neq{\isadigit{0}}{\isacharunderscore}{\kern0pt}conv\ x{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}x{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ is{\isacharunderscore}{\kern0pt}singletonE\ is{\isacharunderscore}{\kern0pt}singleton{\isacharunderscore}{\kern0pt}altdef\ singletonD\ x{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isacharequal}{\kern0pt}\ w{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ w{\isacharunderscore}{\kern0pt}unique\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{1}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}w{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ pass{\isacharunderscore}{\kern0pt}two{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}two{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ {\isadigit{2}}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isasymle}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ profA{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ limitA{\isacharunderscore}{\kern0pt}order{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ a\ \isakeyword{where}\isanewline
-\ \ \ \ a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\ finA\ profA\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}b\ {\isasymin}\ A{\isachardot}{\kern0pt}\ let\ q\ {\isacharequal}{\kern0pt}\ limit\ A\ r\ in\ {\isacharparenleft}{\kern0pt}b\ {\isasympreceq}\isactrlsub q\ a{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limitA{\isacharunderscore}{\kern0pt}order\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}subset\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \ \ \ \ \ \ order\ connex{\isacharunderscore}{\kern0pt}def\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}best{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}b\ {\isasymin}\ A{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}b{\isacharcomma}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}b\ {\isasymin}\ A{\isachardot}{\kern0pt}\ a\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ a\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{2}}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ CollectI\ Suc{\isacharunderscore}{\kern0pt}leI\ not{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}A\ a{\isacharunderscore}{\kern0pt}above\ card{\isacharunderscore}{\kern0pt}UNIV{\isacharunderscore}{\kern0pt}bool\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ empty{\isacharunderscore}{\kern0pt}iff\ finA\isanewline
-\ \ \ \ \ \ \ \ \ \ finite{\isachardot}{\kern0pt}emptyI\ insert{\isacharunderscore}{\kern0pt}iff\ limitA{\isacharunderscore}{\kern0pt}order\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \ \ \ \ \ \ UNIV{\isacharunderscore}{\kern0pt}bool\ nat{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ zero{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}Suc\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ finA{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\ Diff{\isacharunderscore}{\kern0pt}empty\ Diff{\isacharunderscore}{\kern0pt}idemp\ Diff{\isacharunderscore}{\kern0pt}insert{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \ \ \ \ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}A\ card{\isachardot}{\kern0pt}insert{\isacharunderscore}{\kern0pt}remove\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ finite{\isachardot}{\kern0pt}emptyI\ insert{\isacharunderscore}{\kern0pt}Diff\isanewline
-\ \ \ \ \ \ \ \ \ \ numeral{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}iff\ semiring{\isacharunderscore}{\kern0pt}norm{\isacharparenleft}{\kern0pt}{\isadigit{6}}{\isadigit{9}}{\isacharparenright}{\kern0pt}\ card{\isachardot}{\kern0pt}empty\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ limitAa{\isacharunderscore}{\kern0pt}order{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order{\isacharunderscore}{\kern0pt}on\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}lin{\isacharunderscore}{\kern0pt}ord\ order\ top{\isacharunderscore}{\kern0pt}greatest\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{obtain}\isamarkupfalse%
-\ b\ \isakeyword{where}\ b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ let\ q\ {\isacharequal}{\kern0pt}\ limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r\ in\ {\isacharparenleft}{\kern0pt}c\ {\isasympreceq}\isactrlsub q\ b{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limitAa{\isacharunderscore}{\kern0pt}order\ pref{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}above\ empty{\isacharunderscore}{\kern0pt}iff\ insert{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}subset\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\ connex{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}limit{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}c{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ {\isacharparenleft}{\kern0pt}A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}best{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}c{\isacharcomma}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isasymin}\ limit\ A\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ c{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ c\ {\isasymnotin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ b\ Diff{\isacharunderscore}{\kern0pt}iff\ Diff{\isacharunderscore}{\kern0pt}insert{\isadigit{2}}\ subset{\isacharunderscore}{\kern0pt}UNIV\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}subset\ order\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}above{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}subset{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ b\ b{\isacharunderscore}{\kern0pt}best\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \ \ \ \ \ \ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\ order\ insert{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}ab{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ b{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ A{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}a\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}limit\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}b\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b\ above{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}best\ \isacommand{have}\isamarkupfalse%
-\ b{\isacharunderscore}{\kern0pt}above{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ b\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}def\ mem{\isacharunderscore}{\kern0pt}Collect{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ limitA{\isacharunderscore}{\kern0pt}order\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isachardot}{\kern0pt}\ c\ {\isasymin}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ above{\isacharunderscore}{\kern0pt}connex{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharcomma}{\kern0pt}\ c{\isacharbraceright}{\kern0pt}\ {\isasymsubseteq}\ above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}above\ b{\isacharunderscore}{\kern0pt}above\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharcomma}{\kern0pt}\ c{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ DiffE\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}ab\ card{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ above{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}disjoint\ finA\ finite{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \ \ \ \ \ \ insert{\isacharunderscore}{\kern0pt}commute\ numeral{\isacharunderscore}{\kern0pt}{\isadigit{3}}{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{3}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{3}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ finA\ finite{\isacharunderscore}{\kern0pt}subset\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ c{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}trans\ numeral{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}iff\ order{\isacharunderscore}{\kern0pt}refl\ semiring{\isacharunderscore}{\kern0pt}norm{\isacharparenleft}{\kern0pt}{\isadigit{7}}{\isadigit{9}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}c\ {\isasymin}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardot}{\kern0pt}\ c\ {\isasymnotin}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ not{\isacharunderscore}{\kern0pt}le{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ a{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer\ b{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}defer\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharcomma}{\kern0pt}\ b{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}ab\ card{\isacharunderscore}{\kern0pt}above{\isacharunderscore}{\kern0pt}b{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}zero{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}zero{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}rejects\ {\isadigit{0}}\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ rejects{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ card{\isacharunderscore}{\kern0pt}pos{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymle}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ finite{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}connex\ UNIV\ r{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ lin{\isacharunderscore}{\kern0pt}ord{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}connex\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\ aa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Collect\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isasymnot}\ p\ a{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Collect\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymlongrightarrow}\ p\ {\isacharparenleft}{\kern0pt}aa\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ moura\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}a{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}\ {\isasymnotin}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ connex{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}connex\ A\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ limit{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}connex\ subset{\isacharunderscore}{\kern0pt}UNIV\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ rej{\isacharunderscore}{\kern0pt}drop{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}pass{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}a\ Aa{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymnot}\ connex\ Aa\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ {\isasymor}\ a\ {\isasymnotin}\ Aa\ {\isasymor}\ a\ {\isasymnotin}\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ card\ {\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ above{\isacharunderscore}{\kern0pt}connex\ above{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}limit\ bot{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isachardot}{\kern0pt}extremum{\isacharunderscore}{\kern0pt}uniqueI\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}eq\ emptyE\ finite{\isacharunderscore}{\kern0pt}A\ order\ rev{\isacharunderscore}{\kern0pt}finite{\isacharunderscore}{\kern0pt}subset\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{0}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ connex\ f{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharbraceleft}{\kern0pt}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ card{\isacharparenleft}{\kern0pt}above\ {\isacharparenleft}{\kern0pt}limit\ A\ r{\isacharparenright}{\kern0pt}\ a{\isacharparenright}{\kern0pt}\ {\isasymle}\ {\isadigit{0}}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isachardot}{\kern0pt}empty\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{0}}\ r{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}two{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}two{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ r{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ rej{\isacharunderscore}{\kern0pt}drop{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}pass{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ defer\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ AA\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ rrs\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}x{\isadigit{0}}\ x{\isadigit{1}}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymexists}v{\isadigit{2}}\ v{\isadigit{3}}{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}x{\isadigit{1}}\ {\isasymle}\ card\ v{\isadigit{2}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ v{\isadigit{2}}\ v{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ x{\isadigit{0}}\ v{\isadigit{2}}\ v{\isadigit{3}}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isadigit{1}}\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}AA\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ x{\isadigit{0}}\ {\isacharparenleft}{\kern0pt}AA\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ x{\isadigit{0}}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ x{\isadigit{1}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ moura\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}n\ f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnot}\ rejects\ n\ f\ {\isasymor}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnot}\ n\ {\isasymle}\ card\ A\ {\isasymor}\ infinite\ A\ {\isasymor}\ {\isasymnot}\ profile\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}rejects\ n\ f\ {\isasymor}\ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ {\isacharparenleft}{\kern0pt}n\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rejects{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ force\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}n\ f{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymnot}\ rejects\ n\ f\ {\isasymor}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isasymnot}\ n\ {\isasymle}\ card\ A\ {\isasymor}\ infinite\ A\ {\isasymor}\ {\isasymnot}\ profile\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}rejects\ n\ f\ {\isasymor}\ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\ n\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\ profile\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ f\ {\isacharparenleft}{\kern0pt}AA\ f\ n{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ presburger\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ {\isadigit{2}}\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isasymnot}\ profile\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AA\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}rrs\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ r{\isacharparenright}{\kern0pt}\ {\isadigit{2}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rej{\isacharunderscore}{\kern0pt}drop{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}pass\ defers{\isacharunderscore}{\kern0pt}def\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ pass{\isacharunderscore}{\kern0pt}two{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}two\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{1}}\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Composition_Rules/Result_Facts.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%16=6%:%
-%:%17=7%:%
-%:%18=8%:%
-%:%19=9%:%
-%:%20=10%:%
-%:%21=11%:%
-%:%26=11%:%
-%:%29=12%:%
-%:%30=13%:%
-%:%31=13%:%
-%:%34=14%:%
-%:%38=14%:%
-%:%39=14%:%
-%:%40=15%:%
-%:%41=15%:%
-%:%46=15%:%
-%:%49=16%:%
-%:%50=17%:%
-%:%51=17%:%
-%:%53=19%:%
-%:%60=20%:%
-%:%61=20%:%
-%:%62=21%:%
-%:%63=21%:%
-%:%64=22%:%
-%:%65=23%:%
-%:%66=24%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%70=26%:%
-%:%71=27%:%
-%:%72=28%:%
-%:%73=28%:%
-%:%74=29%:%
-%:%75=29%:%
-%:%76=30%:%
-%:%77=31%:%
-%:%78=31%:%
-%:%79=32%:%
-%:%80=32%:%
-%:%81=32%:%
-%:%82=33%:%
-%:%83=34%:%
-%:%84=34%:%
-%:%85=35%:%
-%:%86=36%:%
-%:%87=36%:%
-%:%88=37%:%
-%:%89=37%:%
-%:%90=38%:%
-%:%91=39%:%
-%:%92=39%:%
-%:%93=40%:%
-%:%94=40%:%
-%:%95=40%:%
-%:%96=41%:%
-%:%97=42%:%
-%:%98=42%:%
-%:%99=43%:%
-%:%100=43%:%
-%:%101=44%:%
-%:%102=44%:%
-%:%103=44%:%
-%:%104=45%:%
-%:%105=46%:%
-%:%106=46%:%
-%:%107=47%:%
-%:%108=47%:%
-%:%109=48%:%
-%:%110=49%:%
-%:%111=49%:%
-%:%112=50%:%
-%:%113=50%:%
-%:%114=51%:%
-%:%115=52%:%
-%:%116=52%:%
-%:%117=53%:%
-%:%118=53%:%
-%:%119=54%:%
-%:%120=54%:%
-%:%121=55%:%
-%:%122=55%:%
-%:%123=56%:%
-%:%129=56%:%
-%:%132=57%:%
-%:%133=58%:%
-%:%134=59%:%
-%:%135=59%:%
-%:%142=60%:%
-%:%143=60%:%
-%:%144=61%:%
-%:%145=61%:%
-%:%146=62%:%
-%:%147=63%:%
-%:%148=63%:%
-%:%149=64%:%
-%:%150=64%:%
-%:%151=65%:%
-%:%152=65%:%
-%:%153=66%:%
-%:%154=66%:%
-%:%155=67%:%
-%:%156=67%:%
-%:%157=68%:%
-%:%158=68%:%
-%:%159=69%:%
-%:%160=69%:%
-%:%161=70%:%
-%:%162=70%:%
-%:%163=71%:%
-%:%164=71%:%
-%:%165=72%:%
-%:%166=72%:%
-%:%167=73%:%
-%:%173=73%:%
-%:%176=74%:%
-%:%177=75%:%
-%:%178=75%:%
-%:%181=76%:%
-%:%185=76%:%
-%:%186=76%:%
-%:%187=77%:%
-%:%188=77%:%
-%:%193=77%:%
-%:%196=78%:%
-%:%197=79%:%
-%:%198=80%:%
-%:%199=81%:%
-%:%200=81%:%
-%:%201=82%:%
-%:%202=83%:%
-%:%205=84%:%
-%:%209=84%:%
-%:%210=84%:%
-%:%215=84%:%
-%:%218=85%:%
-%:%219=86%:%
-%:%220=86%:%
-%:%221=87%:%
-%:%222=88%:%
-%:%225=89%:%
-%:%229=89%:%
-%:%230=89%:%
-%:%235=89%:%
-%:%238=90%:%
-%:%239=91%:%
-%:%240=91%:%
-%:%241=92%:%
-%:%242=93%:%
-%:%245=94%:%
-%:%249=94%:%
-%:%250=94%:%
-%:%251=95%:%
-%:%252=95%:%
-%:%257=95%:%
-%:%260=96%:%
-%:%261=97%:%
-%:%262=97%:%
-%:%263=98%:%
-%:%264=99%:%
-%:%271=100%:%
-%:%272=100%:%
-%:%273=101%:%
-%:%274=101%:%
-%:%275=102%:%
-%:%276=102%:%
-%:%277=103%:%
-%:%278=103%:%
-%:%279=104%:%
-%:%280=104%:%
-%:%281=105%:%
-%:%287=105%:%
-%:%290=106%:%
-%:%291=107%:%
-%:%292=107%:%
-%:%293=108%:%
-%:%294=109%:%
-%:%301=110%:%
-%:%302=110%:%
-%:%303=111%:%
-%:%304=111%:%
-%:%305=112%:%
-%:%306=112%:%
-%:%307=113%:%
-%:%308=113%:%
-%:%309=114%:%
-%:%310=114%:%
-%:%311=115%:%
-%:%317=115%:%
-%:%320=116%:%
-%:%321=117%:%
-%:%322=117%:%
-%:%323=118%:%
-%:%324=119%:%
-%:%331=120%:%
-%:%332=120%:%
-%:%333=121%:%
-%:%334=121%:%
-%:%335=122%:%
-%:%336=122%:%
-%:%337=123%:%
-%:%338=123%:%
-%:%339=124%:%
-%:%340=124%:%
-%:%341=125%:%
-%:%347=125%:%
-%:%350=126%:%
-%:%351=127%:%
-%:%352=127%:%
-%:%353=128%:%
-%:%354=129%:%
-%:%361=130%:%
-%:%362=130%:%
-%:%363=131%:%
-%:%364=131%:%
-%:%365=132%:%
-%:%366=132%:%
-%:%367=133%:%
-%:%368=133%:%
-%:%369=134%:%
-%:%370=134%:%
-%:%371=135%:%
-%:%377=135%:%
-%:%380=136%:%
-%:%381=137%:%
-%:%382=137%:%
-%:%383=138%:%
-%:%384=139%:%
-%:%391=140%:%
-%:%392=140%:%
-%:%393=141%:%
-%:%394=141%:%
-%:%395=142%:%
-%:%396=142%:%
-%:%397=143%:%
-%:%398=143%:%
-%:%399=144%:%
-%:%400=144%:%
-%:%401=145%:%
-%:%407=145%:%
-%:%410=146%:%
-%:%411=147%:%
-%:%412=148%:%
-%:%413=148%:%
-%:%414=149%:%
-%:%415=150%:%
-%:%418=151%:%
-%:%422=151%:%
-%:%423=151%:%
-%:%428=151%:%
-%:%431=152%:%
-%:%432=153%:%
-%:%433=154%:%
-%:%434=154%:%
-%:%435=155%:%
-%:%436=156%:%
-%:%439=157%:%
-%:%443=157%:%
-%:%444=157%:%
-%:%449=157%:%
-%:%452=158%:%
-%:%453=159%:%
-%:%454=160%:%
-%:%455=160%:%
-%:%456=161%:%
-%:%457=162%:%
-%:%458=163%:%
-%:%461=164%:%
-%:%465=164%:%
-%:%466=164%:%
-%:%467=165%:%
-%:%468=165%:%
-%:%469=166%:%
-%:%470=166%:%
-%:%471=167%:%
-%:%472=167%:%
-%:%473=168%:%
-%:%474=168%:%
-%:%475=169%:%
-%:%476=169%:%
-%:%477=170%:%
-%:%478=170%:%
-%:%479=171%:%
-%:%480=172%:%
-%:%481=173%:%
-%:%482=174%:%
-%:%483=174%:%
-%:%484=175%:%
-%:%485=176%:%
-%:%486=177%:%
-%:%487=178%:%
-%:%490=181%:%
-%:%491=182%:%
-%:%492=183%:%
-%:%493=183%:%
-%:%494=184%:%
-%:%495=185%:%
-%:%496=185%:%
-%:%497=186%:%
-%:%498=186%:%
-%:%499=187%:%
-%:%500=187%:%
-%:%501=188%:%
-%:%502=189%:%
-%:%503=190%:%
-%:%504=190%:%
-%:%505=191%:%
-%:%506=191%:%
-%:%507=192%:%
-%:%508=192%:%
-%:%509=193%:%
-%:%510=194%:%
-%:%511=194%:%
-%:%512=195%:%
-%:%513=196%:%
-%:%514=196%:%
-%:%515=197%:%
-%:%516=197%:%
-%:%517=198%:%
-%:%518=198%:%
-%:%519=199%:%
-%:%520=199%:%
-%:%521=200%:%
-%:%522=200%:%
-%:%523=201%:%
-%:%524=201%:%
-%:%525=202%:%
-%:%531=202%:%
-%:%534=203%:%
-%:%535=204%:%
-%:%536=204%:%
-%:%537=205%:%
-%:%538=206%:%
-%:%541=207%:%
-%:%545=207%:%
-%:%546=207%:%
-%:%547=208%:%
-%:%548=208%:%
-%:%549=209%:%
-%:%550=209%:%
-%:%551=210%:%
-%:%552=210%:%
-%:%553=211%:%
-%:%554=211%:%
-%:%555=212%:%
-%:%556=212%:%
-%:%557=213%:%
-%:%558=213%:%
-%:%559=214%:%
-%:%560=215%:%
-%:%561=216%:%
-%:%562=216%:%
-%:%563=217%:%
-%:%564=218%:%
-%:%565=219%:%
-%:%566=220%:%
-%:%567=220%:%
-%:%568=221%:%
-%:%569=222%:%
-%:%570=222%:%
-%:%571=223%:%
-%:%572=223%:%
-%:%573=224%:%
-%:%574=225%:%
-%:%575=225%:%
-%:%576=226%:%
-%:%577=226%:%
-%:%578=227%:%
-%:%579=227%:%
-%:%580=228%:%
-%:%581=228%:%
-%:%582=229%:%
-%:%583=229%:%
-%:%584=230%:%
-%:%585=230%:%
-%:%586=231%:%
-%:%587=232%:%
-%:%588=233%:%
-%:%589=234%:%
-%:%590=234%:%
-%:%591=235%:%
-%:%592=235%:%
-%:%593=236%:%
-%:%594=236%:%
-%:%595=237%:%
-%:%596=237%:%
-%:%597=238%:%
-%:%598=239%:%
-%:%599=240%:%
-%:%600=240%:%
-%:%601=241%:%
-%:%602=242%:%
-%:%603=242%:%
-%:%604=243%:%
-%:%605=243%:%
-%:%606=244%:%
-%:%607=244%:%
-%:%608=245%:%
-%:%609=245%:%
-%:%610=246%:%
-%:%611=246%:%
-%:%612=247%:%
-%:%613=247%:%
-%:%614=248%:%
-%:%615=248%:%
-%:%616=249%:%
-%:%617=249%:%
-%:%618=250%:%
-%:%619=250%:%
-%:%620=251%:%
-%:%621=251%:%
-%:%622=252%:%
-%:%628=252%:%
-%:%631=253%:%
-%:%632=259%:%
-%:%633=260%:%
-%:%634=260%:%
-%:%635=261%:%
-%:%636=262%:%
-%:%639=263%:%
-%:%643=263%:%
-%:%644=263%:%
-%:%645=264%:%
-%:%646=264%:%
-%:%647=265%:%
-%:%648=265%:%
-%:%649=266%:%
-%:%650=266%:%
-%:%651=267%:%
-%:%652=267%:%
-%:%653=268%:%
-%:%654=268%:%
-%:%655=269%:%
-%:%656=269%:%
-%:%657=270%:%
-%:%658=271%:%
-%:%659=272%:%
-%:%660=272%:%
-%:%661=273%:%
-%:%662=274%:%
-%:%663=275%:%
-%:%664=276%:%
-%:%665=276%:%
-%:%666=277%:%
-%:%667=278%:%
-%:%668=278%:%
-%:%669=279%:%
-%:%670=279%:%
-%:%671=280%:%
-%:%672=280%:%
-%:%673=281%:%
-%:%674=281%:%
-%:%675=282%:%
-%:%676=282%:%
-%:%677=282%:%
-%:%678=283%:%
-%:%679=284%:%
-%:%680=284%:%
-%:%681=285%:%
-%:%682=285%:%
-%:%683=286%:%
-%:%684=286%:%
-%:%685=286%:%
-%:%686=287%:%
-%:%687=288%:%
-%:%688=289%:%
-%:%689=289%:%
-%:%690=290%:%
-%:%691=290%:%
-%:%692=291%:%
-%:%693=291%:%
-%:%694=291%:%
-%:%695=292%:%
-%:%696=293%:%
-%:%697=294%:%
-%:%698=294%:%
-%:%699=295%:%
-%:%700=295%:%
-%:%701=296%:%
-%:%702=296%:%
-%:%703=297%:%
-%:%704=297%:%
-%:%705=298%:%
-%:%706=298%:%
-%:%707=299%:%
-%:%708=300%:%
-%:%709=301%:%
-%:%710=301%:%
-%:%711=302%:%
-%:%712=302%:%
-%:%713=303%:%
-%:%714=303%:%
-%:%715=304%:%
-%:%716=304%:%
-%:%717=305%:%
-%:%718=305%:%
-%:%719=306%:%
-%:%720=306%:%
-%:%721=307%:%
-%:%722=307%:%
-%:%723=307%:%
-%:%724=308%:%
-%:%725=309%:%
-%:%726=309%:%
-%:%727=310%:%
-%:%728=310%:%
-%:%729=311%:%
-%:%730=312%:%
-%:%731=312%:%
-%:%732=313%:%
-%:%733=314%:%
-%:%734=314%:%
-%:%735=315%:%
-%:%736=315%:%
-%:%737=316%:%
-%:%738=316%:%
-%:%739=317%:%
-%:%740=318%:%
-%:%741=318%:%
-%:%742=319%:%
-%:%743=319%:%
-%:%744=320%:%
-%:%745=320%:%
-%:%746=321%:%
-%:%747=321%:%
-%:%748=322%:%
-%:%749=323%:%
-%:%750=323%:%
-%:%751=324%:%
-%:%752=324%:%
-%:%753=325%:%
-%:%754=325%:%
-%:%755=326%:%
-%:%756=326%:%
-%:%757=327%:%
-%:%758=327%:%
-%:%759=328%:%
-%:%760=328%:%
-%:%761=329%:%
-%:%762=329%:%
-%:%763=330%:%
-%:%764=330%:%
-%:%765=331%:%
-%:%766=331%:%
-%:%767=332%:%
-%:%768=332%:%
-%:%769=333%:%
-%:%770=333%:%
-%:%771=334%:%
-%:%772=334%:%
-%:%773=335%:%
-%:%774=335%:%
-%:%775=336%:%
-%:%776=336%:%
-%:%777=336%:%
-%:%778=337%:%
-%:%779=338%:%
-%:%780=338%:%
-%:%781=339%:%
-%:%782=340%:%
-%:%783=340%:%
-%:%784=341%:%
-%:%785=341%:%
-%:%786=341%:%
-%:%787=342%:%
-%:%788=343%:%
-%:%789=343%:%
-%:%790=344%:%
-%:%791=344%:%
-%:%792=345%:%
-%:%793=345%:%
-%:%794=346%:%
-%:%795=346%:%
-%:%796=347%:%
-%:%797=347%:%
-%:%798=348%:%
-%:%799=348%:%
-%:%800=349%:%
-%:%801=349%:%
-%:%802=350%:%
-%:%803=350%:%
-%:%804=351%:%
-%:%805=351%:%
-%:%806=352%:%
-%:%807=352%:%
-%:%808=353%:%
-%:%809=353%:%
-%:%810=353%:%
-%:%811=354%:%
-%:%812=355%:%
-%:%813=355%:%
-%:%814=356%:%
-%:%815=356%:%
-%:%816=357%:%
-%:%817=357%:%
-%:%818=358%:%
-%:%819=358%:%
-%:%820=359%:%
-%:%821=359%:%
-%:%822=360%:%
-%:%823=360%:%
-%:%824=361%:%
-%:%825=361%:%
-%:%826=362%:%
-%:%827=362%:%
-%:%828=363%:%
-%:%829=363%:%
-%:%830=364%:%
-%:%836=364%:%
-%:%839=365%:%
-%:%840=366%:%
-%:%841=366%:%
-%:%842=367%:%
-%:%843=368%:%
-%:%846=369%:%
-%:%850=369%:%
-%:%851=369%:%
-%:%852=370%:%
-%:%853=370%:%
-%:%854=371%:%
-%:%855=371%:%
-%:%856=372%:%
-%:%857=372%:%
-%:%858=373%:%
-%:%859=373%:%
-%:%860=374%:%
-%:%861=374%:%
-%:%862=375%:%
-%:%863=375%:%
-%:%864=376%:%
-%:%865=377%:%
-%:%866=378%:%
-%:%867=378%:%
-%:%868=379%:%
-%:%869=380%:%
-%:%870=381%:%
-%:%871=382%:%
-%:%872=382%:%
-%:%873=383%:%
-%:%874=383%:%
-%:%875=384%:%
-%:%876=384%:%
-%:%877=385%:%
-%:%878=385%:%
-%:%879=385%:%
-%:%880=386%:%
-%:%881=387%:%
-%:%882=387%:%
-%:%883=388%:%
-%:%884=388%:%
-%:%885=389%:%
-%:%886=389%:%
-%:%887=389%:%
-%:%888=390%:%
-%:%889=391%:%
-%:%890=391%:%
-%:%891=392%:%
-%:%892=392%:%
-%:%893=393%:%
-%:%894=393%:%
-%:%895=394%:%
-%:%896=394%:%
-%:%897=395%:%
-%:%898=396%:%
-%:%899=397%:%
-%:%900=397%:%
-%:%901=398%:%
-%:%902=398%:%
-%:%903=399%:%
-%:%904=399%:%
-%:%905=400%:%
-%:%906=400%:%
-%:%907=401%:%
-%:%908=401%:%
-%:%909=402%:%
-%:%910=402%:%
-%:%911=402%:%
-%:%912=403%:%
-%:%913=403%:%
-%:%914=404%:%
-%:%915=405%:%
-%:%916=406%:%
-%:%917=407%:%
-%:%918=407%:%
-%:%919=408%:%
-%:%920=408%:%
-%:%921=409%:%
-%:%922=409%:%
-%:%923=410%:%
-%:%924=410%:%
-%:%925=411%:%
-%:%926=411%:%
-%:%927=412%:%
-%:%928=412%:%
-%:%929=412%:%
-%:%930=413%:%
-%:%931=413%:%
-%:%932=414%:%
-%:%933=415%:%
-%:%934=416%:%
-%:%935=417%:%
-%:%936=417%:%
-%:%937=418%:%
-%:%938=418%:%
-%:%939=418%:%
-%:%940=419%:%
-%:%941=420%:%
-%:%942=420%:%
-%:%943=421%:%
-%:%944=421%:%
-%:%945=422%:%
-%:%946=422%:%
-%:%947=422%:%
-%:%948=423%:%
-%:%949=423%:%
-%:%950=424%:%
-%:%951=424%:%
-%:%952=425%:%
-%:%953=425%:%
-%:%954=426%:%
-%:%955=426%:%
-%:%956=427%:%
-%:%957=428%:%
-%:%958=429%:%
-%:%959=429%:%
-%:%960=430%:%
-%:%961=430%:%
-%:%962=431%:%
-%:%963=431%:%
-%:%964=432%:%
-%:%965=432%:%
-%:%966=433%:%
-%:%967=433%:%
-%:%968=434%:%
-%:%969=434%:%
-%:%970=435%:%
-%:%971=435%:%
-%:%972=436%:%
-%:%973=437%:%
-%:%974=437%:%
-%:%975=438%:%
-%:%976=438%:%
-%:%977=438%:%
-%:%978=439%:%
-%:%979=439%:%
-%:%980=440%:%
-%:%981=440%:%
-%:%982=441%:%
-%:%983=441%:%
-%:%984=441%:%
-%:%985=442%:%
-%:%986=442%:%
-%:%987=443%:%
-%:%988=444%:%
-%:%989=444%:%
-%:%990=445%:%
-%:%991=445%:%
-%:%992=445%:%
-%:%993=446%:%
-%:%994=446%:%
-%:%995=447%:%
-%:%996=447%:%
-%:%997=448%:%
-%:%998=448%:%
-%:%999=449%:%
-%:%1000=449%:%
-%:%1001=450%:%
-%:%1002=450%:%
-%:%1003=451%:%
-%:%1004=451%:%
-%:%1005=452%:%
-%:%1006=452%:%
-%:%1007=453%:%
-%:%1008=453%:%
-%:%1009=454%:%
-%:%1010=454%:%
-%:%1011=454%:%
-%:%1012=455%:%
-%:%1013=456%:%
-%:%1014=456%:%
-%:%1015=457%:%
-%:%1016=457%:%
-%:%1017=458%:%
-%:%1018=458%:%
-%:%1019=459%:%
-%:%1020=459%:%
-%:%1021=460%:%
-%:%1022=460%:%
-%:%1023=461%:%
-%:%1024=461%:%
-%:%1025=462%:%
-%:%1026=462%:%
-%:%1027=463%:%
-%:%1028=463%:%
-%:%1029=464%:%
-%:%1030=464%:%
-%:%1031=465%:%
-%:%1032=465%:%
-%:%1033=466%:%
-%:%1034=466%:%
-%:%1035=466%:%
-%:%1036=467%:%
-%:%1037=467%:%
-%:%1038=468%:%
-%:%1039=469%:%
-%:%1040=470%:%
-%:%1041=470%:%
-%:%1042=471%:%
-%:%1043=471%:%
-%:%1044=471%:%
-%:%1045=472%:%
-%:%1046=473%:%
-%:%1047=473%:%
-%:%1048=474%:%
-%:%1049=474%:%
-%:%1050=475%:%
-%:%1051=475%:%
-%:%1052=476%:%
-%:%1053=476%:%
-%:%1054=477%:%
-%:%1055=477%:%
-%:%1056=478%:%
-%:%1057=478%:%
-%:%1058=479%:%
-%:%1059=479%:%
-%:%1060=480%:%
-%:%1061=480%:%
-%:%1062=480%:%
-%:%1063=481%:%
-%:%1064=481%:%
-%:%1065=482%:%
-%:%1066=482%:%
-%:%1067=482%:%
-%:%1068=483%:%
-%:%1069=483%:%
-%:%1070=484%:%
-%:%1071=484%:%
-%:%1072=484%:%
-%:%1073=485%:%
-%:%1074=486%:%
-%:%1075=486%:%
-%:%1076=487%:%
-%:%1077=487%:%
-%:%1078=488%:%
-%:%1079=488%:%
-%:%1080=489%:%
-%:%1081=489%:%
-%:%1082=490%:%
-%:%1088=490%:%
-%:%1091=491%:%
-%:%1092=492%:%
-%:%1093=492%:%
-%:%1094=493%:%
-%:%1095=494%:%
-%:%1098=495%:%
-%:%1102=495%:%
-%:%1103=495%:%
-%:%1104=496%:%
-%:%1105=496%:%
-%:%1106=497%:%
-%:%1107=497%:%
-%:%1108=498%:%
-%:%1109=498%:%
-%:%1110=499%:%
-%:%1111=499%:%
-%:%1112=500%:%
-%:%1113=500%:%
-%:%1114=501%:%
-%:%1115=501%:%
-%:%1116=502%:%
-%:%1117=503%:%
-%:%1118=504%:%
-%:%1119=504%:%
-%:%1120=505%:%
-%:%1121=506%:%
-%:%1122=507%:%
-%:%1123=508%:%
-%:%1124=508%:%
-%:%1125=509%:%
-%:%1126=509%:%
-%:%1127=510%:%
-%:%1128=510%:%
-%:%1129=511%:%
-%:%1130=511%:%
-%:%1131=512%:%
-%:%1132=513%:%
-%:%1133=514%:%
-%:%1134=515%:%
-%:%1135=515%:%
-%:%1136=516%:%
-%:%1137=516%:%
-%:%1138=517%:%
-%:%1139=517%:%
-%:%1140=518%:%
-%:%1141=518%:%
-%:%1142=519%:%
-%:%1143=519%:%
-%:%1144=520%:%
-%:%1145=521%:%
-%:%1146=521%:%
-%:%1147=522%:%
-%:%1148=522%:%
-%:%1149=523%:%
-%:%1150=523%:%
-%:%1151=524%:%
-%:%1152=525%:%
-%:%1153=525%:%
-%:%1154=526%:%
-%:%1155=526%:%
-%:%1156=527%:%
-%:%1158=529%:%
-%:%1159=530%:%
-%:%1160=530%:%
-%:%1161=531%:%
-%:%1162=532%:%
-%:%1163=532%:%
-%:%1164=533%:%
-%:%1165=533%:%
-%:%1166=534%:%
-%:%1167=534%:%
-%:%1168=535%:%
-%:%1169=535%:%
-%:%1170=536%:%
-%:%1171=536%:%
-%:%1172=537%:%
-%:%1173=537%:%
-%:%1174=538%:%
-%:%1175=538%:%
-%:%1176=539%:%
-%:%1177=539%:%
-%:%1178=540%:%
-%:%1179=540%:%
-%:%1180=541%:%
-%:%1186=541%:%
-%:%1189=542%:%
-%:%1190=546%:%
-%:%1191=547%:%
-%:%1192=547%:%
-%:%1193=548%:%
-%:%1194=549%:%
-%:%1201=550%:%
-%:%1202=550%:%
-%:%1203=551%:%
-%:%1204=551%:%
-%:%1205=552%:%
-%:%1206=553%:%
-%:%1207=553%:%
-%:%1208=554%:%
-%:%1209=554%:%
-%:%1210=555%:%
-%:%1211=555%:%
-%:%1212=556%:%
-%:%1213=556%:%
-%:%1214=557%:%
-%:%1215=558%:%
-%:%1216=559%:%
-%:%1220=563%:%
-%:%1221=564%:%
-%:%1222=564%:%
-%:%1223=565%:%
-%:%1224=565%:%
-%:%1225=566%:%
-%:%1230=571%:%
-%:%1231=572%:%
-%:%1232=572%:%
-%:%1233=573%:%
-%:%1234=573%:%
-%:%1235=574%:%
-%:%1236=574%:%
-%:%1237=575%:%
-%:%1242=580%:%
-%:%1243=581%:%
-%:%1244=581%:%
-%:%1245=582%:%
-%:%1246=582%:%
-%:%1247=583%:%
-%:%1251=587%:%
-%:%1252=588%:%
-%:%1253=588%:%
-%:%1254=589%:%
-%:%1255=590%:%
-%:%1256=590%:%
-%:%1257=591%:%
-%:%1258=591%:%
-%:%1259=592%:%
-%:%1260=592%:%
-%:%1261=593%:%
-%:%1262=593%:%
-%:%1263=594%:%
-%:%1264=594%:%
-%:%1265=595%:%
-%:%1271=595%:%
-%:%1276=596%:%
-%:%1281=597%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Properties.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Properties.tex
deleted file mode 100644
index 3f1f97b..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Properties.tex
+++ /dev/null
@@ -1,350 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Result{\isacharunderscore}{\kern0pt}Properties}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Result{\isacharunderscore}{\kern0pt}Properties\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ electing\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}electing\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}alt{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ one{\isacharunderscore}{\kern0pt}alt{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}right\ Int{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}right\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\isanewline
-\ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ electing\ electing{\isacharunderscore}{\kern0pt}def\ inf{\isachardot}{\kern0pt}orderE\isanewline
-\ \ \ \ \ \ \ \ one{\isacharunderscore}{\kern0pt}alt\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ decrementing\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}decrementing\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isasymforall}\ A\ p\ {\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}card\ A\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ reject\ m\ A\ p\ {\isasymnoteq}\ A{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ elects\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}elects\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defers\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defers\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ rejects\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}rejects\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ eliminates\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}eliminates\ n\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isachargreater}{\kern0pt}\ n\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ eliminating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ leftover{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymsubset}\ A{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}iff\ Diff{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ eliminates{\isacharunderscore}{\kern0pt}def\ eliminating\ eq{\isacharunderscore}{\kern0pt}iff\ leftover{\isacharunderscore}{\kern0pt}alternatives\isanewline
-\ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}zero\ f{\isacharunderscore}{\kern0pt}prof\ psubsetI\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ rejecting{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ finite{\isacharunderscore}{\kern0pt}subset\ non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}empty\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ rejecting\isanewline
-\ \ \ \ \ \ \ \ rejects{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ eliminating{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}subset\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ finite{\isacharunderscore}{\kern0pt}subset\ non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ not{\isacharunderscore}{\kern0pt}empty\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ eliminating\ eliminates{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}deciding\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}deciding\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ m\ {\isasymand}\ defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Properties/Result_Properties.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%19=4%:%
-%:%22=5%:%
-%:%23=9%:%
-%:%24=10%:%
-%:%25=10%:%
-%:%26=11%:%
-%:%28=13%:%
-%:%29=14%:%
-%:%30=15%:%
-%:%31=15%:%
-%:%32=16%:%
-%:%33=17%:%
-%:%34=18%:%
-%:%35=19%:%
-%:%36=20%:%
-%:%39=21%:%
-%:%43=21%:%
-%:%44=21%:%
-%:%45=22%:%
-%:%46=23%:%
-%:%47=24%:%
-%:%48=24%:%
-%:%53=24%:%
-%:%56=25%:%
-%:%57=26%:%
-%:%58=30%:%
-%:%59=31%:%
-%:%60=31%:%
-%:%61=32%:%
-%:%62=33%:%
-%:%63=34%:%
-%:%64=38%:%
-%:%65=39%:%
-%:%66=39%:%
-%:%67=40%:%
-%:%70=43%:%
-%:%71=44%:%
-%:%72=48%:%
-%:%73=49%:%
-%:%74=49%:%
-%:%75=50%:%
-%:%78=53%:%
-%:%79=54%:%
-%:%80=55%:%
-%:%81=59%:%
-%:%82=60%:%
-%:%83=60%:%
-%:%84=61%:%
-%:%86=63%:%
-%:%87=64%:%
-%:%88=68%:%
-%:%89=69%:%
-%:%90=69%:%
-%:%91=70%:%
-%:%94=73%:%
-%:%95=74%:%
-%:%96=78%:%
-%:%97=79%:%
-%:%98=79%:%
-%:%99=80%:%
-%:%101=82%:%
-%:%102=83%:%
-%:%103=87%:%
-%:%104=88%:%
-%:%105=88%:%
-%:%106=89%:%
-%:%108=91%:%
-%:%109=92%:%
-%:%110=93%:%
-%:%111=93%:%
-%:%112=94%:%
-%:%113=95%:%
-%:%114=96%:%
-%:%115=97%:%
-%:%116=98%:%
-%:%119=99%:%
-%:%123=99%:%
-%:%124=99%:%
-%:%125=100%:%
-%:%126=101%:%
-%:%127=102%:%
-%:%128=102%:%
-%:%133=102%:%
-%:%136=103%:%
-%:%137=104%:%
-%:%138=104%:%
-%:%139=105%:%
-%:%140=106%:%
-%:%141=107%:%
-%:%142=108%:%
-%:%143=109%:%
-%:%144=110%:%
-%:%147=111%:%
-%:%151=111%:%
-%:%152=111%:%
-%:%153=112%:%
-%:%154=113%:%
-%:%155=114%:%
-%:%156=115%:%
-%:%157=115%:%
-%:%162=115%:%
-%:%165=116%:%
-%:%166=117%:%
-%:%167=117%:%
-%:%168=118%:%
-%:%169=119%:%
-%:%170=120%:%
-%:%171=121%:%
-%:%172=122%:%
-%:%173=123%:%
-%:%176=124%:%
-%:%180=124%:%
-%:%181=124%:%
-%:%182=125%:%
-%:%183=126%:%
-%:%184=127%:%
-%:%185=128%:%
-%:%186=128%:%
-%:%191=128%:%
-%:%194=129%:%
-%:%195=136%:%
-%:%196=137%:%
-%:%197=137%:%
-%:%198=138%:%
-%:%199=139%:%
-%:%202=140%:%
-%:%207=141%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Rules.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Rules.tex
deleted file mode 100644
index 77c129c..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Result_Rules.tex
+++ /dev/null
@@ -1,2466 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Result{\isacharunderscore}{\kern0pt}Rules}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Result{\isacharunderscore}{\kern0pt}Rules\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Result{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Elect{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ Result{\isacharunderscore}{\kern0pt}Facts\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ electing{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}disjoint\ Diff{\isacharunderscore}{\kern0pt}empty\ Int{\isacharunderscore}{\kern0pt}absorb{\isadigit{2}}\ electing\isanewline
-\ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ reject\ m\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ A\ {\isacharminus}{\kern0pt}\ reject\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}empty{\isacharunderscore}{\kern0pt}iff\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ elect\ m\ A\ p\ {\isasymunion}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ elec{\isacharunderscore}{\kern0pt}and{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}rej\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isasymunion}\ defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}input{\isacharunderscore}{\kern0pt}sound\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymor}\ defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ Diff{\isacharunderscore}{\kern0pt}subset{\isacharunderscore}{\kern0pt}conv\ Un{\isacharunderscore}{\kern0pt}empty\ fst{\isacharunderscore}{\kern0pt}conv\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ inf{\isachardot}{\kern0pt}absorb{\isadigit{1}}\ sup{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}right\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ subset{\isacharunderscore}{\kern0pt}antisym\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ elector{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}block{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electing\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ AA\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rrs\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electing\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ elect\ f\ {\isacharparenleft}{\kern0pt}AA\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ profile\ {\isacharparenleft}{\kern0pt}AA\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrs\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}AA\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ AA\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ elect\ f\ A\ rs\ {\isasymor}\ {\isasymnot}\ profile\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ infinite\ A\ {\isasymor}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electing\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}block{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}module{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}\ Profile\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}\ Result{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electing{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ AAa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ rrsa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ f{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}f{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electing\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ elect\ f\ {\isacharparenleft}{\kern0pt}AAa\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ profile\ {\isacharparenleft}{\kern0pt}AAa\ f{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ finite\ {\isacharparenleft}{\kern0pt}AAa\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ AAa\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electoral{\isacharunderscore}{\kern0pt}module\ f{\isacharparenright}{\kern0pt}\ {\isasymand}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ elect\ f\ A\ rs\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ profile\ A\ rs\ {\isasymor}\ infinite\ A\ {\isasymor}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ A{\isacharparenright}{\kern0pt}\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ {\isasymnot}\ electing\ f{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ AAb\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ AAc\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ AAd\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ \ \ \ \ f{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AAb\ p{\isacharcomma}{\kern0pt}\ AAc\ p{\isacharcomma}{\kern0pt}\ AAd\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isadigit{3}}{\isachardot}{\kern0pt}cases\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elector{\isacharunderscore}{\kern0pt}sound\ module{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ f{\isadigit{4}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isasymforall}p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ p{\isacharcomma}{\kern0pt}\ AAc\ p{\isacharcomma}{\kern0pt}\ AAd\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}finite\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ profile\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ AAd\ {\isacharparenleft}{\kern0pt}elector\ m\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ reject\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ AAc\ {\isacharparenleft}{\kern0pt}elector\ m\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electing\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{2}}\ f{\isadigit{1}}\ Diff{\isacharunderscore}{\kern0pt}empty\ elector{\isachardot}{\kern0pt}simps\ non{\isacharunderscore}{\kern0pt}block{\isacharunderscore}{\kern0pt}m\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}block\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{{\isacharbraceleft}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymnoteq}\ AAd\ {\isacharparenleft}{\kern0pt}elector\ m\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isasymnot}\ profile\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rrsa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ infinite\ {\isacharparenleft}{\kern0pt}AAa\ {\isacharparenleft}{\kern0pt}elector\ m{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{4}}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \isacommand{{\isacharbraceright}{\kern0pt}}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isadigit{4}}\ f{\isadigit{3}}\ f{\isadigit{1}}\ fst{\isacharunderscore}{\kern0pt}conv\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isadigit{1}}{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isadigit{1}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ electing{\isacharunderscore}{\kern0pt}m{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ m{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electing\ {\isacharparenleft}{\kern0pt}m{\isadigit{1}}\ {\isasymtriangleright}\ m{\isadigit{2}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}card\ A\ {\isasymge}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m{\isadigit{1}}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ m{\isadigit{1}}\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ zero{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isadigit{1}}\ defers{\isacharunderscore}{\kern0pt}def\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ electing{\isacharunderscore}{\kern0pt}m{\isadigit{2}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ conserv{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ conservative{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ a{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ agg{\isacharunderscore}{\kern0pt}a{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}aggregator\ a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ conservative\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}m\ emod{\isacharunderscore}{\kern0pt}n\ agg{\isacharunderscore}{\kern0pt}a\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}wins{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub a\ n{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ emod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ c\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}a\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ in\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect{\isacharunderscore}{\kern0pt}r\ c\ {\isasymsubseteq}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ conservative\ agg{\isacharunderscore}{\kern0pt}conservative{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ emod{\isacharunderscore}{\kern0pt}m\ emod{\isacharunderscore}{\kern0pt}n\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}result{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}SIG{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}wins\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}x\ {\isasymin}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sup{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}right\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ conserv{\isacharunderscore}{\kern0pt}max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}empty\ assms\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prod{\isachardot}{\kern0pt}sel\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ elect\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ n\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}less\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ x{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}helper\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ empty{\isacharunderscore}{\kern0pt}iff\ fin{\isacharunderscore}{\kern0pt}A\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}code\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}elect\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ {\isachardoublequoteopen}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ electing{\isacharunderscore}{\kern0pt}def\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ no{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ no{\isacharunderscore}{\kern0pt}elect\ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\ fin{\isacharunderscore}{\kern0pt}A\ electing{\isacharunderscore}{\kern0pt}def\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ Diff{\isacharunderscore}{\kern0pt}disjoint\ Int{\isacharunderscore}{\kern0pt}absorb{\isadigit{2}}\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elect\ assms\ electing{\isacharunderscore}{\kern0pt}def\ empty{\isacharunderscore}{\kern0pt}iff\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prof{\isacharunderscore}{\kern0pt}A\ x{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ pos{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isasymle}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ pos{\isacharunderscore}{\kern0pt}card\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ \isacommand{have}\isamarkupfalse%
-\ m{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}blocking{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}reject\ m\ A\ p\ {\isasymnoteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a{\isachardot}{\kern0pt}\ a\ {\isasymin}\ A\ {\isasymand}\ a\ {\isasymnotin}\ reject\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ pos{\isacharunderscore}{\kern0pt}card\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ subset{\isacharunderscore}{\kern0pt}antisym\ subset{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ subsetI\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ empty{\isacharunderscore}{\kern0pt}iff\ pos{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}empty{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ One{\isacharunderscore}{\kern0pt}nat{\isacharunderscore}{\kern0pt}def\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ pos{\isacharunderscore}{\kern0pt}card\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}fun{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n\ defers{\isacharunderscore}{\kern0pt}def\ fin{\isacharunderscore}{\kern0pt}A\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}n\ f{\isachardot}{\kern0pt}\ defers\ n\ f\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ f\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ rs{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymnot}\ n\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ {\isasymor}\ infinite\ A\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ {\isasymnot}\ profile\ A\ rs{\isacharparenright}{\kern0pt}\ {\isasymor}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ f\ A\ rs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}non{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \ \ \ \ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}blocking{\isacharunderscore}{\kern0pt}m\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}fun\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ n\ {\isasymge}\ x\ {\isasymand}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\isanewline
-\ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}induct\ n\ arbitrary{\isacharcolon}{\kern0pt}\ acc\ rule{\isacharcolon}{\kern0pt}\ less{\isacharunderscore}{\kern0pt}induct{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{case}\isamarkupfalse%
-{\isacharparenleft}{\kern0pt}less\ n{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ subset{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ step{\isacharunderscore}{\kern0pt}reduces{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ term{\isacharunderscore}{\kern0pt}satisfied{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ term{\isacharunderscore}{\kern0pt}satisfied\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ term{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}acc\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymnoteq}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ rec{\isacharunderscore}{\kern0pt}step{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}\ {\isasymand}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ step{\isacharunderscore}{\kern0pt}reduces{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}set\ term{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}satisfied\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}case\isanewline
-\ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}too{\isacharunderscore}{\kern0pt}small{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ not{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ old{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}at{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharless}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{obtain}\isamarkupfalse%
-\ i\ \isakeyword{where}\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}i\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{with}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}x\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}too{\isacharunderscore}{\kern0pt}big{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nat{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}iff\ old{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}at{\isacharunderscore}{\kern0pt}least{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}leftover{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ f{\isacharunderscore}{\kern0pt}prof{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ step{\isacharunderscore}{\kern0pt}profile{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ limit{\isacharunderscore}{\kern0pt}profile{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isadigit{2}}\ enough{\isacharunderscore}{\kern0pt}leftover\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ snd{\isacharunderscore}{\kern0pt}conv\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isasymge}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}too{\isacharunderscore}{\kern0pt}big\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ x\ {\isasymle}\ i{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ acc\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}leftover\ f{\isacharunderscore}{\kern0pt}prof\ subset\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ i\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ step{\isacharunderscore}{\kern0pt}profile\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}i\ {\isacharequal}{\kern0pt}\ x\ {\isasymor}\ i\ {\isachargreater}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ nat{\isacharunderscore}{\kern0pt}less{\isacharunderscore}{\kern0pt}le\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough\isanewline
-\ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isachargreater}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ {\isadigit{1}}\ {\isacharless}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ x\ {\isasymle}\ i{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isacharless}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ subset\ step{\isacharunderscore}{\kern0pt}profile\ enough{\isacharunderscore}{\kern0pt}leftover\ f{\isacharunderscore}{\kern0pt}prof\ psubset{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}new{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ acc\ {\isasymlongrightarrow}\ non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}n\ {\isacharequal}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\ non{\isacharunderscore}{\kern0pt}electing\ acc\ {\isasymand}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ acc{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ less{\isachardot}{\kern0pt}hyps\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ f{\isacharunderscore}{\kern0pt}prof\ rec{\isacharunderscore}{\kern0pt}step\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}{\isacharparenleft}{\kern0pt}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymlongrightarrow}\ i\ {\isachargreater}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\ i\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\ t\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \ \ \ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ acc\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ {\isacharparenleft}{\kern0pt}acc\ {\isasymtriangleright}\ m{\isacharparenright}{\kern0pt}\ m\ t\ A\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ i{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}x\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}order\ i{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}new{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ new{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}still{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof\ rec{\isacharunderscore}{\kern0pt}step\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\isanewline
-\ \ \ \ \ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ acc{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}enough{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ acc\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ x{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}acc{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ acc{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ acc\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ acc{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}enough\ gr{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}not{\isadigit{0}}\ le{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}implies{\isacharunderscore}{\kern0pt}less\isanewline
-\ \ \ \ \ \ \ \ less{\isacharunderscore}{\kern0pt}one\ linorder{\isacharunderscore}{\kern0pt}neqE{\isacharunderscore}{\kern0pt}nat\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isachardot}{\kern0pt}simps{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}acc\isanewline
-\ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ single{\isacharunderscore}{\kern0pt}elimination\ nat{\isacharunderscore}{\kern0pt}neq{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\ less{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}x\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ enough{\isacharunderscore}{\kern0pt}alternatives{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isasymge}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}x{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymnot}\ card\ A\ {\isacharequal}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}card\ A\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}alternatives\ not{\isacharunderscore}{\kern0pt}le\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymnot}card\ A\ {\isacharless}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isachargreater}{\kern0pt}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}x\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}decr{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isadigit{2}}\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ fastforce\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymge}\ x{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper\ m\ m\ t{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ card{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}x\ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}big{\isacharunderscore}{\kern0pt}enough{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}helper{\isacharunderscore}{\kern0pt}iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ single{\isacharunderscore}{\kern0pt}elimination{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymforall}\ r{\isachardot}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}t\ r{\isacharparenright}{\kern0pt}\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}card\ {\isacharparenleft}{\kern0pt}defer{\isacharunderscore}{\kern0pt}r\ r{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}n\ {\isachargreater}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defers\ n\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}\ A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymand}\ card\ A\ {\isasymge}\ n\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ iter{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}helper\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \ \ \ \ \ \ terminate{\isacharunderscore}{\kern0pt}if{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}left\ x{\isacharunderscore}{\kern0pt}greater{\isacharunderscore}{\kern0pt}zero\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ eliminates{\isacharunderscore}{\kern0pt}def\ single{\isacharunderscore}{\kern0pt}elimination\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ calculation\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isanewline
-\isacommand{theorem}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ defers{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ rejec{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ disj{\isacharunderscore}{\kern0pt}comp{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ m\ n{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ eliminates{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ rejec{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ rejects{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}m\ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharless}{\kern0pt}\ card\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isasymge}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\ dual{\isacharunderscore}{\kern0pt}order{\isachardot}{\kern0pt}strict{\isacharunderscore}{\kern0pt}trans{\isadigit{2}}\ less{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}le{\isacharunderscore}{\kern0pt}nat\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ elec{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ non{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}m\ card{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defers{\isacharunderscore}{\kern0pt}m{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ module\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ card\ A\ {\isacharminus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ fin{\isacharunderscore}{\kern0pt}A{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ reject\ m\ A\ p{\isacharcomma}{\kern0pt}\ defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ module\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}card\ A\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ result{\isacharunderscore}{\kern0pt}count\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{1}}\ elec{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}{\isadigit{0}}\isanewline
-\ \ \ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ case{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ A\ {\isasymge}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ min{\isacharunderscore}{\kern0pt}{\isadigit{2}}{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ case{\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\ rejec{\isacharunderscore}{\kern0pt}n{\isacharunderscore}{\kern0pt}{\isadigit{2}}\ rejects{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}m\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharplus}{\kern0pt}\ card\ {\isacharparenleft}{\kern0pt}reject\ n\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ card\ A\ {\isacharplus}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}geq{\isacharunderscore}{\kern0pt}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ linarith\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ disj{\isacharunderscore}{\kern0pt}comp\ prof{\isacharunderscore}{\kern0pt}A\ fin{\isacharunderscore}{\kern0pt}A\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}m\ card{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}reject\ {\isacharparenleft}{\kern0pt}m\ {\isasymparallel}\isactrlsub {\isasymup}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}card\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Composition_Rules/Result_Rules.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%15=5%:%
-%:%16=6%:%
-%:%17=7%:%
-%:%22=7%:%
-%:%25=8%:%
-%:%26=9%:%
-%:%27=9%:%
-%:%28=10%:%
-%:%29=11%:%
-%:%32=12%:%
-%:%36=12%:%
-%:%37=12%:%
-%:%38=13%:%
-%:%39=14%:%
-%:%40=15%:%
-%:%41=15%:%
-%:%46=15%:%
-%:%49=16%:%
-%:%50=17%:%
-%:%51=18%:%
-%:%52=18%:%
-%:%53=19%:%
-%:%54=20%:%
-%:%55=21%:%
-%:%56=22%:%
-%:%63=23%:%
-%:%64=23%:%
-%:%65=24%:%
-%:%66=24%:%
-%:%67=25%:%
-%:%68=26%:%
-%:%69=27%:%
-%:%70=27%:%
-%:%71=28%:%
-%:%72=28%:%
-%:%73=28%:%
-%:%74=29%:%
-%:%75=30%:%
-%:%76=30%:%
-%:%77=31%:%
-%:%78=31%:%
-%:%79=31%:%
-%:%80=32%:%
-%:%81=33%:%
-%:%82=33%:%
-%:%83=34%:%
-%:%84=35%:%
-%:%85=35%:%
-%:%86=36%:%
-%:%87=36%:%
-%:%88=36%:%
-%:%89=37%:%
-%:%90=38%:%
-%:%91=38%:%
-%:%92=39%:%
-%:%93=39%:%
-%:%94=40%:%
-%:%95=41%:%
-%:%96=42%:%
-%:%97=42%:%
-%:%98=43%:%
-%:%99=43%:%
-%:%100=44%:%
-%:%101=44%:%
-%:%102=44%:%
-%:%103=45%:%
-%:%104=46%:%
-%:%105=46%:%
-%:%106=47%:%
-%:%107=47%:%
-%:%108=48%:%
-%:%109=48%:%
-%:%110=49%:%
-%:%111=49%:%
-%:%112=50%:%
-%:%113=50%:%
-%:%114=51%:%
-%:%115=51%:%
-%:%116=52%:%
-%:%117=53%:%
-%:%118=54%:%
-%:%119=55%:%
-%:%120=55%:%
-%:%121=56%:%
-%:%127=56%:%
-%:%130=57%:%
-%:%131=58%:%
-%:%132=58%:%
-%:%133=59%:%
-%:%134=60%:%
-%:%135=61%:%
-%:%136=62%:%
-%:%143=63%:%
-%:%144=63%:%
-%:%145=64%:%
-%:%146=64%:%
-%:%147=65%:%
-%:%148=66%:%
-%:%149=67%:%
-%:%150=68%:%
-%:%158=76%:%
-%:%159=77%:%
-%:%160=77%:%
-%:%161=78%:%
-%:%162=78%:%
-%:%163=79%:%
-%:%164=79%:%
-%:%165=80%:%
-%:%166=81%:%
-%:%167=82%:%
-%:%168=82%:%
-%:%169=83%:%
-%:%170=83%:%
-%:%171=84%:%
-%:%171=88%:%
-%:%172=89%:%
-%:%173=89%:%
-%:%174=90%:%
-%:%175=90%:%
-%:%176=91%:%
-%:%177=92%:%
-%:%178=93%:%
-%:%179=94%:%
-%:%185=100%:%
-%:%186=101%:%
-%:%187=101%:%
-%:%188=102%:%
-%:%189=102%:%
-%:%190=103%:%
-%:%191=103%:%
-%:%192=104%:%
-%:%193=105%:%
-%:%194=106%:%
-%:%195=107%:%
-%:%196=108%:%
-%:%197=109%:%
-%:%198=109%:%
-%:%199=110%:%
-%:%200=110%:%
-%:%201=111%:%
-%:%202=111%:%
-%:%203=112%:%
-%:%204=113%:%
-%:%205=113%:%
-%:%206=114%:%
-%:%207=114%:%
-%:%208=115%:%
-%:%209=115%:%
-%:%210=116%:%
-%:%211=117%:%
-%:%212=117%:%
-%:%213=118%:%
-%:%214=118%:%
-%:%215=119%:%
-%:%216=119%:%
-%:%217=120%:%
-%:%223=126%:%
-%:%224=127%:%
-%:%225=127%:%
-%:%226=128%:%
-%:%227=129%:%
-%:%228=130%:%
-%:%229=130%:%
-%:%230=131%:%
-%:%231=131%:%
-%:%232=132%:%
-%:%233=132%:%
-%:%234=133%:%
-%:%235=133%:%
-%:%236=134%:%
-%:%237=135%:%
-%:%238=135%:%
-%:%239=136%:%
-%:%240=137%:%
-%:%241=138%:%
-%:%242=138%:%
-%:%243=139%:%
-%:%244=139%:%
-%:%245=140%:%
-%:%246=140%:%
-%:%247=141%:%
-%:%248=141%:%
-%:%249=141%:%
-%:%250=142%:%
-%:%251=142%:%
-%:%252=143%:%
-%:%253=143%:%
-%:%254=144%:%
-%:%255=144%:%
-%:%256=145%:%
-%:%262=145%:%
-%:%265=146%:%
-%:%266=151%:%
-%:%267=152%:%
-%:%268=152%:%
-%:%269=153%:%
-%:%270=154%:%
-%:%271=155%:%
-%:%278=156%:%
-%:%279=156%:%
-%:%280=157%:%
-%:%281=157%:%
-%:%282=158%:%
-%:%283=159%:%
-%:%284=160%:%
-%:%285=160%:%
-%:%286=161%:%
-%:%287=161%:%
-%:%288=162%:%
-%:%289=162%:%
-%:%290=163%:%
-%:%291=164%:%
-%:%292=165%:%
-%:%293=165%:%
-%:%294=166%:%
-%:%295=167%:%
-%:%296=167%:%
-%:%297=168%:%
-%:%298=168%:%
-%:%299=169%:%
-%:%300=169%:%
-%:%301=170%:%
-%:%302=171%:%
-%:%303=172%:%
-%:%304=172%:%
-%:%305=173%:%
-%:%311=173%:%
-%:%314=174%:%
-%:%315=178%:%
-%:%316=179%:%
-%:%317=179%:%
-%:%318=180%:%
-%:%319=181%:%
-%:%320=182%:%
-%:%321=183%:%
-%:%322=184%:%
-%:%325=185%:%
-%:%329=185%:%
-%:%330=185%:%
-%:%331=186%:%
-%:%332=186%:%
-%:%333=187%:%
-%:%334=187%:%
-%:%335=188%:%
-%:%336=188%:%
-%:%337=189%:%
-%:%338=189%:%
-%:%339=190%:%
-%:%340=190%:%
-%:%341=191%:%
-%:%342=191%:%
-%:%343=192%:%
-%:%344=192%:%
-%:%345=193%:%
-%:%346=193%:%
-%:%347=194%:%
-%:%348=194%:%
-%:%349=195%:%
-%:%350=195%:%
-%:%351=196%:%
-%:%352=196%:%
-%:%353=197%:%
-%:%354=197%:%
-%:%355=198%:%
-%:%356=198%:%
-%:%357=199%:%
-%:%358=199%:%
-%:%359=200%:%
-%:%360=200%:%
-%:%361=201%:%
-%:%362=202%:%
-%:%363=203%:%
-%:%364=204%:%
-%:%365=204%:%
-%:%366=205%:%
-%:%367=206%:%
-%:%368=207%:%
-%:%369=208%:%
-%:%370=208%:%
-%:%371=209%:%
-%:%372=209%:%
-%:%373=210%:%
-%:%374=210%:%
-%:%375=211%:%
-%:%376=211%:%
-%:%377=212%:%
-%:%378=212%:%
-%:%379=213%:%
-%:%380=213%:%
-%:%381=214%:%
-%:%382=214%:%
-%:%383=215%:%
-%:%384=216%:%
-%:%385=217%:%
-%:%386=217%:%
-%:%387=218%:%
-%:%388=219%:%
-%:%389=220%:%
-%:%390=220%:%
-%:%391=221%:%
-%:%392=221%:%
-%:%393=222%:%
-%:%394=222%:%
-%:%395=223%:%
-%:%396=223%:%
-%:%397=224%:%
-%:%398=224%:%
-%:%399=225%:%
-%:%400=225%:%
-%:%401=226%:%
-%:%402=227%:%
-%:%403=227%:%
-%:%404=228%:%
-%:%410=228%:%
-%:%413=229%:%
-%:%414=233%:%
-%:%415=234%:%
-%:%416=234%:%
-%:%417=235%:%
-%:%418=236%:%
-%:%419=237%:%
-%:%420=238%:%
-%:%423=239%:%
-%:%427=239%:%
-%:%428=239%:%
-%:%429=240%:%
-%:%430=240%:%
-%:%435=240%:%
-%:%438=241%:%
-%:%439=242%:%
-%:%440=243%:%
-%:%441=243%:%
-%:%442=244%:%
-%:%443=245%:%
-%:%444=246%:%
-%:%445=247%:%
-%:%448=248%:%
-%:%452=248%:%
-%:%453=248%:%
-%:%454=249%:%
-%:%455=250%:%
-%:%456=250%:%
-%:%461=250%:%
-%:%464=251%:%
-%:%465=252%:%
-%:%466=252%:%
-%:%467=253%:%
-%:%468=254%:%
-%:%469=255%:%
-%:%470=256%:%
-%:%471=257%:%
-%:%472=258%:%
-%:%479=259%:%
-%:%480=259%:%
-%:%481=260%:%
-%:%482=260%:%
-%:%483=261%:%
-%:%484=261%:%
-%:%485=262%:%
-%:%486=262%:%
-%:%487=263%:%
-%:%488=264%:%
-%:%489=265%:%
-%:%490=265%:%
-%:%491=266%:%
-%:%497=266%:%
-%:%500=267%:%
-%:%501=268%:%
-%:%502=269%:%
-%:%503=269%:%
-%:%504=270%:%
-%:%505=271%:%
-%:%508=272%:%
-%:%512=272%:%
-%:%513=272%:%
-%:%514=273%:%
-%:%515=273%:%
-%:%516=274%:%
-%:%517=274%:%
-%:%518=275%:%
-%:%519=275%:%
-%:%520=276%:%
-%:%521=276%:%
-%:%522=277%:%
-%:%523=277%:%
-%:%524=278%:%
-%:%525=278%:%
-%:%526=279%:%
-%:%527=280%:%
-%:%528=281%:%
-%:%529=282%:%
-%:%530=282%:%
-%:%531=283%:%
-%:%532=284%:%
-%:%533=285%:%
-%:%534=286%:%
-%:%535=286%:%
-%:%536=287%:%
-%:%537=287%:%
-%:%538=288%:%
-%:%539=289%:%
-%:%540=290%:%
-%:%541=290%:%
-%:%542=291%:%
-%:%548=291%:%
-%:%551=292%:%
-%:%552=296%:%
-%:%553=297%:%
-%:%554=297%:%
-%:%555=298%:%
-%:%556=299%:%
-%:%559=300%:%
-%:%563=300%:%
-%:%564=300%:%
-%:%565=301%:%
-%:%566=301%:%
-%:%567=302%:%
-%:%568=302%:%
-%:%569=303%:%
-%:%570=303%:%
-%:%571=304%:%
-%:%572=304%:%
-%:%573=305%:%
-%:%574=305%:%
-%:%575=306%:%
-%:%576=306%:%
-%:%577=307%:%
-%:%578=308%:%
-%:%579=309%:%
-%:%580=310%:%
-%:%581=310%:%
-%:%582=311%:%
-%:%583=312%:%
-%:%584=313%:%
-%:%585=314%:%
-%:%586=315%:%
-%:%587=315%:%
-%:%588=315%:%
-%:%589=316%:%
-%:%590=317%:%
-%:%591=317%:%
-%:%592=318%:%
-%:%593=319%:%
-%:%594=319%:%
-%:%595=320%:%
-%:%596=320%:%
-%:%597=321%:%
-%:%598=321%:%
-%:%599=322%:%
-%:%600=323%:%
-%:%601=323%:%
-%:%602=324%:%
-%:%608=324%:%
-%:%611=325%:%
-%:%612=331%:%
-%:%613=332%:%
-%:%614=332%:%
-%:%615=333%:%
-%:%616=334%:%
-%:%617=335%:%
-%:%618=336%:%
-%:%619=337%:%
-%:%622=338%:%
-%:%626=338%:%
-%:%627=338%:%
-%:%628=339%:%
-%:%629=339%:%
-%:%630=340%:%
-%:%631=340%:%
-%:%632=341%:%
-%:%633=341%:%
-%:%634=342%:%
-%:%635=342%:%
-%:%636=343%:%
-%:%637=343%:%
-%:%638=344%:%
-%:%639=344%:%
-%:%640=345%:%
-%:%641=345%:%
-%:%642=346%:%
-%:%643=346%:%
-%:%644=347%:%
-%:%645=347%:%
-%:%646=348%:%
-%:%647=348%:%
-%:%648=349%:%
-%:%649=349%:%
-%:%650=350%:%
-%:%651=350%:%
-%:%652=351%:%
-%:%653=352%:%
-%:%654=353%:%
-%:%655=353%:%
-%:%656=354%:%
-%:%657=355%:%
-%:%658=356%:%
-%:%659=357%:%
-%:%660=357%:%
-%:%661=357%:%
-%:%662=358%:%
-%:%663=359%:%
-%:%664=359%:%
-%:%665=360%:%
-%:%666=360%:%
-%:%667=360%:%
-%:%668=361%:%
-%:%669=362%:%
-%:%670=362%:%
-%:%671=363%:%
-%:%672=363%:%
-%:%673=364%:%
-%:%674=364%:%
-%:%675=365%:%
-%:%676=366%:%
-%:%677=366%:%
-%:%678=367%:%
-%:%679=368%:%
-%:%680=369%:%
-%:%681=369%:%
-%:%682=370%:%
-%:%683=370%:%
-%:%684=371%:%
-%:%685=371%:%
-%:%686=372%:%
-%:%687=373%:%
-%:%688=373%:%
-%:%689=374%:%
-%:%690=374%:%
-%:%691=375%:%
-%:%692=376%:%
-%:%693=376%:%
-%:%694=377%:%
-%:%695=378%:%
-%:%696=378%:%
-%:%697=379%:%
-%:%698=379%:%
-%:%699=380%:%
-%:%700=381%:%
-%:%701=382%:%
-%:%702=382%:%
-%:%703=383%:%
-%:%704=384%:%
-%:%705=384%:%
-%:%706=385%:%
-%:%707=385%:%
-%:%708=386%:%
-%:%713=391%:%
-%:%714=392%:%
-%:%715=392%:%
-%:%716=393%:%
-%:%717=393%:%
-%:%718=394%:%
-%:%719=394%:%
-%:%720=395%:%
-%:%721=396%:%
-%:%722=397%:%
-%:%723=397%:%
-%:%724=398%:%
-%:%725=399%:%
-%:%726=400%:%
-%:%727=400%:%
-%:%728=401%:%
-%:%729=401%:%
-%:%730=402%:%
-%:%731=402%:%
-%:%732=403%:%
-%:%733=403%:%
-%:%734=404%:%
-%:%740=404%:%
-%:%743=405%:%
-%:%744=406%:%
-%:%745=406%:%
-%:%746=407%:%
-%:%747=408%:%
-%:%748=409%:%
-%:%749=410%:%
-%:%750=411%:%
-%:%751=412%:%
-%:%752=413%:%
-%:%753=414%:%
-%:%755=416%:%
-%:%762=417%:%
-%:%763=417%:%
-%:%764=418%:%
-%:%765=418%:%
-%:%766=419%:%
-%:%767=419%:%
-%:%768=420%:%
-%:%769=421%:%
-%:%770=422%:%
-%:%771=422%:%
-%:%772=423%:%
-%:%773=423%:%
-%:%774=424%:%
-%:%775=424%:%
-%:%776=425%:%
-%:%777=426%:%
-%:%778=427%:%
-%:%779=427%:%
-%:%780=428%:%
-%:%781=428%:%
-%:%782=429%:%
-%:%783=429%:%
-%:%784=430%:%
-%:%785=430%:%
-%:%786=431%:%
-%:%787=431%:%
-%:%788=432%:%
-%:%789=432%:%
-%:%790=433%:%
-%:%791=433%:%
-%:%792=434%:%
-%:%793=434%:%
-%:%794=435%:%
-%:%795=435%:%
-%:%796=436%:%
-%:%797=436%:%
-%:%798=437%:%
-%:%799=437%:%
-%:%800=438%:%
-%:%801=438%:%
-%:%802=439%:%
-%:%803=439%:%
-%:%804=440%:%
-%:%805=440%:%
-%:%806=441%:%
-%:%807=441%:%
-%:%808=442%:%
-%:%810=444%:%
-%:%811=445%:%
-%:%812=445%:%
-%:%813=446%:%
-%:%814=447%:%
-%:%815=447%:%
-%:%816=448%:%
-%:%817=448%:%
-%:%818=449%:%
-%:%819=449%:%
-%:%820=450%:%
-%:%821=450%:%
-%:%822=451%:%
-%:%823=451%:%
-%:%824=452%:%
-%:%825=452%:%
-%:%826=453%:%
-%:%827=453%:%
-%:%828=454%:%
-%:%829=454%:%
-%:%830=455%:%
-%:%831=455%:%
-%:%832=456%:%
-%:%833=456%:%
-%:%834=457%:%
-%:%835=457%:%
-%:%836=458%:%
-%:%837=458%:%
-%:%838=458%:%
-%:%839=459%:%
-%:%840=460%:%
-%:%841=460%:%
-%:%842=461%:%
-%:%843=461%:%
-%:%844=462%:%
-%:%845=462%:%
-%:%846=463%:%
-%:%847=463%:%
-%:%848=464%:%
-%:%849=464%:%
-%:%850=465%:%
-%:%851=465%:%
-%:%852=466%:%
-%:%853=466%:%
-%:%854=467%:%
-%:%855=467%:%
-%:%856=468%:%
-%:%858=470%:%
-%:%859=471%:%
-%:%860=471%:%
-%:%861=472%:%
-%:%862=472%:%
-%:%863=473%:%
-%:%864=473%:%
-%:%865=474%:%
-%:%868=477%:%
-%:%869=478%:%
-%:%870=478%:%
-%:%871=479%:%
-%:%872=480%:%
-%:%873=480%:%
-%:%874=481%:%
-%:%875=481%:%
-%:%876=482%:%
-%:%877=482%:%
-%:%878=483%:%
-%:%879=483%:%
-%:%880=484%:%
-%:%881=484%:%
-%:%882=485%:%
-%:%883=485%:%
-%:%884=486%:%
-%:%885=486%:%
-%:%886=487%:%
-%:%887=487%:%
-%:%888=488%:%
-%:%889=488%:%
-%:%890=489%:%
-%:%891=489%:%
-%:%892=490%:%
-%:%893=491%:%
-%:%894=492%:%
-%:%895=492%:%
-%:%896=493%:%
-%:%897=493%:%
-%:%898=494%:%
-%:%899=494%:%
-%:%900=495%:%
-%:%901=496%:%
-%:%902=497%:%
-%:%903=497%:%
-%:%904=498%:%
-%:%905=498%:%
-%:%906=499%:%
-%:%907=499%:%
-%:%908=500%:%
-%:%909=501%:%
-%:%910=501%:%
-%:%911=502%:%
-%:%912=502%:%
-%:%913=503%:%
-%:%914=503%:%
-%:%915=504%:%
-%:%916=504%:%
-%:%917=505%:%
-%:%918=505%:%
-%:%919=506%:%
-%:%920=506%:%
-%:%921=507%:%
-%:%922=507%:%
-%:%923=508%:%
-%:%924=508%:%
-%:%925=509%:%
-%:%926=509%:%
-%:%927=509%:%
-%:%928=510%:%
-%:%929=511%:%
-%:%930=511%:%
-%:%931=512%:%
-%:%932=512%:%
-%:%933=513%:%
-%:%934=513%:%
-%:%935=513%:%
-%:%936=514%:%
-%:%937=515%:%
-%:%938=516%:%
-%:%939=516%:%
-%:%940=517%:%
-%:%941=518%:%
-%:%942=518%:%
-%:%943=519%:%
-%:%944=519%:%
-%:%945=519%:%
-%:%946=520%:%
-%:%947=521%:%
-%:%948=522%:%
-%:%949=522%:%
-%:%950=523%:%
-%:%951=523%:%
-%:%952=524%:%
-%:%953=524%:%
-%:%954=524%:%
-%:%955=525%:%
-%:%956=526%:%
-%:%957=526%:%
-%:%958=527%:%
-%:%959=527%:%
-%:%960=527%:%
-%:%961=528%:%
-%:%963=530%:%
-%:%964=531%:%
-%:%965=531%:%
-%:%966=532%:%
-%:%967=532%:%
-%:%968=533%:%
-%:%969=533%:%
-%:%970=534%:%
-%:%971=534%:%
-%:%972=535%:%
-%:%973=535%:%
-%:%974=536%:%
-%:%975=536%:%
-%:%976=537%:%
-%:%977=537%:%
-%:%978=538%:%
-%:%979=538%:%
-%:%980=539%:%
-%:%981=539%:%
-%:%982=540%:%
-%:%983=540%:%
-%:%984=541%:%
-%:%985=541%:%
-%:%986=542%:%
-%:%987=542%:%
-%:%988=543%:%
-%:%989=543%:%
-%:%990=544%:%
-%:%991=544%:%
-%:%992=545%:%
-%:%993=546%:%
-%:%994=547%:%
-%:%995=547%:%
-%:%996=548%:%
-%:%997=548%:%
-%:%998=549%:%
-%:%999=549%:%
-%:%1000=550%:%
-%:%1001=550%:%
-%:%1002=551%:%
-%:%1003=552%:%
-%:%1004=553%:%
-%:%1005=553%:%
-%:%1006=554%:%
-%:%1007=554%:%
-%:%1008=555%:%
-%:%1009=555%:%
-%:%1010=556%:%
-%:%1011=556%:%
-%:%1012=557%:%
-%:%1018=557%:%
-%:%1021=558%:%
-%:%1022=559%:%
-%:%1023=559%:%
-%:%1024=560%:%
-%:%1025=561%:%
-%:%1026=562%:%
-%:%1027=563%:%
-%:%1028=564%:%
-%:%1029=565%:%
-%:%1030=566%:%
-%:%1031=567%:%
-%:%1032=568%:%
-%:%1035=569%:%
-%:%1039=569%:%
-%:%1040=569%:%
-%:%1041=570%:%
-%:%1042=571%:%
-%:%1043=572%:%
-%:%1044=573%:%
-%:%1045=574%:%
-%:%1046=574%:%
-%:%1051=574%:%
-%:%1054=575%:%
-%:%1055=576%:%
-%:%1056=576%:%
-%:%1057=577%:%
-%:%1058=578%:%
-%:%1059=579%:%
-%:%1060=580%:%
-%:%1061=581%:%
-%:%1062=582%:%
-%:%1063=583%:%
-%:%1064=584%:%
-%:%1071=585%:%
-%:%1072=585%:%
-%:%1073=586%:%
-%:%1074=586%:%
-%:%1075=587%:%
-%:%1076=587%:%
-%:%1077=588%:%
-%:%1078=588%:%
-%:%1079=589%:%
-%:%1080=589%:%
-%:%1081=590%:%
-%:%1082=590%:%
-%:%1083=591%:%
-%:%1084=591%:%
-%:%1085=592%:%
-%:%1086=592%:%
-%:%1087=593%:%
-%:%1088=593%:%
-%:%1089=594%:%
-%:%1090=594%:%
-%:%1091=595%:%
-%:%1092=595%:%
-%:%1093=596%:%
-%:%1094=596%:%
-%:%1095=597%:%
-%:%1096=597%:%
-%:%1097=598%:%
-%:%1098=598%:%
-%:%1099=599%:%
-%:%1100=599%:%
-%:%1101=600%:%
-%:%1102=600%:%
-%:%1103=601%:%
-%:%1104=601%:%
-%:%1105=602%:%
-%:%1106=602%:%
-%:%1107=603%:%
-%:%1108=603%:%
-%:%1109=604%:%
-%:%1110=605%:%
-%:%1111=605%:%
-%:%1112=606%:%
-%:%1113=606%:%
-%:%1114=607%:%
-%:%1115=607%:%
-%:%1116=608%:%
-%:%1117=608%:%
-%:%1118=609%:%
-%:%1119=609%:%
-%:%1120=610%:%
-%:%1121=610%:%
-%:%1122=611%:%
-%:%1123=611%:%
-%:%1124=612%:%
-%:%1125=612%:%
-%:%1126=613%:%
-%:%1127=614%:%
-%:%1128=614%:%
-%:%1129=615%:%
-%:%1130=615%:%
-%:%1131=616%:%
-%:%1137=616%:%
-%:%1140=617%:%
-%:%1141=618%:%
-%:%1142=618%:%
-%:%1143=619%:%
-%:%1144=620%:%
-%:%1145=621%:%
-%:%1146=622%:%
-%:%1147=623%:%
-%:%1148=624%:%
-%:%1155=625%:%
-%:%1156=625%:%
-%:%1157=626%:%
-%:%1158=626%:%
-%:%1159=627%:%
-%:%1160=628%:%
-%:%1161=629%:%
-%:%1162=629%:%
-%:%1163=630%:%
-%:%1164=631%:%
-%:%1165=631%:%
-%:%1166=632%:%
-%:%1167=632%:%
-%:%1168=632%:%
-%:%1169=633%:%
-%:%1170=633%:%
-%:%1171=634%:%
-%:%1172=634%:%
-%:%1173=635%:%
-%:%1174=635%:%
-%:%1175=636%:%
-%:%1176=636%:%
-%:%1177=637%:%
-%:%1183=637%:%
-%:%1186=638%:%
-%:%1187=644%:%
-%:%1188=645%:%
-%:%1189=645%:%
-%:%1190=646%:%
-%:%1191=647%:%
-%:%1192=648%:%
-%:%1193=649%:%
-%:%1194=650%:%
-%:%1195=651%:%
-%:%1198=652%:%
-%:%1202=652%:%
-%:%1203=652%:%
-%:%1204=653%:%
-%:%1205=653%:%
-%:%1206=654%:%
-%:%1207=654%:%
-%:%1208=655%:%
-%:%1209=655%:%
-%:%1210=656%:%
-%:%1211=656%:%
-%:%1212=657%:%
-%:%1213=657%:%
-%:%1214=658%:%
-%:%1215=658%:%
-%:%1216=659%:%
-%:%1217=659%:%
-%:%1218=660%:%
-%:%1219=660%:%
-%:%1220=661%:%
-%:%1221=661%:%
-%:%1222=662%:%
-%:%1223=662%:%
-%:%1224=663%:%
-%:%1225=663%:%
-%:%1226=664%:%
-%:%1227=664%:%
-%:%1228=665%:%
-%:%1229=666%:%
-%:%1230=667%:%
-%:%1231=667%:%
-%:%1232=668%:%
-%:%1233=669%:%
-%:%1234=670%:%
-%:%1235=671%:%
-%:%1236=671%:%
-%:%1237=672%:%
-%:%1238=672%:%
-%:%1239=673%:%
-%:%1240=673%:%
-%:%1241=674%:%
-%:%1242=674%:%
-%:%1243=675%:%
-%:%1244=675%:%
-%:%1245=676%:%
-%:%1246=676%:%
-%:%1247=677%:%
-%:%1248=677%:%
-%:%1249=678%:%
-%:%1250=678%:%
-%:%1251=679%:%
-%:%1252=679%:%
-%:%1253=680%:%
-%:%1254=680%:%
-%:%1255=681%:%
-%:%1256=681%:%
-%:%1257=681%:%
-%:%1258=682%:%
-%:%1259=683%:%
-%:%1260=683%:%
-%:%1261=684%:%
-%:%1262=684%:%
-%:%1263=685%:%
-%:%1264=685%:%
-%:%1265=685%:%
-%:%1266=686%:%
-%:%1267=687%:%
-%:%1268=687%:%
-%:%1269=688%:%
-%:%1270=688%:%
-%:%1271=689%:%
-%:%1272=689%:%
-%:%1273=690%:%
-%:%1274=690%:%
-%:%1275=690%:%
-%:%1276=691%:%
-%:%1277=692%:%
-%:%1278=693%:%
-%:%1279=693%:%
-%:%1280=694%:%
-%:%1281=694%:%
-%:%1282=695%:%
-%:%1283=695%:%
-%:%1284=695%:%
-%:%1285=696%:%
-%:%1287=698%:%
-%:%1288=699%:%
-%:%1289=699%:%
-%:%1290=700%:%
-%:%1291=700%:%
-%:%1292=701%:%
-%:%1293=701%:%
-%:%1294=702%:%
-%:%1295=702%:%
-%:%1296=703%:%
-%:%1297=703%:%
-%:%1298=704%:%
-%:%1299=704%:%
-%:%1300=705%:%
-%:%1301=705%:%
-%:%1302=706%:%
-%:%1303=706%:%
-%:%1304=707%:%
-%:%1305=707%:%
-%:%1306=708%:%
-%:%1307=708%:%
-%:%1308=708%:%
-%:%1309=709%:%
-%:%1310=710%:%
-%:%1311=710%:%
-%:%1312=711%:%
-%:%1313=711%:%
-%:%1314=712%:%
-%:%1315=712%:%
-%:%1316=713%:%
-%:%1317=713%:%
-%:%1318=714%:%
-%:%1319=715%:%
-%:%1320=716%:%
-%:%1321=716%:%
-%:%1322=717%:%
-%:%1323=717%:%
-%:%1324=718%:%
-%:%1325=718%:%
-%:%1326=719%:%
-%:%1327=719%:%
-%:%1328=720%:%
-%:%1329=721%:%
-%:%1330=721%:%
-%:%1331=722%:%
-%:%1332=722%:%
-%:%1333=723%:%
-%:%1339=723%:%
-%:%1344=724%:%
-%:%1349=725%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Revision_Composition.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Revision_Composition.tex
deleted file mode 100644
index 6d18707..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Revision_Composition.tex
+++ /dev/null
@@ -1,233 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Revision{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Revision Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Revision{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-A revised electoral module rejects all originally rejected or deferred
-alternatives, and defers the originally elected alternatives.
-It does not elect any alternatives.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ revision{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}revision{\isacharunderscore}{\kern0pt}composition\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharcomma}{\kern0pt}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ rev\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-{\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ {\isacharparenleft}{\kern0pt}{\isachardoublequoteopen}{\isacharunderscore}{\kern0pt}{\isasymdown}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m{\isasymdown}\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ revision{\isacharunderscore}{\kern0pt}composition\ m{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}revision{\isacharunderscore}{\kern0pt}composition\ m{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ elect\ m\ A\ p\ {\isasymsubseteq}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ unity{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}revision{\isacharunderscore}{\kern0pt}composition\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ {\isacharparenleft}{\kern0pt}A\ {\isacharminus}{\kern0pt}\ elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasyminter}\ elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\ disjoint{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}revision{\isacharunderscore}{\kern0pt}composition\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ unity\ disjoint\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}modI{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Composition Rules%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Compositional_Structures/Revision_Composition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%53=19%:%
-%:%63=21%:%
-%:%64=21%:%
-%:%65=22%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=24%:%
-%:%69=25%:%
-%:%70=26%:%
-%:%77=28%:%
-%:%87=30%:%
-%:%88=30%:%
-%:%89=31%:%
-%:%90=32%:%
-%:%97=33%:%
-%:%98=33%:%
-%:%99=34%:%
-%:%100=34%:%
-%:%101=34%:%
-%:%102=35%:%
-%:%103=35%:%
-%:%104=36%:%
-%:%105=36%:%
-%:%106=37%:%
-%:%107=37%:%
-%:%108=38%:%
-%:%109=38%:%
-%:%110=39%:%
-%:%111=39%:%
-%:%112=40%:%
-%:%113=41%:%
-%:%114=42%:%
-%:%115=42%:%
-%:%116=43%:%
-%:%117=43%:%
-%:%118=44%:%
-%:%119=44%:%
-%:%120=45%:%
-%:%121=45%:%
-%:%122=46%:%
-%:%123=47%:%
-%:%124=47%:%
-%:%125=48%:%
-%:%126=48%:%
-%:%127=48%:%
-%:%128=49%:%
-%:%129=49%:%
-%:%130=50%:%
-%:%145=52%:%
-%:%161=58%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Schwartz_Rule.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Schwartz_Rule.tex
deleted file mode 100644
index 638e0c1..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Schwartz_Rule.tex
+++ /dev/null
@@ -1,107 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Schwartz{\isacharunderscore}{\kern0pt}Rule}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-%
-\isamarkupsection{Schwartz Rule%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Schwartz{\isacharunderscore}{\kern0pt}Rule\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Elimination{\isacharunderscore}{\kern0pt}Modules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-This is the Schwartz voting rule. Confusingly, it is sometimes also referred as
-Nanson's rule. The Schwartz rule proceeds as in the classic Nanson's rule, but
-excludes alternatives with a Borda score that is strictly less than the
-average Borda score.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ schwartz{\isacharunderscore}{\kern0pt}rule\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}schwartz{\isacharunderscore}{\kern0pt}rule\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}less{\isacharunderscore}{\kern0pt}average{\isacharunderscore}{\kern0pt}eliminator\ borda{\isacharunderscore}{\kern0pt}score{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Voting_Rules/Schwartz_Rule.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%13=7%:%
-%:%29=9%:%
-%:%30=9%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%55=21%:%
-%:%65=23%:%
-%:%66=23%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%71=26%:%
-%:%76=27%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Sequential_Composition.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Sequential_Composition.tex
deleted file mode 100644
index 5a67aac..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Sequential_Composition.tex
+++ /dev/null
@@ -1,1373 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Sequential{\isacharunderscore}{\kern0pt}Composition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Sequential Composition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Sequential{\isacharunderscore}{\kern0pt}Composition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Properties{\isacharslash}{\kern0pt}Result{\isacharunderscore}{\kern0pt}Properties{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The sequential composition creates a new electoral module from
-two electoral modules. In a sequential composition, the second
-electoral module makes decisions over alternatives deferred by
-the first electoral module.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\isanewline
-\ \ \ \ \ \ \ \ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}sequential{\isacharunderscore}{\kern0pt}composition\ m\ n\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}let\ new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p{\isacharsemicolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ limit{\isacharunderscore}{\kern0pt}profile\ new{\isacharunderscore}{\kern0pt}A\ p\ in\ {\isacharparenleft}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}elect\ n\ new{\isacharunderscore}{\kern0pt}A\ new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\ {\isacharparenleft}{\kern0pt}reject\ n\ new{\isacharunderscore}{\kern0pt}A\ new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ defer\ n\ new{\isacharunderscore}{\kern0pt}A\ new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{abbreviation}\isamarkupfalse%
-\ sequence\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ {\isacharparenleft}{\kern0pt}\isakeyword{infix}\ {\isachardoublequoteopen}{\isasymtriangleright}{\isachardoublequoteclose}\ {\isadigit{5}}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}m\ {\isasymtriangleright}\ n\ {\isacharequal}{\kern0pt}{\isacharequal}{\kern0pt}\ sequential{\isacharunderscore}{\kern0pt}composition\ m\ n{\isachardoublequoteclose}\isanewline
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}disj{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ well{\isacharunderscore}{\kern0pt}formed{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ well{\isacharunderscore}{\kern0pt}formed{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasyminter}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ subset{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasyminter}\ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\isanewline
-\ \ \ \ \ \ \ \ \ \ rev{\isacharunderscore}{\kern0pt}subsetD\ result{\isacharunderscore}{\kern0pt}disj\ distrib{\isacharunderscore}{\kern0pt}imp{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}distrib{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}neutr{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}iff\ sup{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}right\ {\isachardoublequoteopen}{\isadigit{0}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ del{\isacharunderscore}{\kern0pt}insts{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasyminter}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ set{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}mp\ result{\isacharunderscore}{\kern0pt}disj\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ Un{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}Int\ boolean{\isacharunderscore}{\kern0pt}algebra{\isacharunderscore}{\kern0pt}cancel{\isachardot}{\kern0pt}inf{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ inf{\isachardot}{\kern0pt}order{\isacharunderscore}{\kern0pt}iff\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}aci{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ subsetD\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasyminter}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ disjoint{\isacharunderscore}{\kern0pt}iff{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}equal\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ set{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}mp\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\ Int{\isacharunderscore}{\kern0pt}commute\ boolean{\isacharunderscore}{\kern0pt}algebra{\isacharunderscore}{\kern0pt}cancel{\isachardot}{\kern0pt}inf{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ defer{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}rej\ inf{\isachardot}{\kern0pt}commute\ inf{\isachardot}{\kern0pt}orderE\ inf{\isacharunderscore}{\kern0pt}commute\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}aci{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ inf{\isacharunderscore}{\kern0pt}sup{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ result{\isacharunderscore}{\kern0pt}disj\ sup{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}absorb\ sup{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}distrib{\isadigit{1}}\isanewline
-\ \ \ \ \ \ \ \ \ \ distrib{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ sup{\isacharunderscore}{\kern0pt}eq{\isacharunderscore}{\kern0pt}bot{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ ccfv{\isacharunderscore}{\kern0pt}threshold{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\ disjoint{\isacharunderscore}{\kern0pt}n\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ Un{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{moreover}\isamarkupfalse%
-\ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\ f{\isacharunderscore}{\kern0pt}prof\ disjoint{\isacharunderscore}{\kern0pt}m\ disjoint{\isacharunderscore}{\kern0pt}n\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasyminter}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}distrib{\isadigit{2}}\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ distrib{\isacharunderscore}{\kern0pt}imp{\isadigit{2}}\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\ subset{\isacharunderscore}{\kern0pt}Un{\isacharunderscore}{\kern0pt}eq\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}inf{\isacharunderscore}{\kern0pt}distrib{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{ultimately}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}disjoint{\isadigit{3}}\ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ {\isacharparenleft}{\kern0pt}m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isasymunion}\ reject\ m\ A\ p\ {\isasymunion}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}n\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ f{\isacharunderscore}{\kern0pt}prof\ module{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharparenleft}{\kern0pt}n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\ well{\isacharunderscore}{\kern0pt}formed{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p\ {\isacharequal}{\kern0pt}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p\ {\isasymunion}\ elect\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ reject\ m\ A\ p\ {\isasymunion}\ reject\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharcomma}{\kern0pt}\isanewline
-\ \ \ \ \ \ defer\ n\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}A\ {\isacharquery}{\kern0pt}new{\isacharunderscore}{\kern0pt}p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound{\isacharbrackleft}{\kern0pt}simp{\isacharbrackright}{\kern0pt}{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ \ \ \ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}safe{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}r{\isachardot}{\kern0pt}\ well{\isacharunderscore}{\kern0pt}formed\ {\isacharparenleft}{\kern0pt}A{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set{\isacharparenright}{\kern0pt}\ r\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}disjoint{\isadigit{3}}\ r\ {\isasymand}\ set{\isacharunderscore}{\kern0pt}equals{\isacharunderscore}{\kern0pt}partition\ A\ r{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}alts\ fin{\isacharunderscore}{\kern0pt}A\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Lemmata%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}dec{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ empty{\isacharunderscore}{\kern0pt}defer{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ \ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Int{\isacharunderscore}{\kern0pt}lower{\isadigit{1}}\ Un{\isacharunderscore}{\kern0pt}absorb{\isadigit{2}}\ bot{\isachardot}{\kern0pt}extremum{\isacharunderscore}{\kern0pt}uniqueI\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ empty{\isacharunderscore}{\kern0pt}defer\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ prod{\isachardot}{\kern0pt}collapse\isanewline
-\ \ \ \ \ \ \ \ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ result{\isacharunderscore}{\kern0pt}disj\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ electing{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ cases\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ {\isachardoublequoteopen}A\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}n\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ bot{\isachardot}{\kern0pt}extremum{\isacharunderscore}{\kern0pt}uniqueI\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ electing{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ assm{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}A\ {\isasymnoteq}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ ele{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}elect\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ assm\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ finite\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp\ add{\isacharcolon}{\kern0pt}\ Suc{\isacharunderscore}{\kern0pt}leI\ card{\isacharunderscore}{\kern0pt}gt{\isacharunderscore}{\kern0pt}{\isadigit{0}}{\isacharunderscore}{\kern0pt}iff\ defers{\isacharunderscore}{\kern0pt}def{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ def{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ defer\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}{\isadigit{1}}{\isacharunderscore}{\kern0pt}singletonE\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ singletonI\ subsetCE\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ ele\ def\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ \isacommand{have}\isamarkupfalse%
-\ rej{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ reject\ m\ A\ p\ {\isacharequal}{\kern0pt}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ def{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}m\ defers{\isacharunderscore}{\kern0pt}def\ f{\isacharunderscore}{\kern0pt}prof\ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ ele\ rej\ def\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\ res{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ m\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharbraceleft}{\kern0pt}{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ A{\isacharminus}{\kern0pt}{\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}empty\ combine{\isacharunderscore}{\kern0pt}ele{\isacharunderscore}{\kern0pt}rej{\isacharunderscore}{\kern0pt}def\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ reject{\isacharunderscore}{\kern0pt}not{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}or{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ elect\ n\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ def{\isacharunderscore}{\kern0pt}card\ def\ electing{\isacharunderscore}{\kern0pt}n\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ \isacommand{have}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}{\isasymexists}a\ {\isasymin}\ A{\isachardot}{\kern0pt}\ elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\ {\isacharbraceleft}{\kern0pt}a{\isacharbraceright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ electing{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}alt\ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prod{\isachardot}{\kern0pt}sel\isanewline
-\ \ \ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ \ \ sup{\isacharunderscore}{\kern0pt}bot{\isachardot}{\kern0pt}left{\isacharunderscore}{\kern0pt}neutral\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ def\ def{\isacharunderscore}{\kern0pt}card\ electing{\isacharunderscore}{\kern0pt}n\ n{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}m\ f{\isacharunderscore}{\kern0pt}prof\ res{\isacharunderscore}{\kern0pt}m\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ Diff{\isacharunderscore}{\kern0pt}disjoint\ Diff{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}absorb\ Int{\isacharunderscore}{\kern0pt}insert{\isacharunderscore}{\kern0pt}right\isanewline
-\ \ \ \ \ \ \ \ \ \ Un{\isacharunderscore}{\kern0pt}Diff{\isacharunderscore}{\kern0pt}Int\ electing{\isacharunderscore}{\kern0pt}for{\isacharunderscore}{\kern0pt}only{\isacharunderscore}{\kern0pt}alt\ empty{\isacharunderscore}{\kern0pt}iff\isanewline
-\ \ \ \ \ \ \ \ \ \ non{\isacharunderscore}{\kern0pt}electing{\isacharunderscore}{\kern0pt}def\ prod{\isachardot}{\kern0pt}sel\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ singletonI\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}smt\ {\isacharparenleft}{\kern0pt}verit{\isacharcomma}{\kern0pt}\ best{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}card{\isacharunderscore}{\kern0pt}bounded{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ card{\isacharunderscore}{\kern0pt}mono\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ prod{\isachardot}{\kern0pt}sel{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}then{\isacharunderscore}{\kern0pt}elect{\isacharunderscore}{\kern0pt}elec{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}elect\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ elect\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isasymunion}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}elect\ m\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ Un{\isacharunderscore}{\kern0pt}commute\ fst{\isacharunderscore}{\kern0pt}conv\ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}m{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ module{\isacharunderscore}{\kern0pt}n{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ f{\isacharunderscore}{\kern0pt}prof{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ enough{\isacharunderscore}{\kern0pt}leftover{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}card\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isachargreater}{\kern0pt}\ {\isadigit{1}}{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubset}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ enough{\isacharunderscore}{\kern0pt}leftover\ module{\isacharunderscore}{\kern0pt}m\ module{\isacharunderscore}{\kern0pt}n\ f{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ sequential{\isacharunderscore}{\kern0pt}composition{\isachardot}{\kern0pt}simps\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \ \ \ \ single{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}imp{\isacharunderscore}{\kern0pt}red{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\ snd{\isacharunderscore}{\kern0pt}conv\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ metis%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p\ {\isasymsubseteq}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isachardoublequoteopen}{\isasymforall}A\ p{\isachardot}{\kern0pt}\ finite{\isacharunderscore}{\kern0pt}profile\ A\ p\ {\isasymlongrightarrow}\ well{\isacharunderscore}{\kern0pt}formed\ A\ {\isacharparenleft}{\kern0pt}n\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\isanewline
-\ \ \ \ \ \ \ \ well{\isacharunderscore}{\kern0pt}formed\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{hence}\isamarkupfalse%
-\isanewline
-\ \ \ \ {\isachardoublequoteopen}well{\isacharunderscore}{\kern0pt}formed\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ def{\isacharunderscore}{\kern0pt}presv{\isacharunderscore}{\kern0pt}fin{\isacharunderscore}{\kern0pt}prof\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ assms\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}trans{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ {\isacharparenleft}{\kern0pt}defer\ {\isacharparenleft}{\kern0pt}m\ {\isasymtriangleright}\ n{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\ electoral{\isacharunderscore}{\kern0pt}module\ n{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ {\isachardoublequoteopen}finite{\isacharunderscore}{\kern0pt}profile\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\isanewline
-\ \ \ \ {\isachardoublequoteopen}a\ {\isasymin}\ defer\ n\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}limit{\isacharunderscore}{\kern0pt}profile\ {\isacharparenleft}{\kern0pt}defer\ m\ A\ p{\isacharparenright}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isasymand}\isanewline
-\ \ \ \ \ \ a\ {\isasymin}\ defer\ m\ A\ p{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set{\isacharunderscore}{\kern0pt}bounded\ assms{\isacharparenleft}{\kern0pt}{\isadigit{1}}{\isacharparenright}{\kern0pt}\ assms{\isacharparenleft}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ assms{\isacharparenleft}{\kern0pt}{\isadigit{3}}{\isacharparenright}{\kern0pt}\ in{\isacharunderscore}{\kern0pt}mono\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}defers{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}set\isanewline
-\ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ hide{\isacharunderscore}{\kern0pt}lams{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimML
-\isanewline
-%
-\endisadelimML
-%
-\isatagML
-\isacommand{ML}\isamarkupfalse%
-\ {\isacartoucheopen}\isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharasterisk}{\kern0pt}\ val\ w\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}all{\isacharunderscore}{\kern0pt}specifications{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Theory{\isachardot}{\kern0pt}defs{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ x\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ String{\isachardot}{\kern0pt}isPrefix\ {\isachardoublequote}{\kern0pt}Seq{\isachardoublequote}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ snd\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}all{\isacharunderscore}{\kern0pt}specifications{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Theory{\isachardot}{\kern0pt}defs{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ y\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}Sequential{\isacharunderscore}{\kern0pt}Composition{\isachardot}{\kern0pt}sequential{\isacharunderscore}{\kern0pt}composition{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}all{\isacharunderscore}{\kern0pt}specifications{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Theory{\isachardot}{\kern0pt}defs{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ z\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}Sequential{\isacharunderscore}{\kern0pt}Composition{\isachardot}{\kern0pt}sequential{\isacharunderscore}{\kern0pt}composition{\isacharunderscore}{\kern0pt}def{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharhash}{\kern0pt}description\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}hd\ {\isacharparenleft}{\kern0pt}snd\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ y{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ val\ def\ {\isacharequal}{\kern0pt}\ hd\ z\isanewline
-\ \ \ \ val\ def{\isacharunderscore}{\kern0pt}rhs\ {\isacharequal}{\kern0pt}\ {\isacharhash}{\kern0pt}rhs\ def\isanewline
-\isanewline
-\ \ \ \ val\ def{\isacharunderscore}{\kern0pt}funs\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}fun{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}def{\isacharunderscore}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ def{\isacharunderscore}{\kern0pt}consts\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ not\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}fun{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}def{\isacharunderscore}{\kern0pt}rhs{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ val\ fun{\isacharunderscore}{\kern0pt}type\ {\isacharequal}{\kern0pt}\ Defs{\isachardot}{\kern0pt}pretty{\isacharunderscore}{\kern0pt}entry\ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}global{\isacharunderscore}{\kern0pt}context\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}List{\isachardot}{\kern0pt}last\ def{\isacharunderscore}{\kern0pt}funs{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ {\isacharparenleft}{\kern0pt}{\isacharasterisk}{\kern0pt}\ val\ a\ {\isacharequal}{\kern0pt}\ Defs{\isachardot}{\kern0pt}pretty{\isacharunderscore}{\kern0pt}entry\ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}global{\isacharunderscore}{\kern0pt}context\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}nth\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}{\isacharhash}{\kern0pt}lhs\ {\isacharparenleft}{\kern0pt}hd\ z{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}\ {\isacharhash}{\kern0pt}rhs\ {\isacharparenleft}{\kern0pt}hd\ z{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenright}{\kern0pt}{\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isacharasterisk}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ {\isacartoucheclose}\isanewline
-\isanewline
-\isacommand{ML}\isamarkupfalse%
-\ {\isacartoucheopen}\ val\ facts\ {\isacharequal}{\kern0pt}\ Global{\isacharunderscore}{\kern0pt}Theory{\isachardot}{\kern0pt}facts{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}\ \isanewline
-\ \ \ \ \ val\ filtered\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ true{\isacharparenright}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacartoucheclose}\isanewline
-\isanewline
-\isacommand{ML{\isacharunderscore}{\kern0pt}val}\isamarkupfalse%
-\ {\isacartoucheopen}val\ seq{\isacharunderscore}{\kern0pt}consts\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isPrefix\ {\isacharparenleft}{\kern0pt}Context{\isachardot}{\kern0pt}theory{\isacharunderscore}{\kern0pt}name\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ {\isacharparenleft}{\kern0pt}Defs{\isachardot}{\kern0pt}all{\isacharunderscore}{\kern0pt}specifications{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}Theory{\isachardot}{\kern0pt}defs{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ names\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharhash}{\kern0pt}description\ {\isacharparenleft}{\kern0pt}hd\ {\isacharparenleft}{\kern0pt}snd{\isacharparenleft}{\kern0pt}x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ seq{\isacharunderscore}{\kern0pt}consts\isanewline
-\ \ \ \ val\ seq{\isacharunderscore}{\kern0pt}defs\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isSuffix\ {\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}{\isacharunderscore}{\kern0pt}def{\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ names\ \isanewline
-\ \ \ \ val\ strip{\isacharunderscore}{\kern0pt}fun\ {\isacharequal}{\kern0pt}\ fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ String{\isachardot}{\kern0pt}substring\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}x{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}{\isadigit{0}}{\isacharcomma}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}size\ x{\isacharparenright}{\kern0pt}{\isacharminus}{\kern0pt}{\isadigit{4}}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ seq{\isacharunderscore}{\kern0pt}defs{\isacharunderscore}{\kern0pt}stripped\ {\isacharequal}{\kern0pt}\ map\ strip{\isacharunderscore}{\kern0pt}fun\ seq{\isacharunderscore}{\kern0pt}defs\isanewline
-\isanewline
-\ \ \ \ fun\ member{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}item{\isacharcomma}{\kern0pt}\ list{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ List{\isachardot}{\kern0pt}exists\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ x\ {\isacharequal}{\kern0pt}\ item{\isacharparenright}{\kern0pt}\ list\isanewline
-\ \ \ \ val\ filter{\isacharunderscore}{\kern0pt}fun\ {\isacharequal}{\kern0pt}\ fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ member{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}x{\isacharcomma}{\kern0pt}\ seq{\isacharunderscore}{\kern0pt}defs{\isacharunderscore}{\kern0pt}stripped{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ val\ constants\ {\isacharequal}{\kern0pt}\ {\isacharhash}{\kern0pt}constants\ {\isacharparenleft}{\kern0pt}Consts{\isachardot}{\kern0pt}dest\ {\isacharparenleft}{\kern0pt}Sign{\isachardot}{\kern0pt}consts{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ filtered{\isacharunderscore}{\kern0pt}constants\ {\isacharequal}{\kern0pt}\ filter\ {\isacharparenleft}{\kern0pt}filter{\isacharunderscore}{\kern0pt}fun{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}map\ fst\ constants{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ val\ reduce{\isacharunderscore}{\kern0pt}noise\ {\isacharequal}{\kern0pt}\ fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ not\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isSuffix\ {\isachardoublequote}{\kern0pt}{\isacharunderscore}{\kern0pt}graph{\isachardoublequote}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ orelse\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isSuffix\ {\isachardoublequote}{\kern0pt}{\isacharunderscore}{\kern0pt}sumC{\isachardoublequote}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ orelse\ {\isacharparenleft}{\kern0pt}String{\isachardot}{\kern0pt}isSuffix\ {\isachardoublequote}{\kern0pt}{\isacharunderscore}{\kern0pt}rel{\isachardoublequote}{\kern0pt}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ \isanewline
-\isanewline
-\ \ \ \ val\ actual{\isacharunderscore}{\kern0pt}constants\ {\isacharequal}{\kern0pt}\ filter\ reduce{\isacharunderscore}{\kern0pt}noise\ filtered{\isacharunderscore}{\kern0pt}constants\isanewline
-\isanewline
-\ \ \ \ val\ constants\ {\isacharequal}{\kern0pt}\ {\isacharhash}{\kern0pt}constants\ {\isacharparenleft}{\kern0pt}Consts{\isachardot}{\kern0pt}dest\ {\isacharparenleft}{\kern0pt}Sign{\isachardot}{\kern0pt}consts{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ filter{\isacharunderscore}{\kern0pt}fun{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ member{\isacharunderscore}{\kern0pt}of\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharcomma}{\kern0pt}\ actual{\isacharunderscore}{\kern0pt}constants{\isacharparenright}{\kern0pt}\isanewline
-\ \ \ \ val\ sign\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}snd\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}filter\ filter{\isacharunderscore}{\kern0pt}fun{\isadigit{2}}\ constants{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\isanewline
-\isanewline
-\ \ \ \ val\ x\ {\isacharequal}{\kern0pt}\ hd\ sign\isanewline
-\isanewline
-\ \ \ \ val\ x{\isacharunderscore}{\kern0pt}string\ {\isacharequal}{\kern0pt}\ Syntax{\isachardot}{\kern0pt}string{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}typ{\isacharunderscore}{\kern0pt}global\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}\ x\isanewline
-\isanewline
-\ \ \ \ val\ x{\isacharunderscore}{\kern0pt}pretty\ {\isacharequal}{\kern0pt}\ Syntax{\isachardot}{\kern0pt}pretty{\isacharunderscore}{\kern0pt}typ{\isacharunderscore}{\kern0pt}global\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}\ x\isanewline
-\isanewline
-\ \ \ \ val\ x{\isacharunderscore}{\kern0pt}pretty{\isacharunderscore}{\kern0pt}string\ {\isacharequal}{\kern0pt}\ Pretty{\isachardot}{\kern0pt}string{\isacharunderscore}{\kern0pt}of\ x{\isacharunderscore}{\kern0pt}pretty\isanewline
-\isanewline
-\ \ \ \ val\ y\ {\isacharequal}{\kern0pt}\ Logic{\isachardot}{\kern0pt}mk{\isacharunderscore}{\kern0pt}type\ x\isanewline
-\isanewline
-\ \ \ \ val\ n\ {\isacharequal}{\kern0pt}\ Term{\isachardot}{\kern0pt}size{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}typ\ x\isanewline
-\isanewline
-\ \ \ \ val\ pm\ {\isacharequal}{\kern0pt}\ fn\ Basic{\isacharunderscore}{\kern0pt}Term{\isachardot}{\kern0pt}Type\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ snd\ x\isanewline
-\isanewline
-\ \ \ \ fun\ collapse\ {\isacharparenleft}{\kern0pt}lst{\isacharcolon}{\kern0pt}\ {\isacharprime}{\kern0pt}a\ list\ list{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ case\ lst\ of\isanewline
-\ \ \ \ \ \ \ \ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharbar}{\kern0pt}\ hd{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}tl\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ hd\ {\isacharat}{\kern0pt}\ collapse\ tl\isanewline
-\isanewline
-\ \ \ \ val\ a\ {\isacharequal}{\kern0pt}\ fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\isanewline
-\ \ \ \ \ \ let\isanewline
-\ \ \ \ \ \ \ \ fun\ typ{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}string\ {\isacharparenleft}{\kern0pt}x{\isacharcolon}{\kern0pt}\ Basic{\isacharunderscore}{\kern0pt}Term{\isachardot}{\kern0pt}typ{\isacharparenright}{\kern0pt}{\isacharcolon}{\kern0pt}\ string\ {\isacharequal}{\kern0pt}\ case\ x\ of\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ Type\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}\ {\isacharcircum}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}\ String{\isachardot}{\kern0pt}concat\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}typ{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}string{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}\ {\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharbar}{\kern0pt}\ \ {\isacharunderscore}{\kern0pt}\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ in\isanewline
-\ \ \ \ \ \ \ \ typ{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}string\ x\isanewline
-\ \ \ \ \ \ end\isanewline
-\ \ \ \ \ \ \ \isanewline
-\ \ \ \ val\ x{\isacharunderscore}{\kern0pt}string{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ a\ x\isanewline
-\isanewline
-\ \ \ \ val\ z\ {\isacharequal}{\kern0pt}\ Term{\isachardot}{\kern0pt}is{\isacharunderscore}{\kern0pt}Bound\ y\isanewline
-\isanewline
-\isanewline
-\isanewline
-val\ a\ {\isacharequal}{\kern0pt}\ fn\ thy\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharhash}{\kern0pt}constants\ {\isacharparenleft}{\kern0pt}Consts{\isachardot}{\kern0pt}dest\ {\isacharparenleft}{\kern0pt}Sign{\isachardot}{\kern0pt}consts{\isacharunderscore}{\kern0pt}of\ thy{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharcomma}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ let\ fun\ typ{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}string\ {\isacharparenleft}{\kern0pt}x{\isacharcolon}{\kern0pt}\ Basic{\isacharunderscore}{\kern0pt}Term{\isachardot}{\kern0pt}typ{\isacharparenright}{\kern0pt}{\isacharcolon}{\kern0pt}\ string\ {\isacharequal}{\kern0pt}\ case\ x\ of\isanewline
-Type\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isachardoublequote}{\kern0pt}\ {\isacharcircum}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fst\ x{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}\ String{\isachardot}{\kern0pt}concat\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}typ{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}string{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}snd\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharcircum}{\kern0pt}\ {\isachardoublequote}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}\isanewline
-{\isacharbar}{\kern0pt}\ {\isacharunderscore}{\kern0pt}\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isachardoublequote}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}{\isacharprime}{\kern0pt}a{\isacharparenright}{\kern0pt}{\isachardoublequote}{\kern0pt}\ in\ typ{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}string\ x\ end{\isacharparenright}{\kern0pt}{\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fst\ {\isacharparenleft}{\kern0pt}snd\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isacharhash}{\kern0pt}constants\ {\isacharparenleft}{\kern0pt}Consts{\isachardot}{\kern0pt}dest\ {\isacharparenleft}{\kern0pt}Sign{\isachardot}{\kern0pt}consts{\isacharunderscore}{\kern0pt}of\ thy{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ val\ b\ {\isacharequal}{\kern0pt}\ rev\ {\isacharparenleft}{\kern0pt}ListPair{\isachardot}{\kern0pt}zip\ {\isacharparenleft}{\kern0pt}a\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ {\isacartoucheclose}\isanewline
-\isanewline
-\isacommand{ML}\isamarkupfalse%
-\ {\isacartoucheopen}\isanewline
-\ \ \ \ val\ constants\ {\isacharequal}{\kern0pt}\ {\isacharhash}{\kern0pt}constants\ {\isacharparenleft}{\kern0pt}Consts{\isachardot}{\kern0pt}dest\ {\isacharparenleft}{\kern0pt}Sign{\isachardot}{\kern0pt}consts{\isacharunderscore}{\kern0pt}of\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isanewline
-\ \ \ \ val\ sign{\isadigit{2}}\ {\isacharequal}{\kern0pt}rev\ \ {\isacharparenleft}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fst\ \ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ constants\ {\isacharparenright}{\kern0pt}\ {\isacartoucheclose}\isanewline
-\isanewline
-\isanewline
-\isacommand{ML{\isacharunderscore}{\kern0pt}val}\isamarkupfalse%
-\ {\isacartoucheopen}\isanewline
-val\ thms\ {\isacharequal}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fn\ x\ {\isacharequal}{\kern0pt}{\isachargreater}{\kern0pt}\ fst\ {\isacharparenleft}{\kern0pt}snd\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}Global{\isacharunderscore}{\kern0pt}Theory{\isachardot}{\kern0pt}dest{\isacharunderscore}{\kern0pt}thm{\isacharunderscore}{\kern0pt}names\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-val\ thm\ {\isacharequal}{\kern0pt}\ Global{\isacharunderscore}{\kern0pt}Theory{\isachardot}{\kern0pt}get{\isacharunderscore}{\kern0pt}thm\ {\isacharat}{\kern0pt}{\isacharbraceleft}{\kern0pt}theory{\isacharbraceright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}hd\ {\isacharparenleft}{\kern0pt}rev\ thms{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharsemicolon}{\kern0pt}\isanewline
-{\isacharparenleft}{\kern0pt}Syntax{\isachardot}{\kern0pt}string{\isacharunderscore}{\kern0pt}of{\isacharunderscore}{\kern0pt}term{\isacharunderscore}{\kern0pt}global\ \isactrltheory {\isasymopen}Main{\isasymclose}\ {\isacharparenleft}{\kern0pt}Thm{\isachardot}{\kern0pt}prop{\isacharunderscore}{\kern0pt}of\ thm{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharbar}{\kern0pt}{\isachargreater}{\kern0pt}\ writeln\isanewline
-{\isacartoucheclose}%
-\endisatagML
-{\isafoldML}%
-%
-\isadelimML
-\ \isanewline
-%
-\endisadelimML
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Compositional_Structures/Sequential_Composition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%35=14%:%
-%:%44=17%:%
-%:%45=18%:%
-%:%46=19%:%
-%:%47=20%:%
-%:%56=22%:%
-%:%66=24%:%
-%:%67=24%:%
-%:%68=25%:%
-%:%69=26%:%
-%:%74=31%:%
-%:%75=32%:%
-%:%76=33%:%
-%:%77=33%:%
-%:%78=34%:%
-%:%79=35%:%
-%:%80=36%:%
-%:%81=37%:%
-%:%82=38%:%
-%:%83=38%:%
-%:%84=39%:%
-%:%85=40%:%
-%:%86=41%:%
-%:%87=42%:%
-%:%94=43%:%
-%:%95=43%:%
-%:%96=44%:%
-%:%97=44%:%
-%:%98=45%:%
-%:%99=45%:%
-%:%100=46%:%
-%:%101=46%:%
-%:%102=46%:%
-%:%103=47%:%
-%:%104=47%:%
-%:%105=48%:%
-%:%106=48%:%
-%:%107=49%:%
-%:%108=49%:%
-%:%109=49%:%
-%:%110=50%:%
-%:%111=51%:%
-%:%112=51%:%
-%:%113=52%:%
-%:%114=52%:%
-%:%115=53%:%
-%:%116=53%:%
-%:%117=53%:%
-%:%118=54%:%
-%:%119=55%:%
-%:%120=55%:%
-%:%121=56%:%
-%:%122=57%:%
-%:%123=57%:%
-%:%124=58%:%
-%:%125=58%:%
-%:%126=59%:%
-%:%127=59%:%
-%:%128=60%:%
-%:%129=61%:%
-%:%130=61%:%
-%:%131=62%:%
-%:%132=63%:%
-%:%133=64%:%
-%:%134=65%:%
-%:%135=66%:%
-%:%136=66%:%
-%:%137=67%:%
-%:%138=67%:%
-%:%139=68%:%
-%:%140=68%:%
-%:%141=69%:%
-%:%142=70%:%
-%:%143=70%:%
-%:%144=71%:%
-%:%145=72%:%
-%:%146=73%:%
-%:%147=74%:%
-%:%148=74%:%
-%:%149=75%:%
-%:%150=75%:%
-%:%151=76%:%
-%:%152=76%:%
-%:%153=77%:%
-%:%154=78%:%
-%:%155=78%:%
-%:%156=79%:%
-%:%157=80%:%
-%:%158=81%:%
-%:%159=81%:%
-%:%160=82%:%
-%:%161=82%:%
-%:%162=82%:%
-%:%163=83%:%
-%:%164=84%:%
-%:%165=85%:%
-%:%166=85%:%
-%:%167=86%:%
-%:%168=87%:%
-%:%169=88%:%
-%:%170=88%:%
-%:%171=89%:%
-%:%172=89%:%
-%:%173=89%:%
-%:%174=89%:%
-%:%175=90%:%
-%:%176=91%:%
-%:%177=92%:%
-%:%178=92%:%
-%:%179=93%:%
-%:%180=93%:%
-%:%181=94%:%
-%:%182=94%:%
-%:%183=94%:%
-%:%184=95%:%
-%:%185=95%:%
-%:%186=96%:%
-%:%187=97%:%
-%:%188=98%:%
-%:%189=98%:%
-%:%190=99%:%
-%:%191=100%:%
-%:%192=101%:%
-%:%193=101%:%
-%:%194=102%:%
-%:%195=102%:%
-%:%196=102%:%
-%:%197=103%:%
-%:%199=105%:%
-%:%200=106%:%
-%:%201=106%:%
-%:%202=107%:%
-%:%203=107%:%
-%:%204=108%:%
-%:%205=108%:%
-%:%206=109%:%
-%:%207=109%:%
-%:%208=110%:%
-%:%214=110%:%
-%:%217=111%:%
-%:%218=112%:%
-%:%219=112%:%
-%:%220=113%:%
-%:%221=114%:%
-%:%222=115%:%
-%:%223=116%:%
-%:%230=117%:%
-%:%231=117%:%
-%:%232=118%:%
-%:%233=118%:%
-%:%234=119%:%
-%:%235=119%:%
-%:%236=120%:%
-%:%237=120%:%
-%:%238=120%:%
-%:%239=121%:%
-%:%240=121%:%
-%:%241=122%:%
-%:%242=122%:%
-%:%243=122%:%
-%:%244=123%:%
-%:%245=124%:%
-%:%246=124%:%
-%:%247=125%:%
-%:%248=125%:%
-%:%249=125%:%
-%:%250=126%:%
-%:%251=127%:%
-%:%252=127%:%
-%:%253=128%:%
-%:%254=128%:%
-%:%255=129%:%
-%:%256=129%:%
-%:%257=129%:%
-%:%258=130%:%
-%:%260=132%:%
-%:%261=133%:%
-%:%262=133%:%
-%:%263=134%:%
-%:%264=134%:%
-%:%265=135%:%
-%:%266=135%:%
-%:%267=135%:%
-%:%268=136%:%
-%:%270=138%:%
-%:%271=139%:%
-%:%272=139%:%
-%:%273=140%:%
-%:%274=140%:%
-%:%275=141%:%
-%:%278=144%:%
-%:%279=145%:%
-%:%280=145%:%
-%:%281=146%:%
-%:%282=146%:%
-%:%283=147%:%
-%:%284=147%:%
-%:%285=148%:%
-%:%286=148%:%
-%:%287=149%:%
-%:%302=151%:%
-%:%312=153%:%
-%:%313=153%:%
-%:%314=154%:%
-%:%315=155%:%
-%:%316=156%:%
-%:%319=157%:%
-%:%323=157%:%
-%:%324=157%:%
-%:%325=158%:%
-%:%326=158%:%
-%:%327=159%:%
-%:%328=159%:%
-%:%329=160%:%
-%:%330=161%:%
-%:%331=162%:%
-%:%332=162%:%
-%:%333=163%:%
-%:%334=164%:%
-%:%335=165%:%
-%:%336=165%:%
-%:%337=166%:%
-%:%338=167%:%
-%:%339=167%:%
-%:%340=168%:%
-%:%341=168%:%
-%:%342=169%:%
-%:%343=169%:%
-%:%344=170%:%
-%:%345=171%:%
-%:%346=171%:%
-%:%347=172%:%
-%:%362=174%:%
-%:%372=176%:%
-%:%373=176%:%
-%:%374=177%:%
-%:%375=178%:%
-%:%376=179%:%
-%:%377=180%:%
-%:%378=181%:%
-%:%379=182%:%
-%:%382=183%:%
-%:%386=183%:%
-%:%387=183%:%
-%:%388=184%:%
-%:%389=185%:%
-%:%390=186%:%
-%:%391=187%:%
-%:%392=187%:%
-%:%397=187%:%
-%:%400=188%:%
-%:%401=189%:%
-%:%402=189%:%
-%:%403=190%:%
-%:%404=191%:%
-%:%405=192%:%
-%:%406=193%:%
-%:%407=194%:%
-%:%408=195%:%
-%:%415=196%:%
-%:%416=196%:%
-%:%417=197%:%
-%:%418=197%:%
-%:%419=198%:%
-%:%420=198%:%
-%:%421=198%:%
-%:%422=199%:%
-%:%423=199%:%
-%:%424=200%:%
-%:%425=201%:%
-%:%426=201%:%
-%:%427=202%:%
-%:%428=202%:%
-%:%429=203%:%
-%:%430=203%:%
-%:%431=204%:%
-%:%432=204%:%
-%:%433=204%:%
-%:%434=205%:%
-%:%435=205%:%
-%:%436=206%:%
-%:%437=206%:%
-%:%438=207%:%
-%:%439=207%:%
-%:%440=207%:%
-%:%441=208%:%
-%:%442=209%:%
-%:%443=209%:%
-%:%444=210%:%
-%:%445=210%:%
-%:%446=210%:%
-%:%447=211%:%
-%:%448=212%:%
-%:%449=212%:%
-%:%450=213%:%
-%:%451=214%:%
-%:%452=214%:%
-%:%453=215%:%
-%:%454=215%:%
-%:%455=215%:%
-%:%456=216%:%
-%:%457=217%:%
-%:%458=217%:%
-%:%459=218%:%
-%:%460=218%:%
-%:%461=219%:%
-%:%462=219%:%
-%:%463=219%:%
-%:%464=220%:%
-%:%465=221%:%
-%:%466=221%:%
-%:%467=222%:%
-%:%468=223%:%
-%:%469=223%:%
-%:%470=224%:%
-%:%471=224%:%
-%:%472=225%:%
-%:%473=226%:%
-%:%474=227%:%
-%:%475=227%:%
-%:%476=228%:%
-%:%477=229%:%
-%:%478=229%:%
-%:%479=230%:%
-%:%480=230%:%
-%:%481=230%:%
-%:%482=231%:%
-%:%483=232%:%
-%:%484=232%:%
-%:%485=233%:%
-%:%486=234%:%
-%:%487=235%:%
-%:%488=235%:%
-%:%489=236%:%
-%:%490=236%:%
-%:%491=237%:%
-%:%492=237%:%
-%:%493=238%:%
-%:%494=238%:%
-%:%495=239%:%
-%:%496=240%:%
-%:%497=241%:%
-%:%498=242%:%
-%:%499=242%:%
-%:%500=243%:%
-%:%506=243%:%
-%:%509=244%:%
-%:%510=245%:%
-%:%511=245%:%
-%:%512=246%:%
-%:%513=247%:%
-%:%514=248%:%
-%:%515=249%:%
-%:%516=250%:%
-%:%519=251%:%
-%:%523=251%:%
-%:%524=251%:%
-%:%525=252%:%
-%:%526=253%:%
-%:%527=253%:%
-%:%532=253%:%
-%:%535=254%:%
-%:%536=255%:%
-%:%537=255%:%
-%:%538=256%:%
-%:%539=257%:%
-%:%540=258%:%
-%:%541=259%:%
-%:%542=260%:%
-%:%545=261%:%
-%:%549=261%:%
-%:%550=261%:%
-%:%551=262%:%
-%:%552=263%:%
-%:%553=263%:%
-%:%558=263%:%
-%:%561=264%:%
-%:%562=265%:%
-%:%563=265%:%
-%:%564=266%:%
-%:%565=267%:%
-%:%566=268%:%
-%:%567=269%:%
-%:%568=270%:%
-%:%569=271%:%
-%:%570=272%:%
-%:%573=273%:%
-%:%577=273%:%
-%:%578=273%:%
-%:%579=274%:%
-%:%580=274%:%
-%:%585=274%:%
-%:%588=275%:%
-%:%589=276%:%
-%:%590=276%:%
-%:%591=277%:%
-%:%592=278%:%
-%:%593=279%:%
-%:%594=280%:%
-%:%595=281%:%
-%:%596=282%:%
-%:%598=284%:%
-%:%601=285%:%
-%:%605=285%:%
-%:%606=285%:%
-%:%607=286%:%
-%:%608=286%:%
-%:%613=286%:%
-%:%616=287%:%
-%:%617=288%:%
-%:%618=288%:%
-%:%619=289%:%
-%:%620=290%:%
-%:%621=291%:%
-%:%622=292%:%
-%:%623=293%:%
-%:%624=294%:%
-%:%627=295%:%
-%:%631=295%:%
-%:%632=295%:%
-%:%633=296%:%
-%:%634=297%:%
-%:%635=298%:%
-%:%636=298%:%
-%:%641=298%:%
-%:%644=299%:%
-%:%645=300%:%
-%:%646=300%:%
-%:%647=301%:%
-%:%648=302%:%
-%:%649=303%:%
-%:%650=304%:%
-%:%651=305%:%
-%:%658=306%:%
-%:%659=306%:%
-%:%660=307%:%
-%:%661=307%:%
-%:%662=308%:%
-%:%663=308%:%
-%:%664=309%:%
-%:%665=309%:%
-%:%666=310%:%
-%:%667=310%:%
-%:%668=311%:%
-%:%670=313%:%
-%:%671=314%:%
-%:%672=314%:%
-%:%673=315%:%
-%:%674=315%:%
-%:%675=316%:%
-%:%676=317%:%
-%:%677=318%:%
-%:%678=318%:%
-%:%679=319%:%
-%:%680=319%:%
-%:%681=320%:%
-%:%682=320%:%
-%:%683=321%:%
-%:%684=321%:%
-%:%685=322%:%
-%:%686=322%:%
-%:%687=323%:%
-%:%693=323%:%
-%:%696=324%:%
-%:%697=325%:%
-%:%698=325%:%
-%:%699=326%:%
-%:%700=327%:%
-%:%701=328%:%
-%:%702=329%:%
-%:%703=330%:%
-%:%704=331%:%
-%:%706=333%:%
-%:%709=334%:%
-%:%713=334%:%
-%:%714=334%:%
-%:%715=335%:%
-%:%716=336%:%
-%:%717=336%:%
-%:%722=336%:%
-%:%727=337%:%
-%:%732=338%:%
-%:%733=338%:%
-%:%750=355%:%
-%:%751=356%:%
-%:%752=357%:%
-%:%753=357%:%
-%:%754=358%:%
-%:%755=359%:%
-%:%756=360%:%
-%:%757=360%:%
-%:%821=424%:%
-%:%822=425%:%
-%:%823=426%:%
-%:%824=426%:%
-%:%827=429%:%
-%:%828=430%:%
-%:%829=431%:%
-%:%830=432%:%
-%:%831=432%:%
-%:%840=436%:%
-%:%845=437%:%
-%:%850=438%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Sequential_Majority_Comparison.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Sequential_Majority_Comparison.tex
deleted file mode 100644
index 471c3ab..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Sequential_Majority_Comparison.tex
+++ /dev/null
@@ -1,1341 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Sequential{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Comparison}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Sequential Majority Comparison%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Sequential{\isacharunderscore}{\kern0pt}Majority{\isacharunderscore}{\kern0pt}Comparison\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Plurality{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Pass{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Basic{\isacharunderscore}{\kern0pt}Modules{\isacharslash}{\kern0pt}Drop{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Structures{\isacharslash}{\kern0pt}Revision{\isacharunderscore}{\kern0pt}Composition{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Composites{\isacharslash}{\kern0pt}Composite{\isacharunderscore}{\kern0pt}Structures{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Monotonicity{\isacharunderscore}{\kern0pt}Rules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Result{\isacharunderscore}{\kern0pt}Rules{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isacharunderscore}{\kern0pt}Facts{\isachardoublequoteclose}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Composition{\isacharunderscore}{\kern0pt}Rules{\isacharslash}{\kern0pt}Disjoint{\isacharunderscore}{\kern0pt}Compatibility{\isacharunderscore}{\kern0pt}Rules{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-Sequential majority comparison compares two alternatives by plurality voting.
-The loser gets rejected, and the winner is compared to the next alternative.
-This process is repeated until only a single alternative is left, which is
-then elected.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{fun}\isamarkupfalse%
-\ smc\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Preference{\isacharunderscore}{\kern0pt}Relation\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}smc\ x\ A\ p\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isasymup}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isasymcirclearrowleft}\isactrlsub {\isasymexists}\isactrlsub {\isacharbang}{\kern0pt}\isactrlsub d{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ elect{\isacharunderscore}{\kern0pt}module{\isacharparenright}{\kern0pt}\ A\ p{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Soundness%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ smc{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}smc\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{unfolding}\isamarkupfalse%
-\ electoral{\isacharunderscore}{\kern0pt}module{\isacharunderscore}{\kern0pt}def\isanewline
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}simp{\isacharcomma}{\kern0pt}\ safe{\isacharcomma}{\kern0pt}\ simp{\isacharunderscore}{\kern0pt}all{\isacharparenright}{\kern0pt}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ elect{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ elect{\isacharunderscore}{\kern0pt}xa\ emptyE\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ reject{\isacharunderscore}{\kern0pt}xa\isanewline
-\ \ \ \ \ \ \ \ \ \ plurality{\isacharunderscore}{\kern0pt}sound\ result{\isacharunderscore}{\kern0pt}disj\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}False{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ IntI\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ defer{\isacharunderscore}{\kern0pt}xa\ emptyE\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ reject{\isacharunderscore}{\kern0pt}xa\isanewline
-\ \ \ \ \ \ \ \ \ \ plurality{\isacharunderscore}{\kern0pt}sound\ result{\isacharunderscore}{\kern0pt}disj\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ elect{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ elect{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ elect{\isacharunderscore}{\kern0pt}xa\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}mono\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ plurality{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ defer{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ defer\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ defer{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ defer{\isacharunderscore}{\kern0pt}xa\ fin{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ in{\isacharunderscore}{\kern0pt}mono\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ plurality{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ prof{\isacharunderscore}{\kern0pt}A\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharcomma}{\kern0pt}\ lifting{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ plurality{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}plurality{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}{\isacharprime}{\kern0pt}a\ set\ {\isasymRightarrow}\ {\isacharparenleft}{\kern0pt}{\isacharunderscore}{\kern0pt}\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}{\isacharparenright}{\kern0pt}\ set\ list\ {\isasymRightarrow}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set\ {\isasymtimes}\ {\isacharunderscore}{\kern0pt}\ set{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ par{\isadigit{1}}{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{also}\isamarkupfalse%
-\ \isacommand{have}\isamarkupfalse%
-\ par{\isadigit{2}}{\isacharunderscore}{\kern0pt}sound{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}electoral{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ reject{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}alts\ reject{\isacharunderscore}{\kern0pt}xa\ fin{\isacharunderscore}{\kern0pt}A\ in{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \ \ \ \ \ \ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\ order\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ par{\isadigit{1}}{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ par{\isadigit{2}}{\isacharunderscore}{\kern0pt}sound\ plurality{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}no{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\isanewline
-\ \ \ \ A\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ p\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Profile{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}a\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}t\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}smc\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymtriangleright}\isanewline
-\ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ pass{\isacharunderscore}{\kern0pt}module\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}\ x{\isacharparenright}{\kern0pt}\ {\isasymparallel}\isactrlsub {\isacharquery}{\kern0pt}a\isanewline
-\ \ \ \ \ \ \ \ \ drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x\ {\isasymcirclearrowleft}\isactrlsub {\isacharquery}{\kern0pt}t\ {\isacharparenleft}{\kern0pt}Suc\ {\isadigit{0}}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\isanewline
-\ \ \ \ fin{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}finite\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ prof{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}profile\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}xa\ {\isasymin}\ A{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymnotin}\ defer\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ \ \ not{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}xa{\isacharcolon}{\kern0pt}\isanewline
-\ \ \ \ \ \ {\isachardoublequoteopen}xa\ {\isasymnotin}\ reject\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isachardoublequoteopen}xa\ {\isasymin}\ elect\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}smc{\isacharparenright}{\kern0pt}\ A\ p{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ drop{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ loop{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ max{\isacharunderscore}{\kern0pt}agg{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ order\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ pass{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}sound\ xa{\isacharunderscore}{\kern0pt}in{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \ \ \ \ \ \ plurality{\isacharunderscore}{\kern0pt}sound\ rev{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}sound\isanewline
-\ \ \ \ \ \ \ \ \ \ electoral{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}elem\ fin{\isacharunderscore}{\kern0pt}A\ not{\isacharunderscore}{\kern0pt}defer{\isacharunderscore}{\kern0pt}xa\isanewline
-\ \ \ \ \ \ \ \ \ \ not{\isacharunderscore}{\kern0pt}reject{\isacharunderscore}{\kern0pt}xa\ prof{\isacharunderscore}{\kern0pt}A\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Electing%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ smc{\isacharunderscore}{\kern0pt}electing{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}electing\ {\isacharparenleft}{\kern0pt}smc\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}eliminator\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isasymparallel}\isactrlsub {\isasymup}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}loop\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ t\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isadigit{1}}\ in\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}eliminator\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ pass{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}\ {\isadigit{3}}\ smc{\isacharunderscore}{\kern0pt}sound\ smc{\isachardot}{\kern0pt}simps\ electing{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ iter{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ order\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}electing\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ metis\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{(Weak) Monotonicity Property%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{theorem}\isamarkupfalse%
-\ smc{\isacharunderscore}{\kern0pt}monotone{\isacharcolon}{\kern0pt}\isanewline
-\ \ \isakeyword{assumes}\ order{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}linear{\isacharunderscore}{\kern0pt}order\ x{\isachardoublequoteclose}\isanewline
-\ \ \isakeyword{shows}\ {\isachardoublequoteopen}monotonicity\ {\isacharparenleft}{\kern0pt}smc\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isacharminus}{\kern0pt}\isanewline
-\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}pass{\isacharunderscore}{\kern0pt}module\ {\isadigit{1}}\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isasymtriangleright}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}drop{\isacharunderscore}{\kern0pt}module\ {\isadigit{2}}\ x{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}eliminator\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isasymparallel}\isactrlsub {\isasymup}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}loop\ {\isacharequal}{\kern0pt}\isanewline
-\ \ \ \ {\isachardoublequoteopen}let\ t\ {\isacharequal}{\kern0pt}\ defer{\isacharunderscore}{\kern0pt}equal{\isacharunderscore}{\kern0pt}condition\ {\isadigit{1}}\ in\ {\isacharparenleft}{\kern0pt}{\isacharquery}{\kern0pt}eliminator\ {\isasymcirclearrowleft}\isactrlsub t{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}invariant{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ pass{\isacharunderscore}{\kern0pt}one{\isacharunderscore}{\kern0pt}mod{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}monotonicity\ {\isacharquery}{\kern0pt}tie{\isacharunderscore}{\kern0pt}breaker{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharparenleft}{\kern0pt}plurality{\isasymdown}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{4}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}blocking\ {\isacharquery}{\kern0pt}pass{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}plurality{\isacharunderscore}{\kern0pt}defer{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isadigit{3}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ blast\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}disjoint{\isacharunderscore}{\kern0pt}compatibility\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \ \ \ \ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}agg{\isacharunderscore}{\kern0pt}conservative\ max{\isacharunderscore}{\kern0pt}aggregator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}compare{\isacharunderscore}{\kern0pt}two{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}rejects\ {\isadigit{2}}\ {\isacharquery}{\kern0pt}drop{\isadigit{2}}{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ order\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}def{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}inv\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{2}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}eliminates\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}eliminator{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{1}}\ {\isadigit{0}}{\isadigit{0}}{\isadigit{2}}\ par{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}elim{\isacharunderscore}{\kern0pt}one\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{0}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defer{\isacharunderscore}{\kern0pt}lift{\isacharunderscore}{\kern0pt}invariance\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{1}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}non{\isacharunderscore}{\kern0pt}electing\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}defers\ {\isadigit{1}}\ {\isacharquery}{\kern0pt}loop{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{1}}{\isadigit{0}}\ {\isadigit{2}}{\isadigit{0}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}electing\ elect{\isacharunderscore}{\kern0pt}module{\isachardoublequoteclose}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ simp\ \isanewline
-\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isacharquery}{\kern0pt}thesis\isanewline
-\ \ \ \ \isacommand{using}\isamarkupfalse%
-\ {\isadigit{0}}\ {\isadigit{1}}\ {\isadigit{2}}\ {\isadigit{3}}\isanewline
-\ \ \ \ \ \ \ \ \ \ monotonicity{\isacharunderscore}{\kern0pt}def\isanewline
-\ \ \ \ \ \ \ \ \ \ iter{\isachardot}{\kern0pt}simps\isanewline
-\ \ \ \ \ \ \ \ \ \ smc{\isacharunderscore}{\kern0pt}sound\ smc{\isachardot}{\kern0pt}simps\ order\ seq{\isacharunderscore}{\kern0pt}comp{\isacharunderscore}{\kern0pt}mono\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}{\kern0pt}metis\ {\isacharparenleft}{\kern0pt}full{\isacharunderscore}{\kern0pt}types{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Voting_Rules/Sequential_Majority_Comparison.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%34=13%:%
-%:%35=14%:%
-%:%36=15%:%
-%:%37=16%:%
-%:%38=17%:%
-%:%39=18%:%
-%:%40=19%:%
-%:%41=20%:%
-%:%42=21%:%
-%:%51=24%:%
-%:%52=25%:%
-%:%53=26%:%
-%:%54=27%:%
-%:%63=29%:%
-%:%73=31%:%
-%:%74=31%:%
-%:%75=32%:%
-%:%84=36%:%
-%:%94=43%:%
-%:%95=43%:%
-%:%96=44%:%
-%:%97=45%:%
-%:%100=46%:%
-%:%104=46%:%
-%:%105=46%:%
-%:%106=47%:%
-%:%107=47%:%
-%:%108=48%:%
-%:%109=48%:%
-%:%110=49%:%
-%:%111=50%:%
-%:%112=51%:%
-%:%113=52%:%
-%:%114=52%:%
-%:%115=53%:%
-%:%116=53%:%
-%:%117=54%:%
-%:%118=54%:%
-%:%119=55%:%
-%:%121=57%:%
-%:%122=58%:%
-%:%123=58%:%
-%:%124=59%:%
-%:%125=60%:%
-%:%126=61%:%
-%:%127=62%:%
-%:%128=63%:%
-%:%129=64%:%
-%:%130=65%:%
-%:%131=65%:%
-%:%132=66%:%
-%:%133=66%:%
-%:%134=67%:%
-%:%135=68%:%
-%:%136=69%:%
-%:%137=70%:%
-%:%138=71%:%
-%:%139=71%:%
-%:%140=72%:%
-%:%141=72%:%
-%:%142=73%:%
-%:%143=73%:%
-%:%144=74%:%
-%:%145=75%:%
-%:%146=76%:%
-%:%147=77%:%
-%:%148=77%:%
-%:%149=78%:%
-%:%150=78%:%
-%:%151=79%:%
-%:%152=79%:%
-%:%153=80%:%
-%:%155=82%:%
-%:%156=83%:%
-%:%157=83%:%
-%:%158=84%:%
-%:%159=85%:%
-%:%160=86%:%
-%:%161=87%:%
-%:%162=88%:%
-%:%163=89%:%
-%:%164=90%:%
-%:%165=90%:%
-%:%166=91%:%
-%:%167=91%:%
-%:%168=92%:%
-%:%169=93%:%
-%:%170=94%:%
-%:%171=95%:%
-%:%172=96%:%
-%:%173=96%:%
-%:%174=97%:%
-%:%175=97%:%
-%:%176=98%:%
-%:%177=98%:%
-%:%178=99%:%
-%:%179=100%:%
-%:%180=101%:%
-%:%181=102%:%
-%:%182=102%:%
-%:%183=103%:%
-%:%184=103%:%
-%:%185=104%:%
-%:%186=104%:%
-%:%187=105%:%
-%:%189=107%:%
-%:%190=108%:%
-%:%191=108%:%
-%:%192=109%:%
-%:%193=110%:%
-%:%194=111%:%
-%:%195=112%:%
-%:%196=113%:%
-%:%197=113%:%
-%:%198=114%:%
-%:%199=114%:%
-%:%200=115%:%
-%:%201=116%:%
-%:%202=117%:%
-%:%203=118%:%
-%:%204=118%:%
-%:%205=119%:%
-%:%206=119%:%
-%:%207=120%:%
-%:%208=120%:%
-%:%209=121%:%
-%:%210=122%:%
-%:%211=123%:%
-%:%212=124%:%
-%:%213=124%:%
-%:%214=125%:%
-%:%215=125%:%
-%:%216=126%:%
-%:%217=126%:%
-%:%218=127%:%
-%:%220=129%:%
-%:%221=130%:%
-%:%222=130%:%
-%:%223=131%:%
-%:%224=132%:%
-%:%225=133%:%
-%:%226=134%:%
-%:%227=135%:%
-%:%228=135%:%
-%:%229=136%:%
-%:%230=136%:%
-%:%231=137%:%
-%:%232=138%:%
-%:%233=139%:%
-%:%234=140%:%
-%:%235=140%:%
-%:%236=141%:%
-%:%237=141%:%
-%:%238=142%:%
-%:%239=142%:%
-%:%240=143%:%
-%:%241=144%:%
-%:%242=145%:%
-%:%243=146%:%
-%:%244=146%:%
-%:%245=147%:%
-%:%246=147%:%
-%:%247=148%:%
-%:%248=148%:%
-%:%249=149%:%
-%:%251=151%:%
-%:%252=152%:%
-%:%253=152%:%
-%:%254=153%:%
-%:%255=154%:%
-%:%256=155%:%
-%:%257=156%:%
-%:%258=157%:%
-%:%259=157%:%
-%:%260=158%:%
-%:%261=159%:%
-%:%262=160%:%
-%:%263=160%:%
-%:%264=161%:%
-%:%265=161%:%
-%:%266=162%:%
-%:%267=163%:%
-%:%268=163%:%
-%:%269=164%:%
-%:%270=164%:%
-%:%271=165%:%
-%:%272=165%:%
-%:%273=165%:%
-%:%274=166%:%
-%:%275=167%:%
-%:%276=167%:%
-%:%277=168%:%
-%:%278=168%:%
-%:%279=169%:%
-%:%280=169%:%
-%:%281=170%:%
-%:%282=170%:%
-%:%283=171%:%
-%:%284=172%:%
-%:%285=173%:%
-%:%286=174%:%
-%:%287=175%:%
-%:%288=175%:%
-%:%289=176%:%
-%:%290=176%:%
-%:%291=177%:%
-%:%292=177%:%
-%:%293=178%:%
-%:%294=179%:%
-%:%295=180%:%
-%:%296=181%:%
-%:%297=181%:%
-%:%298=182%:%
-%:%299=182%:%
-%:%300=183%:%
-%:%301=183%:%
-%:%302=184%:%
-%:%304=186%:%
-%:%305=187%:%
-%:%306=187%:%
-%:%307=188%:%
-%:%308=189%:%
-%:%309=190%:%
-%:%310=191%:%
-%:%311=192%:%
-%:%312=193%:%
-%:%313=194%:%
-%:%314=195%:%
-%:%315=195%:%
-%:%316=196%:%
-%:%317=196%:%
-%:%318=197%:%
-%:%319=198%:%
-%:%320=199%:%
-%:%321=200%:%
-%:%322=201%:%
-%:%323=201%:%
-%:%324=202%:%
-%:%339=205%:%
-%:%349=213%:%
-%:%350=213%:%
-%:%351=214%:%
-%:%352=215%:%
-%:%359=216%:%
-%:%360=216%:%
-%:%361=217%:%
-%:%362=217%:%
-%:%363=218%:%
-%:%364=218%:%
-%:%365=219%:%
-%:%366=219%:%
-%:%367=220%:%
-%:%368=220%:%
-%:%369=221%:%
-%:%370=221%:%
-%:%371=222%:%
-%:%372=222%:%
-%:%373=223%:%
-%:%374=223%:%
-%:%375=224%:%
-%:%376=225%:%
-%:%377=226%:%
-%:%378=226%:%
-%:%379=227%:%
-%:%380=227%:%
-%:%381=228%:%
-%:%382=228%:%
-%:%383=229%:%
-%:%384=229%:%
-%:%385=230%:%
-%:%386=230%:%
-%:%387=231%:%
-%:%388=231%:%
-%:%389=232%:%
-%:%390=232%:%
-%:%391=233%:%
-%:%392=233%:%
-%:%393=234%:%
-%:%394=234%:%
-%:%395=235%:%
-%:%396=235%:%
-%:%397=236%:%
-%:%398=237%:%
-%:%399=237%:%
-%:%400=238%:%
-%:%401=238%:%
-%:%402=239%:%
-%:%403=239%:%
-%:%404=240%:%
-%:%405=240%:%
-%:%406=241%:%
-%:%407=241%:%
-%:%408=242%:%
-%:%409=242%:%
-%:%410=243%:%
-%:%411=243%:%
-%:%412=244%:%
-%:%413=244%:%
-%:%414=245%:%
-%:%415=245%:%
-%:%416=246%:%
-%:%417=246%:%
-%:%418=247%:%
-%:%419=247%:%
-%:%420=248%:%
-%:%421=248%:%
-%:%422=249%:%
-%:%423=249%:%
-%:%424=250%:%
-%:%425=250%:%
-%:%426=251%:%
-%:%427=251%:%
-%:%428=252%:%
-%:%429=253%:%
-%:%430=253%:%
-%:%431=254%:%
-%:%432=254%:%
-%:%433=255%:%
-%:%434=255%:%
-%:%435=256%:%
-%:%436=256%:%
-%:%437=257%:%
-%:%438=257%:%
-%:%439=258%:%
-%:%440=258%:%
-%:%441=259%:%
-%:%442=259%:%
-%:%443=260%:%
-%:%444=260%:%
-%:%445=261%:%
-%:%446=261%:%
-%:%447=262%:%
-%:%448=262%:%
-%:%449=263%:%
-%:%450=263%:%
-%:%451=264%:%
-%:%452=264%:%
-%:%453=265%:%
-%:%454=265%:%
-%:%455=266%:%
-%:%456=266%:%
-%:%457=267%:%
-%:%458=267%:%
-%:%459=268%:%
-%:%460=268%:%
-%:%461=269%:%
-%:%462=269%:%
-%:%463=270%:%
-%:%464=271%:%
-%:%465=271%:%
-%:%466=272%:%
-%:%467=272%:%
-%:%468=273%:%
-%:%469=273%:%
-%:%470=274%:%
-%:%471=274%:%
-%:%472=275%:%
-%:%473=275%:%
-%:%474=276%:%
-%:%475=276%:%
-%:%476=277%:%
-%:%477=278%:%
-%:%478=278%:%
-%:%479=279%:%
-%:%480=279%:%
-%:%481=280%:%
-%:%482=280%:%
-%:%483=281%:%
-%:%484=281%:%
-%:%485=282%:%
-%:%486=282%:%
-%:%487=283%:%
-%:%488=284%:%
-%:%489=284%:%
-%:%490=285%:%
-%:%491=285%:%
-%:%492=286%:%
-%:%493=287%:%
-%:%494=288%:%
-%:%495=288%:%
-%:%496=289%:%
-%:%511=291%:%
-%:%521=297%:%
-%:%522=297%:%
-%:%523=298%:%
-%:%524=299%:%
-%:%531=300%:%
-%:%532=300%:%
-%:%533=301%:%
-%:%534=302%:%
-%:%535=302%:%
-%:%536=303%:%
-%:%537=303%:%
-%:%538=304%:%
-%:%539=304%:%
-%:%540=305%:%
-%:%541=305%:%
-%:%542=306%:%
-%:%543=306%:%
-%:%544=307%:%
-%:%545=307%:%
-%:%546=308%:%
-%:%547=308%:%
-%:%548=309%:%
-%:%549=310%:%
-%:%550=311%:%
-%:%551=311%:%
-%:%552=312%:%
-%:%553=312%:%
-%:%554=313%:%
-%:%555=313%:%
-%:%556=314%:%
-%:%557=314%:%
-%:%558=315%:%
-%:%559=315%:%
-%:%560=316%:%
-%:%561=316%:%
-%:%562=317%:%
-%:%563=317%:%
-%:%564=318%:%
-%:%565=318%:%
-%:%566=319%:%
-%:%567=319%:%
-%:%568=320%:%
-%:%569=320%:%
-%:%570=321%:%
-%:%571=321%:%
-%:%572=322%:%
-%:%573=322%:%
-%:%574=323%:%
-%:%575=323%:%
-%:%576=324%:%
-%:%577=324%:%
-%:%578=325%:%
-%:%579=325%:%
-%:%580=326%:%
-%:%581=327%:%
-%:%582=327%:%
-%:%583=328%:%
-%:%584=328%:%
-%:%585=329%:%
-%:%586=329%:%
-%:%587=330%:%
-%:%588=330%:%
-%:%589=331%:%
-%:%590=331%:%
-%:%591=332%:%
-%:%592=332%:%
-%:%593=333%:%
-%:%594=333%:%
-%:%595=334%:%
-%:%596=334%:%
-%:%597=335%:%
-%:%598=335%:%
-%:%599=336%:%
-%:%600=336%:%
-%:%601=337%:%
-%:%602=337%:%
-%:%603=338%:%
-%:%604=338%:%
-%:%605=339%:%
-%:%606=339%:%
-%:%607=340%:%
-%:%608=340%:%
-%:%609=341%:%
-%:%610=341%:%
-%:%611=342%:%
-%:%612=342%:%
-%:%613=343%:%
-%:%614=343%:%
-%:%615=344%:%
-%:%616=344%:%
-%:%617=345%:%
-%:%618=345%:%
-%:%619=346%:%
-%:%620=346%:%
-%:%621=347%:%
-%:%622=347%:%
-%:%623=348%:%
-%:%624=349%:%
-%:%625=349%:%
-%:%626=350%:%
-%:%627=350%:%
-%:%628=351%:%
-%:%629=351%:%
-%:%630=352%:%
-%:%631=352%:%
-%:%632=353%:%
-%:%633=353%:%
-%:%634=354%:%
-%:%635=354%:%
-%:%636=355%:%
-%:%637=355%:%
-%:%638=356%:%
-%:%639=356%:%
-%:%640=357%:%
-%:%641=357%:%
-%:%642=358%:%
-%:%642=359%:%
-%:%643=360%:%
-%:%644=360%:%
-%:%645=361%:%
-%:%646=361%:%
-%:%647=362%:%
-%:%648=362%:%
-%:%649=363%:%
-%:%650=363%:%
-%:%651=364%:%
-%:%652=364%:%
-%:%653=365%:%
-%:%654=365%:%
-%:%655=366%:%
-%:%656=366%:%
-%:%657=367%:%
-%:%658=367%:%
-%:%659=368%:%
-%:%660=368%:%
-%:%661=369%:%
-%:%662=369%:%
-%:%663=370%:%
-%:%664=370%:%
-%:%665=371%:%
-%:%666=371%:%
-%:%667=372%:%
-%:%668=372%:%
-%:%669=373%:%
-%:%670=373%:%
-%:%671=374%:%
-%:%672=375%:%
-%:%673=375%:%
-%:%674=376%:%
-%:%675=376%:%
-%:%676=377%:%
-%:%677=377%:%
-%:%678=378%:%
-%:%679=378%:%
-%:%680=379%:%
-%:%681=379%:%
-%:%682=380%:%
-%:%683=380%:%
-%:%684=381%:%
-%:%685=381%:%
-%:%686=382%:%
-%:%687=382%:%
-%:%688=383%:%
-%:%689=383%:%
-%:%690=384%:%
-%:%691=385%:%
-%:%692=385%:%
-%:%693=386%:%
-%:%694=386%:%
-%:%695=387%:%
-%:%696=387%:%
-%:%697=388%:%
-%:%698=388%:%
-%:%699=389%:%
-%:%700=389%:%
-%:%701=390%:%
-%:%702=390%:%
-%:%703=391%:%
-%:%704=391%:%
-%:%705=392%:%
-%:%706=392%:%
-%:%707=393%:%
-%:%708=393%:%
-%:%709=394%:%
-%:%710=394%:%
-%:%711=395%:%
-%:%712=395%:%
-%:%713=396%:%
-%:%714=397%:%
-%:%715=397%:%
-%:%716=398%:%
-%:%717=398%:%
-%:%718=399%:%
-%:%719=400%:%
-%:%720=401%:%
-%:%721=402%:%
-%:%722=402%:%
-%:%723=403%:%
-%:%729=403%:%
-%:%734=404%:%
-%:%739=405%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Termination_Condition.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Termination_Condition.tex
deleted file mode 100644
index 5434a3b..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Termination_Condition.tex
+++ /dev/null
@@ -1,99 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Termination{\isacharunderscore}{\kern0pt}Condition}%
-%
-\isadelimdocument
-\isanewline
-%
-\endisadelimdocument
-%
-\isatagdocument
-\isanewline
-\isanewline
-%
-\isamarkupsection{Termination Condition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Termination{\isacharunderscore}{\kern0pt}Condition\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Social{\isacharunderscore}{\kern0pt}Choice{\isacharunderscore}{\kern0pt}Types{\isacharslash}{\kern0pt}Result{\isachardoublequoteclose}\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-The termination condition is used in loops. It decides whether or not to
-terminate the loop after each iteration, depending on the current state
-of the loop.%
-\end{isamarkuptext}\isamarkuptrue%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-%
-\isatagdocument
-%
-\isamarkupsubsection{Definition%
-}
-\isamarkuptrue%
-%
-\endisatagdocument
-{\isafolddocument}%
-%
-\isadelimdocument
-%
-\endisadelimdocument
-\isacommand{type{\isacharunderscore}{\kern0pt}synonym}\isamarkupfalse%
-\ {\isacharprime}{\kern0pt}a\ Termination{\isacharunderscore}{\kern0pt}Condition\ {\isacharequal}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Result\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Compositional_Framework/Components/Termination_Condition.thy%:%
-%:%6=3%:%
-%:%11=4%:%
-%:%12=5%:%
-%:%14=8%:%
-%:%30=10%:%
-%:%31=10%:%
-%:%32=11%:%
-%:%33=12%:%
-%:%42=15%:%
-%:%43=16%:%
-%:%44=17%:%
-%:%53=19%:%
-%:%63=21%:%
-%:%64=21%:%
-%:%67=22%:%
-%:%72=23%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Weak_Monotonicity.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Weak_Monotonicity.tex
deleted file mode 100644
index 3a14e3e..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/Weak_Monotonicity.tex
+++ /dev/null
@@ -1,62 +0,0 @@
-%
-\begin{isabellebody}%
-\setisabellecontext{Weak{\isacharunderscore}{\kern0pt}Monotonicity}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Weak{\isacharunderscore}{\kern0pt}Monotonicity\isanewline
-\ \ \isakeyword{imports}\ {\isachardoublequoteopen}{\isachardot}{\kern0pt}{\isachardot}{\kern0pt}{\isacharslash}{\kern0pt}Compositional{\isacharunderscore}{\kern0pt}Framework{\isacharslash}{\kern0pt}Components{\isacharslash}{\kern0pt}Electoral{\isacharunderscore}{\kern0pt}Module{\isachardoublequoteclose}\isanewline
-\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-\isanewline
-\isanewline
-\isacommand{definition}\isamarkupfalse%
-\ monotonicity\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ Electoral{\isacharunderscore}{\kern0pt}Module\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}monotonicity\ m\ {\isasymequiv}\isanewline
-\ \ \ \ electoral{\isacharunderscore}{\kern0pt}module\ m\ {\isasymand}\isanewline
-\ \ \ \ \ \ {\isacharparenleft}{\kern0pt}{\isasymforall}A\ p\ q\ w{\isachardot}{\kern0pt}\isanewline
-\ \ \ \ \ \ \ \ \ \ {\isacharparenleft}{\kern0pt}finite\ A\ {\isasymand}\ w\ {\isasymin}\ elect\ m\ A\ p\ {\isasymand}\ lifted\ A\ p\ q\ w{\isacharparenright}{\kern0pt}\ {\isasymlongrightarrow}\ w\ {\isasymin}\ elect\ m\ A\ q{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
-%
-\isadelimtheory
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\end{isabellebody}%
-\endinput
-%:%file=~/Documents/Studies/VotingRuleGenerator/virage/src/test/resources/verifiedVotingRuleConstruction/theories/Social_Choice_Properties/Weak_Monotonicity.thy%:%
-%:%10=1%:%
-%:%11=1%:%
-%:%12=2%:%
-%:%13=3%:%
-%:%14=4%:%
-%:%19=4%:%
-%:%22=5%:%
-%:%23=9%:%
-%:%24=10%:%
-%:%25=10%:%
-%:%26=11%:%
-%:%29=14%:%
-%:%32=15%:%
-%:%37=16%:%
\ No newline at end of file
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/comment.sty b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/comment.sty
deleted file mode 100644
index 658686f..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/comment.sty
+++ /dev/null
@@ -1,278 +0,0 @@
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Comment.sty   version 3.6, October 1999
-%
-% Purpose:
-% selectively in/exclude pieces of text: the user can define new
-% comment versions, and each is controlled separately.
-% Special comments can be defined where the user specifies the
-% action that is to be taken with each comment line.
-%
-% Author
-%    Victor Eijkhout
-%    Department of Computer Science
-%    University of Tennessee
-%    107 Ayres Hall
-%    Knoxville TN 37996
-%    USA
-%
-%    victor@eijkhout.net
-%
-% This program is free software; you can redistribute it and/or
-% modify it under the terms of the GNU General Public License
-% as published by the Free Software Foundation; either version 2
-% of the License, or (at your option) any later version.
-% 
-% This program is distributed in the hope that it will be useful,
-% but WITHOUT ANY WARRANTY; without even the implied warranty of
-% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
-% GNU General Public License for more details.
-%
-% For a copy of the GNU General Public License, write to the 
-% Free Software Foundation, Inc.,
-% 59 Temple Place - Suite 330, Boston, MA  02111-1307, USA,
-% or find it on the net, for instance at
-% http://www.gnu.org/copyleft/gpl.html
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% This style can be used with plain TeX or LaTeX, and probably
-% most other packages too.
-%
-% Usage: all text included between
-%    \comment ... \endcomment
-% or \begin{comment} ... \end{comment}
-% is discarded. 
-%
-% The opening and closing commands should appear on a line
-% of their own. No starting spaces, nothing after it.
-% This environment should work with arbitrary amounts
-% of comment, and the comment can be arbitrary text.
-%
-% Other `comment' environments are defined by
-% and are selected/deselected with
-% \includecomment{versiona}
-% \excludecoment{versionb}
-%
-% These environments are used as
-% \versiona ... \endversiona
-% or \begin{versiona} ... \end{versiona}
-% with the opening and closing commands again on a line of 
-% their own.
-%
-% LaTeX users note: for an included comment, the
-% \begin and \end lines act as if they don't exist.
-% In particular, they don't imply grouping, so assignments 
-% &c are not local.
-%
-% Special comments are defined as
-% \specialcomment{name}{before commands}{after commands}
-% where the second and third arguments are executed before
-% and after each comment block. You can use this for global
-% formatting commands.
-% To keep definitions &c local, you can include \begingroup
-% in the `before commands' and \endgroup in the `after commands'.
-% ex:
-% \specialcomment{smalltt}
-%     {\begingroup\ttfamily\footnotesize}{\endgroup}
-% You do *not* have to do an additional
-% \includecomment{smalltt}
-% To remove 'smalltt' blocks, give \excludecomment{smalltt}
-% after the definition.
-%
-% Processing comments can apply processing to each line.
-% \processcomment{name}{each-line commands}%
-%    {before commands}{after commands}
-% By defining a control sequence 
-% \def\Thiscomment##1{...} in the before commands the user can
-% specify what is to be done with each comment line.
-% BUG this does not work quite yet BUG
-%
-% Trick for short in/exclude macros (such as \maybe{this snippet}):
-%\includecomment{cond}
-%\newcommand{\maybe}[1]{}
-%\begin{cond}
-%\renewcommand{\maybe}[1]{#1}
-%\end{cond}
-%
-% Basic approach of the implementation:
-% to comment something out, scoop up  every line in verbatim mode
-% as macro argument, then throw it away.
-% For inclusions, in LaTeX the block is written out to
-% a file \CommentCutFile (default "comment.cut"), which is
-% then included.
-% In plain TeX (and other formats) both the opening and
-% closing comands are defined as noop.
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% Changes in version 3.1
-% - updated author's address
-% - cleaned up some code
-% - trailing contents on \begin{env} line is always discarded
-%  even if you've done \includecomment{env}
-% - comments no longer define grouping!! you can even
-%   \includecomment{env}
-%   \begin{env}
-%   \begin{itemize}
-%   \end{env}
-%  Isn't that something ...
-% - included comments are written to file and input again.
-% Changes in 3.2
-% - \specialcomment brought up to date (thanks to Ivo Welch).
-% Changes in 3.3
-% - updated author's address again
-% - parametrised \CommentCutFile
-% Changes in 3.4
-% - added GNU public license
-% - added \processcomment, because Ivo's fix (above) brought an
-%   inconsistency to light.
-% Changes in 3.5
-% - corrected typo in header.
-% - changed author email
-% - corrected \specialcomment yet again.
-% - fixed excludecomment of an earlier defined environment.
-% Changes in 3.6
-% - The 'cut' file is now written more verbatim, using \meaning;
-%   some people reported having trouble with ISO latin 1, or umlaute.sty.
-% - removed some \newif statements.
-%   Has this suddenly become \outer again?
-%
-% Known bugs:
-% - excludecomment leads to one superfluous space
-% - processcomment leads to a superfluous line break
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\def\makeinnocent#1{\catcode`#1=12 }
-\def\csarg#1#2{\expandafter#1\csname#2\endcsname}
-\def\latexname{lplain}\def\latexename{LaTeX2e}
-\newwrite\CommentStream
-\def\CommentCutFile{comment.cut}
-
-\def\ProcessComment#1% start it all of
-   {\begingroup
-    \def\CurrentComment{#1}%
-    \let\do\makeinnocent \dospecials 
-    \makeinnocent\^^L% and whatever other special cases
-    \endlinechar`\^^M \catcode`\^^M=12 \xComment}
-%\def\ProcessCommentWithArg#1#2% to be used in \leveledcomment
-%   {\begingroup
-%    \def\CurrentComment{#1}%
-%    \let\do\makeinnocent \dospecials 
-%    \makeinnocent\^^L% and whatever other special cases
-%    \endlinechar`\^^M \catcode`\^^M=12 \xComment}
-{\catcode`\^^M=12 \endlinechar=-1 %
- \gdef\xComment#1^^M{%
-    \expandafter\ProcessCommentLine}
- \gdef\ProcessCommentLine#1^^M{\def\test{#1}
-      \csarg\ifx{End\CurrentComment Test}\test
-          \edef\next{\noexpand\EndOfComment{\CurrentComment}}%
-      \else \ThisComment{#1}\let\next\ProcessCommentLine
-      \fi \next}
-}
-
-\def\CSstringmeaning#1{\expandafter\CSgobblearrow\meaning#1}
-\def\CSstringcsnoescape#1{\expandafter\CSgobbleescape\string#1}
-{\escapechar-1
-\expandafter\expandafter\expandafter\gdef
-  \expandafter\expandafter\expandafter\CSgobblearrow
-    \expandafter\string\csname macro:->\endcsname{}
-}
-\def\CSgobbleescape#1{\ifnum`\\=`#1 \else #1\fi}
-\def\WriteCommentLine#1{\def\CStmp{#1}%
-    \immediate\write\CommentStream{\CSstringmeaning\CStmp}}
-
-% 3.1 change: in LaTeX and LaTeX2e prevent grouping
-\if 0%
-\ifx\fmtname\latexename 
-    0%
-\else \ifx\fmtname\latexname 
-          0%
-      \else 
-          1%
-\fi   \fi
-%%%%
-%%%% definitions for LaTeX
-%%%%
-\def\AfterIncludedComment
-   {\immediate\closeout\CommentStream
-    \input{\CommentCutFile}\relax
-    }%
-\def\TossComment{\immediate\closeout\CommentStream}
-\def\BeforeIncludedComment
-   {\immediate\openout\CommentStream=\CommentCutFile
-    \let\ThisComment\WriteCommentLine}
-\def\includecomment
- #1{\message{Include comment '#1'}%
-    \csarg\let{After#1Comment}\AfterIncludedComment
-    \csarg\def{#1}{\BeforeIncludedComment
-        \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\long\def\specialcomment
- #1#2#3{\message{Special comment '#1'}%
-    % note: \AfterIncludedComment does \input, so #2 goes here!
-    \csarg\def{After#1Comment}{#2\AfterIncludedComment#3}%
-    \csarg\def{#1}{\BeforeIncludedComment\relax
-          \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\long\def\processcomment
- #1#2#3#4{\message{Lines-Processing comment '#1'}%
-    \csarg\def{After#1Comment}{#3\AfterIncludedComment#4}%
-    \csarg\def{#1}{\BeforeIncludedComment#2\relax
-          \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\def\leveledcomment
- #1#2{\message{Include comment '#1' up to level '#2'}%
-    %\csname #1IsLeveledCommenttrue\endcsname
-    \csarg\let{After#1Comment}\AfterIncludedComment
-    \csarg\def{#1}{\BeforeIncludedComment
-        \ProcessCommentWithArg{#1}}%
-    \CommentEndDef{#1}}
-\else 
-%%%%
-%%%%plain TeX and other formats
-%%%%
-\def\includecomment
- #1{\message{Including comment '#1'}%
-    \csarg\def{#1}{}%
-    \csarg\def{end#1}{}}
-\long\def\specialcomment
- #1#2#3{\message{Special comment '#1'}%
-    \csarg\def{#1}{\def\ThisComment{}\def\AfterComment{#3}#2%
-           \ProcessComment{#1}}%
-    \CommentEndDef{#1}}
-\fi
-
-%%%%
-%%%% general definition of skipped comment
-%%%%
-\def\excludecomment
- #1{\message{Excluding comment '#1'}%
-    \csarg\def{#1}{\let\AfterComment\relax
-           \def\ThisComment####1{}\ProcessComment{#1}}%
-    \csarg\let{After#1Comment}\TossComment
-    \CommentEndDef{#1}}
-
-\if 0%
-\ifx\fmtname\latexename 
-    0%
-\else \ifx\fmtname\latexname 
-          0%
-      \else 
-          1%
-\fi   \fi
-% latex & latex2e:
-\def\EndOfComment#1{\endgroup\end{#1}%
-    \csname After#1Comment\endcsname}
-\def\CommentEndDef#1{{\escapechar=-1\relax
-    \csarg\xdef{End#1Test}{\string\\end\string\{#1\string\}}%
-    }}
-\else
-% plain & other
-\def\EndOfComment#1{\endgroup\AfterComment}
-\def\CommentEndDef#1{{\escapechar=-1\relax
-    \csarg\xdef{End#1Test}{\string\\end#1}%
-    }}
-\fi
-
-\excludecomment{comment}
-
-\endinput
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabelle.sty b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabelle.sty
deleted file mode 100644
index b6dcb3e..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabelle.sty
+++ /dev/null
@@ -1,267 +0,0 @@
-%%
-%% macros for Isabelle generated LaTeX output
-%%
-
-%%% Simple document preparation (based on theory token language and symbols)
-
-% isabelle environments
-
-\newcommand{\isabellecontext}{UNKNOWN}
-\newcommand{\setisabellecontext}[1]{\def\isabellecontext{#1}}
-
-\newcommand{\isastyle}{\UNDEF}
-\newcommand{\isastylett}{\UNDEF}
-\newcommand{\isastyleminor}{\UNDEF}
-\newcommand{\isastyleminortt}{\UNDEF}
-\newcommand{\isastylescript}{\UNDEF}
-\newcommand{\isastyletext}{\normalsize\normalfont\rmfamily}
-\newcommand{\isastyletxt}{\normalfont\rmfamily}
-\newcommand{\isastylecmt}{\normalfont\rmfamily}
-
-\newcommand{\isaspacing}{%
-  \sfcode 42 1000 % .
-  \sfcode 63 1000 % ?
-  \sfcode 33 1000 % !
-  \sfcode 58 1000 % :
-  \sfcode 59 1000 % ;
-  \sfcode 44 1000 % ,
-}
-
-%symbol markup -- \emph achieves decent spacing via italic corrections
-\newcommand{\isamath}[1]{\emph{$#1$}}
-\newcommand{\isatext}[1]{\emph{#1}}
-\DeclareRobustCommand{\isascriptstyle}{\def\isamath##1{##1}\def\isatext##1{\mbox{\isaspacing\isastylescript##1}}}
-\newcommand{\isactrlsub}[1]{\emph{\isascriptstyle${}\sb{#1}$}}
-\newcommand{\isactrlsup}[1]{\emph{\isascriptstyle${}\sp{#1}$}}
-\DeclareRobustCommand{\isactrlbsub}{\emph\bgroup\math{}\sb\bgroup\mbox\bgroup\isaspacing\isastylescript}
-\DeclareRobustCommand{\isactrlesub}{\egroup\egroup\endmath\egroup}
-\DeclareRobustCommand{\isactrlbsup}{\emph\bgroup\math{}\sp\bgroup\mbox\bgroup\isaspacing\isastylescript}
-\DeclareRobustCommand{\isactrlesup}{\egroup\egroup\endmath\egroup}
-\newcommand{\isactrlbold}[1]{{\bfseries\upshape\boldmath#1}}
-
-\newenvironment{isaantiq}{{\isacharat\isacharbraceleft}}{{\isacharbraceright}}
-
-\newdimen\isa@parindent\newdimen\isa@parskip
-
-\newenvironment{isabellebody}{%
-\isamarkuptrue\par%
-\isa@parindent\parindent\parindent0pt%
-\isa@parskip\parskip\parskip0pt%
-\isaspacing\isastyle}{\par}
-
-\newenvironment{isabellebodytt}{%
-\isamarkuptrue\par%
-\isa@parindent\parindent\parindent0pt%
-\isa@parskip\parskip\parskip0pt%
-\isaspacing\isastylett}{\par}
-
-\newenvironment{isabelle}
-{\begin{trivlist}\begin{isabellebody}\item\relax}
-{\end{isabellebody}\end{trivlist}}
-
-\newenvironment{isabellett}
-{\begin{trivlist}\begin{isabellebodytt}\item\relax}
-{\end{isabellebodytt}\end{trivlist}}
-
-\newcommand{\isa}[1]{\emph{\isaspacing\isastyleminor #1}}
-\newcommand{\isatt}[1]{\emph{\isaspacing\isastyleminortt #1}}
-
-\newcommand{\isaindent}[1]{\hphantom{#1}}
-\newcommand{\isanewline}{\mbox{}\par\mbox{}}
-\newcommand{\isasep}{}
-\newcommand{\isadigit}[1]{#1}
-
-\newcommand{\isachardefaults}{%
-\def\isacharbell{\isamath{\bigbox}}%requires stmaryrd
-\chardef\isacharbang=`\!%
-\chardef\isachardoublequote=`\"%
-\chardef\isachardoublequoteopen=`\"%
-\chardef\isachardoublequoteclose=`\"%
-\chardef\isacharhash=`\#%
-\chardef\isachardollar=`\$%
-\chardef\isacharpercent=`\%%
-\chardef\isacharampersand=`\&%
-\chardef\isacharprime=`\'%
-\chardef\isacharparenleft=`\(%
-\chardef\isacharparenright=`\)%
-\chardef\isacharasterisk=`\*%
-\chardef\isacharplus=`\+%
-\chardef\isacharcomma=`\,%
-\chardef\isacharminus=`\-%
-\chardef\isachardot=`\.%
-\chardef\isacharslash=`\/%
-\chardef\isacharcolon=`\:%
-\chardef\isacharsemicolon=`\;%
-\chardef\isacharless=`\<%
-\chardef\isacharequal=`\=%
-\chardef\isachargreater=`\>%
-\chardef\isacharquery=`\?%
-\chardef\isacharat=`\@%
-\chardef\isacharbrackleft=`\[%
-\chardef\isacharbackslash=`\\%
-\chardef\isacharbrackright=`\]%
-\chardef\isacharcircum=`\^%
-\chardef\isacharunderscore=`\_%
-\def\isacharunderscorekeyword{\_}%
-\chardef\isacharbackquote=`\`%
-\chardef\isacharbackquoteopen=`\`%
-\chardef\isacharbackquoteclose=`\`%
-\chardef\isacharbraceleft=`\{%
-\chardef\isacharbar=`\|%
-\chardef\isacharbraceright=`\}%
-\chardef\isachartilde=`\~%
-\def\isacharverbatimopen{\isacharbraceleft\isacharasterisk}%
-\def\isacharverbatimclose{\isacharasterisk\isacharbraceright}%
-\def\isacartoucheopen{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\langle$}}}%
-\def\isacartoucheclose{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\rangle$}}}%
-}
-
-
-% keyword and section markup
-
-\newcommand{\isakeyword}[1]
-{\emph{\normalfont\bfseries\def\isachardot{.}\def\isacharunderscore{\isacharunderscorekeyword}%
-\def\isacharbraceleft{\{}\def\isacharbraceright{\}}#1}}
-\newcommand{\isacommand}[1]{\isakeyword{#1}}
-
-\newcommand{\isakeywordcontrol}[1]
-{\emph{\normalfont\bfseries\itshape\def\isacharunderscore{\isacharunderscorekeyword}#1\,}}
-
-\newcommand{\isamarkupchapter}[1]{\chapter{#1}}
-\newcommand{\isamarkupsection}[1]{\section{#1}}
-\newcommand{\isamarkupsubsection}[1]{\subsection{#1}}
-\newcommand{\isamarkupsubsubsection}[1]{\subsubsection{#1}}
-\newcommand{\isamarkupparagraph}[1]{\paragraph{#1}}
-\newcommand{\isamarkupsubparagraph}[1]{\subparagraph{#1}}
-
-\newif\ifisamarkup
-\newcommand{\isabeginpar}{\par\ifisamarkup\relax\else\medskip\fi}
-\newcommand{\isaendpar}{\par\medskip}
-\newenvironment{isapar}{\parindent\isa@parindent\parskip\isa@parskip\isabeginpar}{\isaendpar}
-\newenvironment{isamarkuptext}{\par\isastyletext\begin{isapar}}{\end{isapar}}
-\newenvironment{isamarkuptxt}{\par\isastyletxt\begin{isapar}}{\end{isapar}}
-\newcommand{\isamarkupcmt}[1]{{\isastylecmt--- #1}}
-
-
-% styles
-
-\def\isabellestyle#1{\csname isabellestyle#1\endcsname}
-
-\newcommand{\isabellestyledefault}{%
-\def\isastyle{\small\normalfont\ttfamily\slshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\small\normalfont\ttfamily\slshape}%
-\def\isastyleminortt{\small\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\ttfamily\slshape}%
-\isachardefaults%
-}
-\isabellestyledefault
-
-\newcommand{\isabellestylett}{%
-\def\isastyle{\small\normalfont\ttfamily}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\small\normalfont\ttfamily}%
-\def\isastyleminortt{\small\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\ttfamily}%
-\isachardefaults%
-}
-
-\newcommand{\isabellestyleit}{%
-\def\isastyle{\small\normalfont\itshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\normalfont\itshape}%
-\def\isastyleminortt{\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\itshape}%
-\isachardefaults%
-\def\isacharunderscorekeyword{\mbox{-}}%
-\def\isacharbang{\isamath{!}}%
-\def\isachardoublequote{}%
-\def\isachardoublequoteopen{}%
-\def\isachardoublequoteclose{}%
-\def\isacharhash{\isamath{\#}}%
-\def\isachardollar{\isamath{\$}}%
-\def\isacharpercent{\isamath{\%}}%
-\def\isacharampersand{\isamath{\&}}%
-\def\isacharprime{\isamath{\mskip2mu{'}\mskip-2mu}}%
-\def\isacharparenleft{\isamath{(}}%
-\def\isacharparenright{\isamath{)}}%
-\def\isacharasterisk{\isamath{*}}%
-\def\isacharplus{\isamath{+}}%
-\def\isacharcomma{\isamath{\mathord,}}%
-\def\isacharminus{\isamath{-}}%
-\def\isachardot{\isamath{\mathord.}}%
-\def\isacharslash{\isamath{/}}%
-\def\isacharcolon{\isamath{\mathord:}}%
-\def\isacharsemicolon{\isamath{\mathord;}}%
-\def\isacharless{\isamath{<}}%
-\def\isacharequal{\isamath{=}}%
-\def\isachargreater{\isamath{>}}%
-\def\isacharat{\isamath{@}}%
-\def\isacharbrackleft{\isamath{[}}%
-\def\isacharbackslash{\isamath{\backslash}}%
-\def\isacharbrackright{\isamath{]}}%
-\def\isacharunderscore{\mbox{-}}%
-\def\isacharbraceleft{\isamath{\{}}%
-\def\isacharbar{\isamath{\mid}}%
-\def\isacharbraceright{\isamath{\}}}%
-\def\isachartilde{\isamath{{}\sp{\sim}}}%
-\def\isacharbackquoteopen{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\langle$}}}%
-\def\isacharbackquoteclose{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\rangle$}}}%
-}
-
-\newcommand{\isabellestyleliteral}{%
-\isabellestyleit%
-\def\isacharunderscore{\_}%
-\def\isacharunderscorekeyword{\_}%
-\chardef\isacharbackquoteopen=`\`%
-\chardef\isacharbackquoteclose=`\`%
-}
-
-\newcommand{\isabellestyleliteralunderscore}{%
-\isabellestyleliteral%
-\def\isacharunderscore{\textunderscore}%
-\def\isacharunderscorekeyword{\textunderscore}%
-}
-
-\newcommand{\isabellestylesl}{%
-\isabellestyleit%
-\def\isastyle{\small\normalfont\slshape}%
-\def\isastylett{\small\normalfont\ttfamily}%
-\def\isastyleminor{\normalfont\slshape}%
-\def\isastyleminortt{\normalfont\ttfamily}%
-\def\isastylescript{\footnotesize\normalfont\slshape}%
-}
-
-
-% cancel text
-
-\usepackage[normalem]{ulem}
-\newcommand{\isamarkupcancel}[1]{\isa{\xout{#1}}}
-
-
-% tagged regions
-
-%plain TeX version of comment package -- much faster!
-\let\isafmtname\fmtname\def\fmtname{plain}
-\usepackage{comment}
-\let\fmtname\isafmtname
-
-\newcommand{\isafold}[1]{\emph{$\langle\mathord{\mathit{#1}}\rangle$}}
-
-\newcommand{\isakeeptag}[1]%
-{\includecomment{isadelim#1}\includecomment{isatag#1}\csarg\def{isafold#1}{}}
-\newcommand{\isadroptag}[1]%
-{\excludecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{}}
-\newcommand{\isafoldtag}[1]%
-{\includecomment{isadelim#1}\excludecomment{isatag#1}\csarg\def{isafold#1}{\isafold{#1}}}
-
-\isakeeptag{document}
-\isakeeptag{theory}
-\isakeeptag{proof}
-\isakeeptag{ML}
-\isakeeptag{visible}
-\isadroptag{invisible}
-\isakeeptag{important}
-\isakeeptag{unimportant}
-
-\IfFileExists{isabelletags.sty}{\usepackage{isabelletags}}{}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabellesym.sty b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabellesym.sty
deleted file mode 100644
index b0d076c..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabellesym.sty
+++ /dev/null
@@ -1,427 +0,0 @@
-%%
-%% definitions of standard Isabelle symbols
-%%
-
-\newcommand{\isasymzero}{\isamath{\mathbf{0}}}  %requires amssymb
-\newcommand{\isasymone}{\isamath{\mathbf{1}}}  %requires amssymb
-\newcommand{\isasymtwo}{\isamath{\mathbf{2}}}  %requires amssymb
-\newcommand{\isasymthree}{\isamath{\mathbf{3}}}  %requires amssymb
-\newcommand{\isasymfour}{\isamath{\mathbf{4}}}  %requires amssymb
-\newcommand{\isasymfive}{\isamath{\mathbf{5}}}  %requires amssymb
-\newcommand{\isasymsix}{\isamath{\mathbf{6}}}  %requires amssymb
-\newcommand{\isasymseven}{\isamath{\mathbf{7}}}  %requires amssymb
-\newcommand{\isasymeight}{\isamath{\mathbf{8}}}  %requires amssymb
-\newcommand{\isasymnine}{\isamath{\mathbf{9}}}  %requires amssymb
-\newcommand{\isasymA}{\isamath{\mathcal{A}}}
-\newcommand{\isasymB}{\isamath{\mathcal{B}}}
-\newcommand{\isasymC}{\isamath{\mathcal{C}}}
-\newcommand{\isasymD}{\isamath{\mathcal{D}}}
-\newcommand{\isasymE}{\isamath{\mathcal{E}}}
-\newcommand{\isasymF}{\isamath{\mathcal{F}}}
-\newcommand{\isasymG}{\isamath{\mathcal{G}}}
-\newcommand{\isasymH}{\isamath{\mathcal{H}}}
-\newcommand{\isasymI}{\isamath{\mathcal{I}}}
-\newcommand{\isasymJ}{\isamath{\mathcal{J}}}
-\newcommand{\isasymK}{\isamath{\mathcal{K}}}
-\newcommand{\isasymL}{\isamath{\mathcal{L}}}
-\newcommand{\isasymM}{\isamath{\mathcal{M}}}
-\newcommand{\isasymN}{\isamath{\mathcal{N}}}
-\newcommand{\isasymO}{\isamath{\mathcal{O}}}
-\newcommand{\isasymP}{\isamath{\mathcal{P}}}
-\newcommand{\isasymQ}{\isamath{\mathcal{Q}}}
-\newcommand{\isasymR}{\isamath{\mathcal{R}}}
-\newcommand{\isasymS}{\isamath{\mathcal{S}}}
-\newcommand{\isasymT}{\isamath{\mathcal{T}}}
-\newcommand{\isasymU}{\isamath{\mathcal{U}}}
-\newcommand{\isasymV}{\isamath{\mathcal{V}}}
-\newcommand{\isasymW}{\isamath{\mathcal{W}}}
-\newcommand{\isasymX}{\isamath{\mathcal{X}}}
-\newcommand{\isasymY}{\isamath{\mathcal{Y}}}
-\newcommand{\isasymZ}{\isamath{\mathcal{Z}}}
-\newcommand{\isasyma}{\isamath{\mathrm{a}}}
-\newcommand{\isasymb}{\isamath{\mathrm{b}}}
-\newcommand{\isasymc}{\isamath{\mathrm{c}}}
-\newcommand{\isasymd}{\isamath{\mathrm{d}}}
-\newcommand{\isasyme}{\isamath{\mathrm{e}}}
-\newcommand{\isasymf}{\isamath{\mathrm{f}}}
-\newcommand{\isasymg}{\isamath{\mathrm{g}}}
-\newcommand{\isasymh}{\isamath{\mathrm{h}}}
-\newcommand{\isasymi}{\isamath{\mathrm{i}}}
-\newcommand{\isasymj}{\isamath{\mathrm{j}}}
-\newcommand{\isasymk}{\isamath{\mathrm{k}}}
-\newcommand{\isasyml}{\isamath{\mathrm{l}}}
-\newcommand{\isasymm}{\isamath{\mathrm{m}}}
-\newcommand{\isasymn}{\isamath{\mathrm{n}}}
-\newcommand{\isasymo}{\isamath{\mathrm{o}}}
-\newcommand{\isasymp}{\isamath{\mathrm{p}}}
-\newcommand{\isasymq}{\isamath{\mathrm{q}}}
-\newcommand{\isasymr}{\isamath{\mathrm{r}}}
-\newcommand{\isasyms}{\isamath{\mathrm{s}}}
-\newcommand{\isasymt}{\isamath{\mathrm{t}}}
-\newcommand{\isasymu}{\isamath{\mathrm{u}}}
-\newcommand{\isasymv}{\isamath{\mathrm{v}}}
-\newcommand{\isasymw}{\isamath{\mathrm{w}}}
-\newcommand{\isasymx}{\isamath{\mathrm{x}}}
-\newcommand{\isasymy}{\isamath{\mathrm{y}}}
-\newcommand{\isasymz}{\isamath{\mathrm{z}}}
-\newcommand{\isasymAA}{\isamath{\mathfrak{A}}}  %requires eufrak
-\newcommand{\isasymBB}{\isamath{\mathfrak{B}}}  %requires eufrak
-\newcommand{\isasymCC}{\isamath{\mathfrak{C}}}  %requires eufrak
-\newcommand{\isasymDD}{\isamath{\mathfrak{D}}}  %requires eufrak
-\newcommand{\isasymEE}{\isamath{\mathfrak{E}}}  %requires eufrak
-\newcommand{\isasymFF}{\isamath{\mathfrak{F}}}  %requires eufrak
-\newcommand{\isasymGG}{\isamath{\mathfrak{G}}}  %requires eufrak
-\newcommand{\isasymHH}{\isamath{\mathfrak{H}}}  %requires eufrak
-\newcommand{\isasymII}{\isamath{\mathfrak{I}}}  %requires eufrak
-\newcommand{\isasymJJ}{\isamath{\mathfrak{J}}}  %requires eufrak
-\newcommand{\isasymKK}{\isamath{\mathfrak{K}}}  %requires eufrak
-\newcommand{\isasymLL}{\isamath{\mathfrak{L}}}  %requires eufrak
-\newcommand{\isasymMM}{\isamath{\mathfrak{M}}}  %requires eufrak
-\newcommand{\isasymNN}{\isamath{\mathfrak{N}}}  %requires eufrak
-\newcommand{\isasymOO}{\isamath{\mathfrak{O}}}  %requires eufrak
-\newcommand{\isasymPP}{\isamath{\mathfrak{P}}}  %requires eufrak
-\newcommand{\isasymQQ}{\isamath{\mathfrak{Q}}}  %requires eufrak
-\newcommand{\isasymRR}{\isamath{\mathfrak{R}}}  %requires eufrak
-\newcommand{\isasymSS}{\isamath{\mathfrak{S}}}  %requires eufrak
-\newcommand{\isasymTT}{\isamath{\mathfrak{T}}}  %requires eufrak
-\newcommand{\isasymUU}{\isamath{\mathfrak{U}}}  %requires eufrak
-\newcommand{\isasymVV}{\isamath{\mathfrak{V}}}  %requires eufrak
-\newcommand{\isasymWW}{\isamath{\mathfrak{W}}}  %requires eufrak
-\newcommand{\isasymXX}{\isamath{\mathfrak{X}}}  %requires eufrak
-\newcommand{\isasymYY}{\isamath{\mathfrak{Y}}}  %requires eufrak
-\newcommand{\isasymZZ}{\isamath{\mathfrak{Z}}}  %requires eufrak
-\newcommand{\isasymaa}{\isamath{\mathfrak{a}}}  %requires eufrak
-\newcommand{\isasymbb}{\isamath{\mathfrak{b}}}  %requires eufrak
-\newcommand{\isasymcc}{\isamath{\mathfrak{c}}}  %requires eufrak
-\newcommand{\isasymdd}{\isamath{\mathfrak{d}}}  %requires eufrak
-\newcommand{\isasymee}{\isamath{\mathfrak{e}}}  %requires eufrak
-\newcommand{\isasymff}{\isamath{\mathfrak{f}}}  %requires eufrak
-\newcommand{\isasymgg}{\isamath{\mathfrak{g}}}  %requires eufrak
-\newcommand{\isasymhh}{\isamath{\mathfrak{h}}}  %requires eufrak
-\newcommand{\isasymii}{\isamath{\mathfrak{i}}}  %requires eufrak
-\newcommand{\isasymjj}{\isamath{\mathfrak{j}}}  %requires eufrak
-\newcommand{\isasymkk}{\isamath{\mathfrak{k}}}  %requires eufrak
-\newcommand{\isasymll}{\isamath{\mathfrak{l}}}  %requires eufrak
-\newcommand{\isasymmm}{\isamath{\mathfrak{m}}}  %requires eufrak
-\newcommand{\isasymnn}{\isamath{\mathfrak{n}}}  %requires eufrak
-\newcommand{\isasymoo}{\isamath{\mathfrak{o}}}  %requires eufrak
-\newcommand{\isasympp}{\isamath{\mathfrak{p}}}  %requires eufrak
-\newcommand{\isasymqq}{\isamath{\mathfrak{q}}}  %requires eufrak
-\newcommand{\isasymrr}{\isamath{\mathfrak{r}}}  %requires eufrak
-\newcommand{\isasymss}{\isamath{\mathfrak{s}}}  %requires eufrak
-\newcommand{\isasymtt}{\isamath{\mathfrak{t}}}  %requires eufrak
-\newcommand{\isasymuu}{\isamath{\mathfrak{u}}}  %requires eufrak
-\newcommand{\isasymvv}{\isamath{\mathfrak{v}}}  %requires eufrak
-\newcommand{\isasymww}{\isamath{\mathfrak{w}}}  %requires eufrak
-\newcommand{\isasymxx}{\isamath{\mathfrak{x}}}  %requires eufrak
-\newcommand{\isasymyy}{\isamath{\mathfrak{y}}}  %requires eufrak
-\newcommand{\isasymzz}{\isamath{\mathfrak{z}}}  %requires eufrak
-\newcommand{\isasymalpha}{\isamath{\alpha}}
-\newcommand{\isasymbeta}{\isamath{\beta}}
-\newcommand{\isasymgamma}{\isamath{\gamma}}
-\newcommand{\isasymdelta}{\isamath{\delta}}
-\newcommand{\isasymepsilon}{\isamath{\varepsilon}}
-\newcommand{\isasymzeta}{\isamath{\zeta}}
-\newcommand{\isasymeta}{\isamath{\eta}}
-\newcommand{\isasymtheta}{\isamath{\vartheta}}
-\newcommand{\isasymiota}{\isamath{\iota}}
-\newcommand{\isasymkappa}{\isamath{\kappa}}
-\newcommand{\isasymlambda}{\isamath{\lambda}}
-\newcommand{\isasymmu}{\isamath{\mu}}
-\newcommand{\isasymnu}{\isamath{\nu}}
-\newcommand{\isasymxi}{\isamath{\xi}}
-\newcommand{\isasympi}{\isamath{\pi}}
-\newcommand{\isasymrho}{\isamath{\varrho}}
-\newcommand{\isasymsigma}{\isamath{\sigma}}
-\newcommand{\isasymtau}{\isamath{\tau}}
-\newcommand{\isasymupsilon}{\isamath{\upsilon}}
-\newcommand{\isasymphi}{\isamath{\varphi}}
-\newcommand{\isasymchi}{\isamath{\chi}}
-\newcommand{\isasympsi}{\isamath{\psi}}
-\newcommand{\isasymomega}{\isamath{\omega}}
-\newcommand{\isasymGamma}{\isamath{\Gamma}}
-\newcommand{\isasymDelta}{\isamath{\Delta}}
-\newcommand{\isasymTheta}{\isamath{\Theta}}
-\newcommand{\isasymLambda}{\isamath{\Lambda}}
-\newcommand{\isasymXi}{\isamath{\Xi}}
-\newcommand{\isasymPi}{\isamath{\Pi}}
-\newcommand{\isasymSigma}{\isamath{\Sigma}}
-\newcommand{\isasymUpsilon}{\isamath{\Upsilon}}
-\newcommand{\isasymPhi}{\isamath{\Phi}}
-\newcommand{\isasymPsi}{\isamath{\Psi}}
-\newcommand{\isasymOmega}{\isamath{\Omega}}
-\newcommand{\isasymbool}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{B}}}
-\newcommand{\isasymcomplex}{\isamath{\mathrm{C}\mkern-15mu{\phantom{\mathrm{t}}\vrule}\mkern9mu}}
-\newcommand{\isasymnat}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{N}}}
-\newcommand{\isasymrat}{\isamath{\mathrm{Q}\mkern-16mu{\phantom{\mathrm{t}}\vrule}\mkern10mu}}
-\newcommand{\isasymreal}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{R}}}
-\newcommand{\isasymint}{\isamath{\mathsf{Z}\mkern-7.5mu\mathsf{Z}}}
-\newcommand{\isasymleftarrow}{\isamath{\leftarrow}}
-\newcommand{\isasymrightarrow}{\isamath{\rightarrow}}
-\newcommand{\isasymlongleftarrow}{\isamath{\longleftarrow}}
-\newcommand{\isasymlongrightarrow}{\isamath{\longrightarrow}}
-\newcommand{\isasymlonglongleftarrow}{\isamath{\xleftarrow{\hphantom{AAA}}}}  %requires amsmath
-\newcommand{\isasymlonglongrightarrow}{\isamath{\xrightarrow{\hphantom{AAA}}}}  %requires amsmath
-\newcommand{\isasymlonglonglongleftarrow}{\isamath{\xleftarrow{\hphantom{AAAA}}}}  %requires amsmath
-\newcommand{\isasymlonglonglongrightarrow}{\isamath{\xrightarrow{\hphantom{AAAA}}}}  %requires amsmath
-\newcommand{\isasymLeftarrow}{\isamath{\Leftarrow}}
-\newcommand{\isasymRightarrow}{\isamath{\Rightarrow}}
-\newcommand{\isasymLongleftarrow}{\isamath{\Longleftarrow}}
-\newcommand{\isasymLongrightarrow}{\isamath{\Longrightarrow}}
-\newcommand{\isasymLleftarrow}{\isamath{\Lleftarrow}}  %requires amssymb
-\newcommand{\isasymRrightarrow}{\isamath{\Rrightarrow}}  %requires amssymb
-\newcommand{\isasymleftrightarrow}{\isamath{\leftrightarrow}}
-\newcommand{\isasymLeftrightarrow}{\isamath{\Leftrightarrow}}
-\newcommand{\isasymlongleftrightarrow}{\isamath{\longleftrightarrow}}
-\newcommand{\isasymLongleftrightarrow}{\isamath{\Longleftrightarrow}}
-\newcommand{\isasymmapsto}{\isamath{\mapsto}}
-\newcommand{\isasymlongmapsto}{\isamath{\longmapsto}}
-\newcommand{\isasymmidarrow}{\isamath{\relbar}}
-\newcommand{\isasymMidarrow}{\isamath{\Relbar}}
-\newcommand{\isasymhookleftarrow}{\isamath{\hookleftarrow}}
-\newcommand{\isasymhookrightarrow}{\isamath{\hookrightarrow}}
-\newcommand{\isasymleftharpoondown}{\isamath{\leftharpoondown}}
-\newcommand{\isasymrightharpoondown}{\isamath{\rightharpoondown}}
-\newcommand{\isasymleftharpoonup}{\isamath{\leftharpoonup}}
-\newcommand{\isasymrightharpoonup}{\isamath{\rightharpoonup}}
-\newcommand{\isasymrightleftharpoons}{\isamath{\rightleftharpoons}}
-\newcommand{\isasymleadsto}{\isamath{\leadsto}}  %requires amssymb
-\newcommand{\isasymdownharpoonleft}{\isamath{\downharpoonleft}}  %requires amssymb
-\newcommand{\isasymdownharpoonright}{\isamath{\downharpoonright}}  %requires amssymb
-\newcommand{\isasymupharpoonleft}{\isamath{\upharpoonleft}}  %requires amssymb
-\newcommand{\isasymupharpoonright}{\isamath{\upharpoonright}}  %requires amssymb
-\newcommand{\isasymrestriction}{\isamath{\restriction}}  %requires amssymb
-\newcommand{\isasymColon}{\isamath{\mathrel{::}}}
-\newcommand{\isasymup}{\isamath{\uparrow}}
-\newcommand{\isasymUp}{\isamath{\Uparrow}}
-\newcommand{\isasymdown}{\isamath{\downarrow}}
-\newcommand{\isasymDown}{\isamath{\Downarrow}}
-\newcommand{\isasymupdown}{\isamath{\updownarrow}}
-\newcommand{\isasymUpdown}{\isamath{\Updownarrow}}
-\newcommand{\isasymlangle}{\isamath{\langle}}
-\newcommand{\isasymrangle}{\isamath{\rangle}}
-\newcommand{\isasymllangle}{\isamath{\langle\mskip-5mu\langle}}
-\newcommand{\isasymrrangle}{\isamath{\rangle\mskip-5mu\rangle}}
-\newcommand{\isasymlceil}{\isamath{\lceil}}
-\newcommand{\isasymrceil}{\isamath{\rceil}}
-\newcommand{\isasymlfloor}{\isamath{\lfloor}}
-\newcommand{\isasymrfloor}{\isamath{\rfloor}}
-\newcommand{\isasymlparr}{\isamath{\mathopen{(\mkern-3mu\mid}}}
-\newcommand{\isasymrparr}{\isamath{\mathclose{\mid\mkern-3mu)}}}
-\newcommand{\isasymlbrakk}{\isamath{\mathopen{\lbrack\mkern-3mu\lbrack}}}
-\newcommand{\isasymrbrakk}{\isamath{\mathclose{\rbrack\mkern-3mu\rbrack}}}
-\newcommand{\isasymlbrace}{\isamath{\mathopen{\lbrace\mkern-4.5mu\mid}}}
-\newcommand{\isasymrbrace}{\isamath{\mathclose{\mid\mkern-4.5mu\rbrace}}}
-\newcommand{\isasymguillemotleft}{\isatext{\flqq}}  %requires babel
-\newcommand{\isasymguillemotright}{\isatext{\frqq}}  %requires babel
-\newcommand{\isasymbottom}{\isamath{\bot}}
-\newcommand{\isasymtop}{\isamath{\top}}
-\newcommand{\isasymand}{\isamath{\wedge}}
-\newcommand{\isasymAnd}{\isamath{\bigwedge}}
-\newcommand{\isasymor}{\isamath{\vee}}
-\newcommand{\isasymOr}{\isamath{\bigvee}}
-\newcommand{\isasymforall}{\isamath{\forall\,}}
-\newcommand{\isasymexists}{\isamath{\exists\,}}
-\newcommand{\isasymnot}{\isamath{\neg}}
-\newcommand{\isasymnexists}{\isamath{\nexists\,}}  %requires amssymb
-\newcommand{\isasymcircle}{\isamath{\ocircle}}  %requires wasysym
-\newcommand{\isasymbox}{\isamath{\Box}}  %requires amssymb
-\newcommand{\isasymdiamond}{\isamath{\Diamond}}  %requires amssymb
-\newcommand{\isasymdiamondop}{\isamath{\diamond}}
-\newcommand{\isasymsurd}{\isamath{\surd}}
-\newcommand{\isasymturnstile}{\isamath{\vdash}}
-\newcommand{\isasymTurnstile}{\isamath{\models}}
-\newcommand{\isasymtturnstile}{\isamath{\vdash\!\!\!\vdash}}
-\newcommand{\isasymTTurnstile}{\isamath{\mid\!\models}}
-\newcommand{\isasymstileturn}{\isamath{\dashv}}
-\newcommand{\isasymle}{\isamath{\le}}
-\newcommand{\isasymge}{\isamath{\ge}}
-\newcommand{\isasymlless}{\isamath{\ll}}
-\newcommand{\isasymggreater}{\isamath{\gg}}
-\newcommand{\isasymlesssim}{\isamath{\lesssim}}  %requires amssymb
-\newcommand{\isasymgreatersim}{\isamath{\gtrsim}}  %requires amssymb
-\newcommand{\isasymlessapprox}{\isamath{\lessapprox}}  %requires amssymb
-\newcommand{\isasymgreaterapprox}{\isamath{\gtrapprox}}  %requires amssymb
-\newcommand{\isasymin}{\isamath{\in}}
-\newcommand{\isasymnotin}{\isamath{\notin}}
-\newcommand{\isasymsubset}{\isamath{\subset}}
-\newcommand{\isasymsupset}{\isamath{\supset}}
-\newcommand{\isasymsubseteq}{\isamath{\subseteq}}
-\newcommand{\isasymsupseteq}{\isamath{\supseteq}}
-\newcommand{\isasymsqsubset}{\isamath{\sqsubset}}  %requires amssymb
-\newcommand{\isasymsqsupset}{\isamath{\sqsupset}}  %requires amssymb
-\newcommand{\isasymsqsubseteq}{\isamath{\sqsubseteq}}
-\newcommand{\isasymsqsupseteq}{\isamath{\sqsupseteq}}
-\newcommand{\isasyminter}{\isamath{\cap}}
-\newcommand{\isasymInter}{\isamath{\bigcap\,}}
-\newcommand{\isasymunion}{\isamath{\cup}}
-\newcommand{\isasymUnion}{\isamath{\bigcup\,}}
-\newcommand{\isasymsqunion}{\isamath{\sqcup}}
-\newcommand{\isasymSqunion}{\isamath{\bigsqcup\,}}
-\newcommand{\isasymsqinter}{\isamath{\sqcap}}
-\newcommand{\isasymSqinter}{\isamath{\bigsqcap\,}}  %requires stmaryrd
-\newcommand{\isasymsetminus}{\isamath{\setminus}}
-\newcommand{\isasympropto}{\isamath{\propto}}
-\newcommand{\isasymuplus}{\isamath{\uplus}}
-\newcommand{\isasymUplus}{\isamath{\biguplus\,}}
-\newcommand{\isasymnoteq}{\isamath{\not=}}
-\newcommand{\isasymsim}{\isamath{\sim}}
-\newcommand{\isasymdoteq}{\isamath{\doteq}}
-\newcommand{\isasymsimeq}{\isamath{\simeq}}
-\newcommand{\isasymapprox}{\isamath{\approx}}
-\newcommand{\isasymasymp}{\isamath{\asymp}}
-\newcommand{\isasymcong}{\isamath{\cong}}
-\newcommand{\isasymsmile}{\isamath{\smile}}
-\newcommand{\isasymequiv}{\isamath{\equiv}}
-\newcommand{\isasymfrown}{\isamath{\frown}}
-\newcommand{\isasymJoin}{\isamath{\Join}}  %requires amssymb
-\newcommand{\isasymbowtie}{\isamath{\bowtie}}
-\newcommand{\isasymprec}{\isamath{\prec}}
-\newcommand{\isasymsucc}{\isamath{\succ}}
-\newcommand{\isasympreceq}{\isamath{\preceq}}
-\newcommand{\isasymsucceq}{\isamath{\succeq}}
-\newcommand{\isasymparallel}{\isamath{\parallel}}
-\newcommand{\isasymbar}{\isamath{\mid}}
-\newcommand{\isasymbbar}{\isamath{[\mskip-1.5mu]}}
-\newcommand{\isasymplusminus}{\isamath{\pm}}
-\newcommand{\isasymminusplus}{\isamath{\mp}}
-\newcommand{\isasymtimes}{\isamath{\times}}
-\newcommand{\isasymdiv}{\isamath{\div}}
-\newcommand{\isasymcdot}{\isamath{\cdot}}
-\newcommand{\isasymsqdot}{\isamath{\sbox\z@{$\centerdot$}\ht\z@=.33333\ht\z@\vcenter{\box\z@}}}  %requires amssymb
-\newcommand{\isasymstar}{\isamath{\star}}
-\newcommand{\isasymbullet}{\boldmath\isamath{\mathchoice{\displaystyle{\cdot}}{\textstyle{\cdot}}{\scriptstyle{\bullet}}{\scriptscriptstyle{\bullet}}}}
-\newcommand{\isasymcirc}{\isamath{\circ}}
-\newcommand{\isasymdagger}{\isamath{\dagger}}
-\newcommand{\isasymddagger}{\isamath{\ddagger}}
-\newcommand{\isasymlhd}{\isamath{\lhd}}  %requires amssymb
-\newcommand{\isasymrhd}{\isamath{\rhd}}  %requires amssymb
-\newcommand{\isasymunlhd}{\isamath{\unlhd}}  %requires amssymb
-\newcommand{\isasymunrhd}{\isamath{\unrhd}}  %requires amssymb
-\newcommand{\isasymtriangleleft}{\isamath{\triangleleft}}
-\newcommand{\isasymtriangleright}{\isamath{\triangleright}}
-\newcommand{\isasymtriangle}{\isamath{\triangle}}
-\newcommand{\isasymtriangleq}{\isamath{\triangleq}}  %requires amssymb
-\newcommand{\isasymoplus}{\isamath{\oplus}}
-\newcommand{\isasymOplus}{\isamath{\bigoplus\,}}
-\newcommand{\isasymotimes}{\isamath{\otimes}}
-\newcommand{\isasymOtimes}{\isamath{\bigotimes\,}}
-\newcommand{\isasymodot}{\isamath{\odot}}
-\newcommand{\isasymOdot}{\isamath{\bigodot\,}}
-\newcommand{\isasymominus}{\isamath{\ominus}}
-\newcommand{\isasymoslash}{\isamath{\oslash}}
-\newcommand{\isasymdots}{\isamath{\dots}}
-\newcommand{\isasymcdots}{\isamath{\cdots}}
-\newcommand{\isasymSum}{\isamath{\sum\,}}
-\newcommand{\isasymProd}{\isamath{\prod\,}}
-\newcommand{\isasymCoprod}{\isamath{\coprod\,}}
-\newcommand{\isasyminfinity}{\isamath{\infty}}
-\newcommand{\isasymintegral}{\isamath{\int\,}}
-\newcommand{\isasymointegral}{\isamath{\oint\,}}
-\newcommand{\isasymclubsuit}{\isamath{\clubsuit}}
-\newcommand{\isasymdiamondsuit}{\isamath{\diamondsuit}}
-\newcommand{\isasymheartsuit}{\isamath{\heartsuit}}
-\newcommand{\isasymspadesuit}{\isamath{\spadesuit}}
-\newcommand{\isasymaleph}{\isamath{\aleph}}
-\newcommand{\isasymemptyset}{\isamath{\emptyset}}
-\newcommand{\isasymnabla}{\isamath{\nabla}}
-\newcommand{\isasympartial}{\isamath{\partial}}
-\newcommand{\isasymRe}{\isamath{\Re}}
-\newcommand{\isasymIm}{\isamath{\Im}}
-\newcommand{\isasymflat}{\isamath{\flat}}
-\newcommand{\isasymnatural}{\isamath{\natural}}
-\newcommand{\isasymsharp}{\isamath{\sharp}}
-\newcommand{\isasymangle}{\isamath{\angle}}
-\newcommand{\isasymcopyright}{\isatext{\normalfont\rmfamily\copyright}}
-\newcommand{\isasymregistered}{\isatext{\normalfont\rmfamily\textregistered}}
-\newcommand{\isasyminverse}{\isamath{{}^{-1}}}
-\newcommand{\isasymonequarter}{\isatext{\normalfont\rmfamily\textonequarter}}  %requires textcomp
-\newcommand{\isasymonehalf}{\isatext{\normalfont\rmfamily\textonehalf}}  %requires textcomp
-\newcommand{\isasymthreequarters}{\isatext{\normalfont\rmfamily\textthreequarters}}  %requires textcomp
-\newcommand{\isasymordfeminine}{\isatext{\normalfont\rmfamily\textordfeminine}}
-\newcommand{\isasymordmasculine}{\isatext{\normalfont\rmfamily\textordmasculine}}
-\newcommand{\isasymsection}{\isatext{\normalfont\rmfamily\S}}
-\newcommand{\isasymparagraph}{\isatext{\normalfont\rmfamily\P}}
-\newcommand{\isasymexclamdown}{\isatext{\normalfont\rmfamily\textexclamdown}}
-\newcommand{\isasymquestiondown}{\isatext{\normalfont\rmfamily\textquestiondown}}
-\newcommand{\isasymeuro}{\isatext{\euro}}  %requires eurosym
-\newcommand{\isasympounds}{\isamath{\pounds}}
-\newcommand{\isasymyen}{\isatext{\yen}}  %requires amssymb
-\newcommand{\isasymcent}{\isatext{\textcent}}  %requires textcomp
-\newcommand{\isasymcurrency}{\isatext{\textcurrency}} %requires textcomp
-\newcommand{\isasymdegree}{\isatext{\normalfont\rmfamily\textdegree}}  %requires textcomp
-\newcommand{\isasymhyphen}{\isatext{\normalfont\rmfamily-}}
-\newcommand{\isasymamalg}{\isamath{\amalg}}
-\newcommand{\isasymmho}{\isamath{\mho}}  %requires amssymb
-\newcommand{\isasymlozenge}{\isamath{\lozenge}}  %requires amssymb
-\newcommand{\isasymwp}{\isamath{\wp}}
-\newcommand{\isasymwrong}{\isamath{\wr}}
-\newcommand{\isasymacute}{\isatext{\'\relax}}
-\newcommand{\isasymindex}{\isatext{\i}}
-\newcommand{\isasymdieresis}{\isatext{\"\relax}}
-\newcommand{\isasymcedilla}{\isatext{\c\relax}}
-\newcommand{\isasymhungarumlaut}{\isatext{\H\relax}}
-\newcommand{\isasymsome}{\isamath{\epsilon\,}}
-\newcommand{\isasymbind}{\isamath{\mathbin{>\!\!\!>\mkern-6.7mu=}}}
-\newcommand{\isasymthen}{\isamath{\mathbin{>\!\!\!>}}}
-\newcommand{\isasymopen}{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\langle$}}}
-\newcommand{\isasymclose}{\isatext{\raise.3ex\hbox{$\scriptscriptstyle\rangle$}}}
-\newcommand{\isasymhole}{\isatext{\normalfont\rmfamily\wasylozenge}}  %requires wasysym
-\newcommand{\isasymnewline}{\isatext{\fbox{$\hookleftarrow$}}}
-\newcommand{\isasymcomment}{\isatext{\isastylecmt---}}
-\newcommand{\isasymproof}{\isamath{\,\langle\mathit{proof}\rangle}}
-
-\newcommand{\isactrlmarker}{\isatext{\ding{48}}}  %requires pifont
-\newcommand{\isactrlassert}{\isakeywordcontrol{assert}}
-\newcommand{\isactrlcancel}{\isakeywordcontrol{cancel}}
-\newcommand{\isactrlbinding}{\isakeywordcontrol{binding}}
-\newcommand{\isactrlclass}{\isakeywordcontrol{class}}
-\newcommand{\isactrlclassUNDERSCOREsyntax}{\isakeywordcontrol{class{\isacharunderscore}syntax}}
-\newcommand{\isactrlcommandUNDERSCOREkeyword}{\isakeywordcontrol{command{\isacharunderscore}keyword}}
-\newcommand{\isactrlconst}{\isakeywordcontrol{const}}
-\newcommand{\isactrlconstUNDERSCOREabbrev}{\isakeywordcontrol{const{\isacharunderscore}abbrev}}
-\newcommand{\isactrlconstUNDERSCOREname}{\isakeywordcontrol{const{\isacharunderscore}name}}
-\newcommand{\isactrlconstUNDERSCOREsyntax}{\isakeywordcontrol{const{\isacharunderscore}syntax}}
-\newcommand{\isactrlcontext}{\isakeywordcontrol{context}}
-\newcommand{\isactrlcprop}{\isakeywordcontrol{cprop}}
-\newcommand{\isactrlcterm}{\isakeywordcontrol{cterm}}
-\newcommand{\isactrlctyp}{\isakeywordcontrol{ctyp}}
-\newcommand{\isactrldir}{\isakeywordcontrol{dir}}
-\newcommand{\isactrlfile}{\isakeywordcontrol{file}}
-\newcommand{\isactrlhere}{\isakeywordcontrol{here}}
-\newcommand{\isactrlkeyword}{\isakeywordcontrol{keyword}}
-\newcommand{\isactrllatex}{\isakeywordcontrol{latex}}
-\newcommand{\isactrllocale}{\isakeywordcontrol{locale}}
-\newcommand{\isactrlmakeUNDERSCOREstring}{\isakeywordcontrol{make{\isacharunderscore}string}}
-\newcommand{\isactrlmasterUNDERSCOREdir}{\isakeywordcontrol{master{\isacharunderscore}dir}}
-\newcommand{\isactrlmethod}{\isakeywordcontrol{method}}
-\newcommand{\isactrlnamedUNDERSCOREtheorems}{\isakeywordcontrol{named{\isacharunderscore}theorems}}
-\newcommand{\isactrlnonterminal}{\isakeywordcontrol{nonterminal}}
-\newcommand{\isactrloracleUNDERSCOREname}{\isakeywordcontrol{oracle{\isacharunderscore}name}}
-\newcommand{\isactrlpath}{\isakeywordcontrol{path}}
-\newcommand{\isactrlpathUNDERSCOREbinding}{\isakeywordcontrol{path{\isacharunderscore}binding}}
-\newcommand{\isactrlplugin}{\isakeywordcontrol{plugin}}
-\newcommand{\isactrlprint}{\isakeywordcontrol{print}}
-\newcommand{\isactrlprop}{\isakeywordcontrol{prop}}
-\newcommand{\isactrlscala}{\isakeywordcontrol{scala}}
-\newcommand{\isactrlscalaUNDERSCOREfunction}{\isakeywordcontrol{scala{\isacharunderscore}function}}
-\newcommand{\isactrlscalaUNDERSCOREmethod}{\isakeywordcontrol{scala{\isacharunderscore}method}}
-\newcommand{\isactrlscalaUNDERSCOREobject}{\isakeywordcontrol{scala{\isacharunderscore}object}}
-\newcommand{\isactrlscalaUNDERSCOREtype}{\isakeywordcontrol{scala{\isacharunderscore}type}}
-\newcommand{\isactrlsimproc}{\isakeywordcontrol{simproc}}
-\newcommand{\isactrlsort}{\isakeywordcontrol{sort}}
-\newcommand{\isactrlsyntaxUNDERSCOREconst}{\isakeywordcontrol{syntax{\isacharunderscore}const}}
-\newcommand{\isactrlsystemUNDERSCOREoption}{\isakeywordcontrol{system{\isacharunderscore}option}}
-\newcommand{\isactrlterm}{\isakeywordcontrol{term}}
-\newcommand{\isactrltheory}{\isakeywordcontrol{theory}}
-\newcommand{\isactrltheoryUNDERSCOREcontext}{\isakeywordcontrol{theory{\isacharunderscore}context}}
-\newcommand{\isactrltyp}{\isakeywordcontrol{typ}}
-\newcommand{\isactrltypeUNDERSCOREabbrev}{\isakeywordcontrol{type{\isacharunderscore}abbrev}}
-\newcommand{\isactrltypeUNDERSCOREname}{\isakeywordcontrol{type{\isacharunderscore}name}}
-\newcommand{\isactrltypeUNDERSCOREsyntax}{\isakeywordcontrol{type{\isacharunderscore}syntax}}
-\newcommand{\isactrlundefined}{\isakeywordcontrol{undefined}}
-
-\newcommand{\isactrlcode}{\isakeywordcontrol{code}}
-\newcommand{\isactrlcomputation}{\isakeywordcontrol{computation}}
-\newcommand{\isactrlcomputationUNDERSCOREconv}{\isakeywordcontrol{computation{\isacharunderscore}conv}}
-\newcommand{\isactrlcomputationUNDERSCOREcheck}{\isakeywordcontrol{computation{\isacharunderscore}check}}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabelletags.sty b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabelletags.sty
deleted file mode 100644
index 8b13789..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/isabelletags.sty
+++ /dev/null
@@ -1 +0,0 @@
-
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/pdfsetup.sty b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/pdfsetup.sty
deleted file mode 100644
index 2bf5ccf..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/pdfsetup.sty
+++ /dev/null
@@ -1,7 +0,0 @@
-%%
-%% default hyperref setup (both for pdf and dvi output)
-%%
-
-\usepackage{color}
-\definecolor{linkcolor}{rgb}{0,0,0.5}
-\usepackage[colorlinks=true,linkcolor=linkcolor,citecolor=linkcolor,filecolor=linkcolor,urlcolor=linkcolor,pdfpagelabels]{hyperref}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/railsetup.sty b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/railsetup.sty
deleted file mode 100644
index 93cae05..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/railsetup.sty
+++ /dev/null
@@ -1,1202 +0,0 @@
-% rail.sty - style file to support railroad diagrams
-%
-% 09-Jul-90 L. Rooijakkers
-% 08-Oct-90 L. Rooijakkers	fixed centering bug when \rail@tmpc<0.
-% 07-Feb-91 L. Rooijakkers	added \railoptions command, indexing
-% 08-Feb-91 L. Rooijakkers	minor fixes
-% 28-Jun-94 K. Barthelmann	turned into LaTeX2e package
-% 08-Dec-96 K. Barthelmann	replaced \@writefile
-% 13-Dec-96 K. Barthelmann	cleanup
-% 22-Feb-98 K. Barthelmann	fixed catcodes of special characters
-% 18-Apr-98 K. Barthelmann	fixed \par handling
-% 19-May-98 J. Olsson		Added new macros to support arrow heads.
-% 26-Jul-98 K. Barthelmann	changed \par to output newlines
-% 02-May-11 M. Wenzel           default setup for Isabelle
-%
-% This style file needs to be used in conjunction with the 'rail'
-% program. Running LaTeX as 'latex file' produces file.rai, which should be
-% processed by Rail with 'rail file'. This produces file.rao, which will
-% be picked up by LaTeX on the next 'latex file' run.
-%
-% LaTeX will warn if there is no file.rao or it's out of date.
-%
-% The macros in this file thus consist of two parts: those that read and
-% write the .rai and .rao files, and those that do the actual formatting
-% of the railroad diagrams.
-
-\NeedsTeXFormat{LaTeX2e}
-\ProvidesPackage{rail}[1998/05/19]
-
-% railroad diagram formatting parameters (user level)
-% all of these are copied into their internal versions by \railinit
-%
-% \railunit : \unitlength within railroad diagrams
-% \railextra : extra length at outside of diagram
-% \railboxheight : height of ovals and frames
-% \railboxskip : vertical space between lines
-% \railboxleft : space to the left of a box
-% \railboxright : space to the right of a box
-% \railovalspace : extra space around contents of oval
-% \railframespace : extra space around contents of frame
-% \railtextleft : space to the left of text
-% \railtextright : space to the right of text
-% \railtextup : space to lift text up
-% \railjoinsize : circle size of join/split arcs
-% \railjoinadjust : space to adjust join
-%
-% \railnamesep : separator between name and rule body
-
-\newlength\railunit
-\newlength\railextra
-\newlength\railboxheight
-\newlength\railboxskip
-\newlength\railboxleft
-\newlength\railboxright
-\newlength\railovalspace
-\newlength\railframespace
-\newlength\railtextleft
-\newlength\railtextright
-\newlength\railtextup
-\newlength\railjoinsize
-\newlength\railjoinadjust
-\newlength\railnamesep
-
-% initialize the parameters
-
-\setlength\railunit{1sp}
-\setlength\railextra{4ex}
-\setlength\railboxleft{1ex}
-\setlength\railboxright{1ex}
-\setlength\railovalspace{2ex}
-\setlength\railframespace{2ex}
-\setlength\railtextleft{1ex}
-\setlength\railtextright{1ex}
-\setlength\railjoinadjust{0pt}
-\setlength\railnamesep{1ex}
-
-\DeclareOption{10pt}{
-  \setlength\railboxheight{16pt}
-  \setlength\railboxskip{24pt}
-  \setlength\railtextup{5pt}
-  \setlength\railjoinsize{16pt}
-}
-\DeclareOption{11pt}{
-  \setlength\railboxheight{16pt}
-  \setlength\railboxskip{24pt}
-  \setlength\railtextup{5pt}
-  \setlength\railjoinsize{16pt}
-}
-\DeclareOption{12pt}{
-  \setlength\railboxheight{20pt}
-  \setlength\railboxskip{28pt}
-  \setlength\railtextup{6pt}
-  \setlength\railjoinsize{20pt}
-}
-
-\ExecuteOptions{10pt}
-\ProcessOptions
-
-% internal versions of the formatting parameters
-%
-% \rail@extra   : \railextra
-% \rail@boxht   : \railboxheight
-% \rail@boxsp   : \railboxskip
-% \rail@boxlf   : \railboxleft
-% \rail@boxrt   : \railboxright
-% \rail@boxhht  : \railboxheight / 2
-% \rail@ovalsp  : \railovalspace
-% \rail@framesp : \railframespace
-% \rail@textlf  : \railtextleft
-% \rail@textrt  : \railtextright
-% \rail@textup  : \railtextup
-% \rail@joinsz  : \railjoinsize
-% \rail@joinhsz : \railjoinsize / 2
-% \rail@joinadj : \railjoinadjust
-%
-% \railinit : internalize all of the parameters.
-
-\newcount\rail@extra
-\newcount\rail@boxht
-\newcount\rail@boxsp
-\newcount\rail@boxlf
-\newcount\rail@boxrt
-\newcount\rail@boxhht
-\newcount\rail@ovalsp
-\newcount\rail@framesp
-\newcount\rail@textlf
-\newcount\rail@textrt
-\newcount\rail@textup
-\newcount\rail@joinsz
-\newcount\rail@joinhsz
-\newcount\rail@joinadj
-
-\newcommand\railinit{
-\rail@extra=\railextra
-\divide\rail@extra by \railunit
-\rail@boxht=\railboxheight
-\divide\rail@boxht by \railunit
-\rail@boxsp=\railboxskip
-\divide\rail@boxsp by \railunit
-\rail@boxlf=\railboxleft
-\divide\rail@boxlf by \railunit
-\rail@boxrt=\railboxright
-\divide\rail@boxrt by \railunit
-\rail@boxhht=\railboxheight
-\divide\rail@boxhht by \railunit
-\divide\rail@boxhht by 2
-\rail@ovalsp=\railovalspace
-\divide\rail@ovalsp by \railunit
-\rail@framesp=\railframespace
-\divide\rail@framesp by \railunit
-\rail@textlf=\railtextleft
-\divide\rail@textlf by \railunit
-\rail@textrt=\railtextright
-\divide\rail@textrt by \railunit
-\rail@textup=\railtextup
-\divide\rail@textup by \railunit
-\rail@joinsz=\railjoinsize
-\divide\rail@joinsz by \railunit
-\rail@joinhsz=\railjoinsize
-\divide\rail@joinhsz by \railunit
-\divide\rail@joinhsz by 2
-\rail@joinadj=\railjoinadjust
-\divide\rail@joinadj by \railunit
-}
-
-\AtBeginDocument{\railinit}
-
-% \rail@param : declarations for list environment
-%
-% \railparam{TEXT} : sets \rail@param to TEXT
-%
-% \rail@reserved : characters reserved for grammar
-
-\newcommand\railparam[1]{
-\def\rail@param{
-  \setlength\leftmargin{0pt}\setlength\rightmargin{0pt}
-  \setlength\labelwidth{0pt}\setlength\labelsep{0pt}
-  \setlength\itemindent{0pt}\setlength\listparindent{0pt}
-  #1
-}
-}
-\railparam{}
-
-\newtoks\rail@reserved
-\rail@reserved={:;|*+?[]()'"}
-
-% \rail@termfont : format setup for terminals
-%
-% \rail@nontfont : format setup for nonterminals
-%
-% \rail@annofont : format setup for annotations
-%
-% \rail@rulefont : format setup for rule names
-%
-% \rail@indexfont : format setup for index entry
-%
-% \railtermfont{TEXT} : set terminal format setup to TEXT
-%
-% \railnontermfont{TEXT} : set nonterminal format setup to TEXT
-%
-% \railannotatefont{TEXT} : set annotation format setup to TEXT
-%
-% \railnamefont{TEXT} : set rule name format setup to TEXT
-%
-% \railindexfont{TEXT} : set index entry format setup to TEXT
-
-\def\rail@termfont{\ttfamily\upshape}
-\def\rail@nontfont{\rmfamily\upshape}
-\def\rail@annofont{\rmfamily\itshape}
-\def\rail@namefont{\rmfamily\itshape}
-\def\rail@indexfont{\rmfamily\itshape}
-
-\newcommand\railtermfont[1]{
-\def\rail@termfont{#1}
-}
-
-\newcommand\railnontermfont[1]{
-\def\rail@nontfont{#1}
-}
-
-\newcommand\railannotatefont[1]{
-\def\rail@annofont{#1}
-}
-
-\newcommand\railnamefont[1]{
-\def\rail@namefont{#1}
-}
-
-\newcommand\railindexfont[1]{
-\def\rail@indexfont{#1}
-}
-
-% railroad read/write macros
-%
-% \begin{rail} TEXT \end{rail} : TEXT is written out to the .rai file,
-%                                as \rail@i{NR}{TEXT}. Then the matching
-%                                \rail@o{NR}{FMT} from the .rao file is
-%                                executed (if defined).
-%
-% \railoptions{OPTIONS} : OPTIONS are written out to the .rai file,
-%                         as \rail@p{OPTIONS}.
-%
-% \railterm{IDENT,IDENT,...} : format IDENT as terminals. writes out
-%                              \rail@t{IDENT} to the .rai file
-%
-% \railalias{IDENT}{TEXT} : format IDENT as TEXT. defines \rail@t@IDENT as
-%                           TEXT.
-%
-% \railtoken{IDENT}{TEXT} : abbreviates \railalias{IDENT}{TEXT}\railterm{IDENT}
-%                           (for backward compatibility)
-%
-% \rail@setcodes : guards special characters
-%
-% \rail@makeother{CHARACTER} : sets \catcode of CHARACTER to "other"
-%                              used inside a loop for \rail@setcodes
-%
-% \rail@nr : railroad diagram counter
-%
-% \ifrail@match : current \rail@i{NR}{TEXT} matches
-%
-% \rail@first : actions to be done first. read in .rao file,
-%               open .rai file if \@filesw true, undefine \rail@first.
-%               executed from \begin{rail}, \railoptions and \railterm.
-%
-% \rail@i{NR}{TEXT} : defines \rail@i@NR as TEXT. written to the .rai
-%                     file by \rail, read from the .rao file by
-%                     \rail@first
-%
-% \rail@t{IDENT} : tells Rail that IDENT is to be custom formatted,
-%                  written to the .rai file by \railterm.
-%
-% \rail@o{NR}{TEXT} : defines \rail@o@NR as TEXT, read from the .rao
-%                     file by \rail@first.
-%
-% \rail@p{OPTIONS} : pass options to rail, written to the .rai file by
-%                    \railoptions
-%
-% \rail@write{TEXT} : write TEXT to the .rai file
-%
-% \rail@warn : warn user for mismatching diagrams
-%
-% \rail@endwarn : either \relax or \rail@warn
-%
-% \ifrail@all : checked at the end of the document
-
-\def\rail@makeother#1{
-  \expandafter\catcode\expandafter`\csname\string #1\endcsname=12
-}
-
-\def\rail@setcodes{
-\let\par=\relax
-\let\\=\relax
-\expandafter\@tfor\expandafter\rail@symbol\expandafter:\expandafter=%
-  \the\rail@reserved
-\do{\expandafter\rail@makeother\rail@symbol}
-}
-
-\newcount\rail@nr
-
-\newif\ifrail@all
-\rail@alltrue
-
-\newif\ifrail@match
-
-\def\rail@first{
-\begingroup
-\makeatletter
-\rail@setcodes
-\InputIfFileExists{\jobname.rao}{}{\PackageInfo{rail}{No file \jobname.rao}}
-\makeatother
-\endgroup
-\if@filesw
-\newwrite\tf@rai
-\immediate\openout\tf@rai=\jobname.rai
-\fi
-\global\let\rail@first=\relax
-}
-
-\long\def\rail@body#1\end{
-{
-\newlinechar=`^^J
-\def\par{\string\par^^J}
-\rail@write{\string\rail@i{\number\rail@nr}{#1}}
-}
-\xdef\rail@i@{#1}
-\end
-}
-
-\newenvironment{rail}{
-\global\advance\rail@nr by 1
-\rail@first
-\begingroup
-\rail@setcodes
-\rail@body
-}{
-\endgroup
-\rail@matchtrue
-\@ifundefined{rail@o@\number\rail@nr}{\rail@matchfalse}{}
-\expandafter\ifx\csname rail@i@\number\rail@nr\endcsname\rail@i@
-\else
-\rail@matchfalse
-\fi
-\ifrail@match
-\csname rail@o@\number\rail@nr\endcsname
-\else
-\PackageWarning{rail}{Railroad diagram {\number\rail@nr} doesn't match}
-\global\let\rail@endwarn=\rail@warn
-\begin{list}{}{\rail@param}
-\rail@begin{1}{}
-\rail@setbox{\bfseries ???}
-\rail@oval
-\rail@end
-\end{list}
-\fi
-}
-
-\newcommand\railoptions[1]{
-\rail@first
-\rail@write{\string\rail@p{#1}}
-}
-
-\newcommand\railterm[1]{
-\rail@first
-\@for\rail@@:=#1\do{
-\rail@write{\string\rail@t{\rail@@}}
-}
-}
-
-\newcommand\railalias[2]{
-\expandafter\def\csname rail@t@#1\endcsname{#2}
-}
-
-\newcommand\railtoken[2]{\railalias{#1}{#2}\railterm{#1}}
-
-\long\def\rail@i#1#2{
-\expandafter\gdef\csname rail@i@#1\endcsname{#2}
-}
-
-\def\rail@o#1#2{
-\expandafter\gdef\csname rail@o@#1\endcsname{
-\begin{list}{}{\rail@param}
-#2
-\end{list}
-}
-}
-
-\def\rail@t#1{}
-
-\def\rail@p#1{}
-
-\long\def\rail@write#1{\@ifundefined{tf@rai}{}{\immediate\write\tf@rai{#1}}}
-
-\def\rail@warn{
-\PackageWarningNoLine{rail}{Railroad diagram(s) may have changed.
-                            Use 'rail' and rerun}
-}
-
-\let\rail@endwarn=\relax
-
-\AtEndDocument{\rail@endwarn}
-
-% index entry macro
-%
-% \rail@index{IDENT} : add index entry for IDENT
-
-\def\rail@index#1{
-\index{\rail@indexfont#1}
-}
-
-% railroad formatting primitives
-%
-% \rail@x : current x
-% \rail@y : current y
-% \rail@ex : current end x
-% \rail@sx : starting x for \rail@cr
-% \rail@rx : rightmost previous x for \rail@cr
-%
-% \rail@tmpa : temporary count
-% \rail@tmpb : temporary count
-% \rail@tmpc : temporary count
-%
-% \rail@put : put at (\rail@x,\rail@y)
-% \rail@vput : put vector at (\rail@x,\rail@y)
-%
-% \rail@eline : end line by drawing from \rail@ex to \rail@x
-%
-% \rail@vreline : end line by drawing a vector from \rail@x to \rail@ex
-%
-% \rail@vleline : end line by drawing a vector from \rail@ex to \rail@x
-%
-% \rail@sety{LEVEL} : set \rail@y to level LEVEL
-
-\newcount\rail@x
-\newcount\rail@y
-\newcount\rail@ex
-\newcount\rail@sx
-\newcount\rail@rx
-
-\newcount\rail@tmpa
-\newcount\rail@tmpb
-\newcount\rail@tmpc
-
-\def\rail@put{\put(\number\rail@x,\number\rail@y)}
-
-\def\rail@vput{\put(\number\rail@ex,\number\rail@y)}
-
-\def\rail@eline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@put{\line(-1,0){\number\rail@tmpb}}
-}
-
-\def\rail@vreline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@vput{\vector(1,0){\number\rail@tmpb}}
-}
-
-\def\rail@vleline{
-\rail@tmpb=\rail@x
-\advance\rail@tmpb by -\rail@ex
-\rail@put{\vector(-1,0){\number\rail@tmpb}}
-}
-
-\def\rail@sety#1{
-\rail@y=#1
-\multiply\rail@y by -\rail@boxsp
-\advance\rail@y by -\rail@boxht
-}
-
-% \rail@begin{HEIGHT}{NAME} : begin a railroad diagram of height HEIGHT
-%
-% \rail@end : end a railroad diagram
-%
-% \rail@expand{IDENT} : expand IDENT
-
-\def\rail@begin#1#2{
-\item
-\begin{minipage}[t]{\linewidth}
-\ifx\@empty#2\else
-{\rail@namefont \rail@expand{#2}}\\*[\railnamesep]
-\fi
-\unitlength=\railunit
-\rail@tmpa=#1
-\multiply\rail@tmpa by \rail@boxsp
-\begin{picture}(0,\number\rail@tmpa)(0,-\number\rail@tmpa)
-\rail@ex=0
-\rail@rx=0
-\rail@x=\rail@extra
-\rail@sx=\rail@x
-\rail@sety{0}
-}
-
-\def\rail@end{
-\advance\rail@x by \rail@extra
-\rail@eline
-\end{picture}
-\end{minipage}
-}
-
-\def\rail@vend{
-\advance\rail@x by \rail@extra
-\rail@vreline
-\end{picture}
-\end{minipage}
-}
-
-\def\rail@expand#1{\@ifundefined{rail@t@#1}{#1}{\csname rail@t@#1\endcsname}}
-
-% \rail@token{TEXT}[ANNOT] : format token TEXT with annotation
-% \rail@ltoken{TEXT}[ANNOT] : format token TEXT with annotation, arrow left
-% \rail@rtoken{TEXT}[ANNOT] : format token TEXT with annotation, arrow right
-%
-% \rail@ctoken{TEXT}[ANNOT] : format token TEXT centered with annotation
-% \rail@lctoken{TEXT}[ANNOT] : format token TEXT centered with annotation, arrow left
-% \rail@rctoken{TEXT}[ANNOT] : format token TEXT centered with annotation, arrow right
-%
-% \rail@nont{TEXT}[ANNOT] : format nonterminal TEXT with annotation
-% \rail@lnont{TEXT}[ANNOT] : format nonterminal TEXT with annotation, arrow left
-% \rail@rnont{TEXT}[ANNOT] : format nonterminal TEXT with annotation. arrow right
-%
-% \rail@cnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation
-% \rail@lcnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation,
-%                             arrow left
-% \rail@rcnont{TEXT}[ANNOT] : format nonterminal TEXT centered with annotation,
-%                             arrow right
-%
-% \rail@term{TEXT}[ANNOT] : format terminal TEXT with annotation
-% \rail@lterm{TEXT}[ANNOT] : format terminal TEXT with annotation, arrow left
-% \rail@rterm{TEXT}[ANNOT] : format terminal TEXT with annotation, arrow right
-%
-% \rail@cterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation
-% \rail@lcterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation, arrow left
-% \rail@rcterm{TEXT}[ANNOT] : format terminal TEXT centered with annotation,
-%                             arrow right
-%
-% \rail@annote[TEXT] : format TEXT as annotation
-
-\def\rail@token#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@oval
-}
-
-\def\rail@ltoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vloval
-}
-
-\def\rail@rtoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vroval
-}
-
-\def\rail@ctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@coval
-}
-
-\def\rail@lctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcoval
-}
-
-\def\rail@rctoken#1[#2]{
-\rail@setbox{%
-{\rail@termfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcoval
-}
-
-\def\rail@nont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@frame
-}
-
-\def\rail@lnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlframe
-}
-
-\def\rail@rnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrframe
-}
-
-\def\rail@cnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@cframe
-}
-
-\def\rail@lcnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcframe
-}
-
-\def\rail@rcnont#1[#2]{
-\rail@setbox{%
-{\rail@nontfont \rail@expand{#1}}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcframe
-}
-
-\def\rail@term#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@oval
-}
-
-\def\rail@lterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vloval
-}
-
-\def\rail@rterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vroval
-}
-
-\def\rail@cterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@coval
-}
-
-\def\rail@lcterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vlcoval
-}
-
-\def\rail@rcterm#1[#2]{
-\rail@setbox{%
-{\rail@termfont #1}\ifx\@empty#2\else\ {\rail@annofont #2}\fi
-}
-\rail@vrcoval
-}
-
-\def\rail@annote[#1]{
-\rail@setbox{\rail@annofont #1}
-\rail@text
-}
-
-% \rail@box : temporary box for \rail@oval and \rail@frame
-%
-% \rail@setbox{TEXT} : set \rail@box to TEXT, set \rail@tmpa to width
-%
-% \rail@oval : format \rail@box of width \rail@tmpa inside an oval
-% \rail@vloval : format \rail@box of width \rail@tmpa inside an oval, vector left
-% \rail@vroval : format \rail@box of width \rail@tmpa inside an oval, vector right
-%
-% \rail@coval : same as \rail@oval, but centered between \rail@x and
-%               \rail@mx
-% \rail@vlcoval : same as \rail@oval, but centered between \rail@x and
-%                 \rail@mx, vector left
-% \rail@vrcoval : same as \rail@oval, but centered between \rail@x and
-%                 \rail@mx, vector right
-%
-% \rail@frame : format \rail@box of width \rail@tmpa inside a frame
-% \rail@vlframe : format \rail@box of width \rail@tmpa inside a frame, vector left
-% \rail@vrframe : format \rail@box of width \rail@tmpa inside a frame, vector right
-%
-% \rail@cframe : same as \rail@frame, but centered between \rail@x and
-%                \rail@mx
-% \rail@vlcframe : same as \rail@frame, but centered between \rail@x and
-%                  \rail@mx, vector left
-% \rail@vrcframe : same as \rail@frame, but centered between \rail@x and
-%                  \rail@mx, vector right
-%
-% \rail@text : format \rail@box of width \rail@tmpa above the line
-
-\newbox\rail@box
-
-\def\rail@setbox#1{
-\setbox\rail@box\hbox{\strut#1}
-\rail@tmpa=\wd\rail@box
-\divide\rail@tmpa by \railunit
-}
-
-\def\rail@oval{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@vloval{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-\rail@vleline
-}
-
-\def\rail@vroval{
-\advance\rail@x by \rail@boxlf
-\rail@vreline
-\advance\rail@tmpa by \rail@ovalsp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\rail@tmpb=\rail@tmpa
-\divide\rail@tmpb by 2
-\advance\rail@y by -\rail@boxhht
-\rail@put{\makebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpb
-\rail@put{\oval(\number\rail@tmpa,\number\rail@boxht)}
-\advance\rail@x by \rail@tmpb
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@coval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@oval
-}
-
-\def\rail@vlcoval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vloval
-}
-
-\def\rail@vrcoval{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@ovalsp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vroval
-}
-
-\def\rail@frame{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@vlframe{
-\advance\rail@x by \rail@boxlf
-\rail@eline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-\rail@vleline
-}
-
-\def\rail@vrframe{
-\advance\rail@x by \rail@boxlf
-\rail@vreline
-\advance\rail@tmpa by \rail@framesp
-\ifnum\rail@tmpa<\rail@boxht\rail@tmpa=\rail@boxht\fi
-\advance\rail@y by -\rail@boxhht
-\rail@put{\framebox(\number\rail@tmpa,\number\rail@boxht){\box\rail@box}}
-\advance\rail@y by \rail@boxhht
-\advance\rail@x by \rail@tmpa
-\rail@ex=\rail@x
-\advance\rail@x by \rail@boxrt
-}
-
-\def\rail@cframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@frame
-}
-
-\def\rail@vlcframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vlframe
-}
-
-\def\rail@vrcframe{
-\rail@tmpb=\rail@tmpa
-\advance\rail@tmpb by \rail@framesp
-\ifnum\rail@tmpb<\rail@boxht\rail@tmpb=\rail@boxht\fi
-\advance\rail@tmpb by \rail@boxlf
-\advance\rail@tmpb by \rail@boxrt
-\rail@tmpc=\rail@mx
-\advance\rail@tmpc by -\rail@x
-\advance\rail@tmpc by -\rail@tmpb
-\divide\rail@tmpc by 2
-\ifnum\rail@tmpc>0
-\advance\rail@x by \rail@tmpc
-\fi
-\rail@vrframe
-}
-
-\def\rail@text{
-\advance\rail@x by \rail@textlf
-\advance\rail@y by \rail@textup
-\rail@put{\box\rail@box}
-\advance\rail@y by -\rail@textup
-\advance\rail@x by \rail@tmpa
-\advance\rail@x by \rail@textrt
-}
-
-% alternatives
-%
-% \rail@jx \rail@jy : current join point
-%
-% \rail@gx \rail@gy \rail@gex \rail@grx : global versions of \rail@x etc,
-%                                         to pass values over group closings
-%
-% \rail@mx : maximum x so far
-%
-% \rail@sy : starting \rail@y for alternatives
-%
-% \rail@jput : put at (\rail@jx,\rail@jy)
-%
-% \rail@joval[PART] : put \oval[PART] with adjust
-
-\newcount\rail@jx
-\newcount\rail@jy
-
-\newcount\rail@gx
-\newcount\rail@gy
-\newcount\rail@gex
-\newcount\rail@grx
-
-\newcount\rail@sy
-\newcount\rail@mx
-
-\def\rail@jput{
-\put(\number\rail@jx,\number\rail@jy)
-}
-
-\def\rail@joval[#1]{
-\advance\rail@jx by \rail@joinadj
-\rail@jput{\oval(\number\rail@joinsz,\number\rail@joinsz)[#1]}
-\advance\rail@jx by -\rail@joinadj
-}
-
-% \rail@barsplit : incoming split for '|'
-%
-% \rail@plussplit : incoming split for '+'
-%
-
-\def\rail@barsplit{
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-}
-
-\def\rail@plussplit{
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by \rail@joinsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-}
-
-% \rail@alt{SPLIT} : start alternatives with incoming split SPLIT
-%
-
-\def\rail@alt#1{
-\rail@sy=\rail@y
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@x by \rail@joinsz
-\rail@mx=0
-\let\rail@list=\@empty
-\let\rail@comma=\@empty
-\let\rail@split=#1
-\begingroup
-\rail@sx=\rail@x
-\rail@rx=0
-}
-
-% \rail@nextalt{FIX}{Y} : start next alternative at vertical position Y
-%                         and fix-up FIX
-%
-
-\def\rail@nextalt#1#2{
-\global\rail@gx=\rail@x
-\global\rail@gy=\rail@y
-\global\rail@gex=\rail@ex
-\global\rail@grx=\rail@rx
-\endgroup
-#1
-\ifnum\rail@gx>\rail@mx\rail@mx=\rail@gx\fi
-\ifnum\rail@grx>\rail@mx\rail@mx=\rail@grx\fi
-\edef\rail@list{\rail@list\rail@comma\number\rail@gex:\number\rail@gy}
-\def\rail@comma{,}
-\rail@split
-\let\rail@split=\@empty
-\rail@sety{#2}
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[bl]
-\advance\rail@jx by -\rail@joinhsz
-\rail@ex=\rail@x
-\begingroup
-\rail@sx=\rail@x
-\rail@rx=0
-}
-
-% \rail@barjoin : outgoing join for first '|' alternative
-%
-% \rail@plusjoin : outgoing join for first '+' alternative
-%
-% \rail@altjoin : join for subsequent alternative
-%
-
-\def\rail@barjoin{
-\ifnum\rail@y<\rail@sy
-\global\rail@gex=\rail@jx
-\else
-\global\rail@gex=\rail@ex
-\fi
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-\ifnum\rail@y<\rail@sy
-\rail@altjoin
-\fi
-}
-
-\def\rail@plusjoin{
-\global\rail@gex=\rail@ex
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by -\rail@joinsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-}
-
-\def\rail@altjoin{
-\rail@eline
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by -\rail@joinhsz
-\rail@joval[br]
-\advance\rail@jx by \rail@joinhsz
-}
-
-% \rail@eltsplit EX:Y; : split EX:Y into \rail@ex \rail@y
-%
-% \rail@endalt{JOIN} : end alternatives with outgoing join JOIN
-
-\def\rail@eltsplit#1:#2;{\rail@ex=#1\rail@y=#2}
-
-\def\rail@endalt#1{
-\global\rail@gx=\rail@x
-\global\rail@gy=\rail@y
-\global\rail@gex=\rail@ex
-\global\rail@grx=\rail@rx
-\endgroup
-\ifnum\rail@gx>\rail@mx\rail@mx=\rail@gx\fi
-\ifnum\rail@grx>\rail@mx\rail@mx=\rail@grx\fi
-\edef\rail@list{\rail@list\rail@comma\number\rail@gex:\number\rail@gy}
-\rail@x=\rail@mx
-\rail@jx=\rail@x
-\rail@jy=\rail@sy
-\advance\rail@jx by \rail@joinsz
-\let\rail@join=#1
-\@for\rail@elt:=\rail@list\do{
-\expandafter\rail@eltsplit\rail@elt;
-\rail@join
-\let\rail@join=\rail@altjoin
-}
-\rail@x=\rail@mx
-\rail@y=\rail@sy
-\rail@ex=\rail@gex
-\advance\rail@x by \rail@joinsz
-}
-
-% \rail@bar : start '|' alternatives
-%
-% \rail@nextbar : next '|' alternative
-%
-% \rail@endbar : end '|' alternatives
-%
-
-\def\rail@bar{
-\rail@alt\rail@barsplit
-}
-
-\def\rail@nextbar{
-\rail@nextalt\relax
-}
-
-\def\rail@endbar{
-\rail@endalt\rail@barjoin
-}
-
-% \rail@plus : start '+' alternatives
-%
-% \rail@nextplus: next '+' alternative
-%
-% \rail@endplus : end '+' alternatives
-%
-
-\def\rail@plus{
-\rail@alt\rail@plussplit
-}
-
-\def\rail@nextplus{
-\rail@nextalt\rail@fixplus
-}
-
-\def\rail@fixplus{
-\ifnum\rail@gy<\rail@sy
-\begingroup
-\rail@x=\rail@gx
-\rail@y=\rail@gy
-\rail@ex=\rail@gex
-\rail@rx=\rail@grx
-\ifnum\rail@x<\rail@rx
-\rail@x=\rail@rx
-\fi
-\rail@eline
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@joinhsz
-\rail@joval[br]
-\advance\rail@jx by \rail@joinhsz
-\rail@tmpa=\rail@sy
-\advance\rail@tmpa by -\rail@joinhsz
-\advance\rail@tmpa by -\rail@jy
-\rail@jput{\line(0,1){\number\rail@tmpa}}
-\rail@jy=\rail@sy
-\advance\rail@jy by -\rail@joinhsz
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jy by \rail@joinhsz
-\global\rail@gx=\rail@jx
-\global\rail@gy=\rail@jy
-\global\rail@gex=\rail@gx
-\global\rail@grx=\rail@rx
-\endgroup
-\fi
-}
-
-\def\rail@endplus{
-\rail@endalt\rail@plusjoin
-}
-
-% \rail@cr{Y} : carriage return to vertical position Y
-
-\def\rail@cr#1{
-\rail@tmpa=\rail@sx
-\advance\rail@tmpa by \rail@joinsz
-\ifnum\rail@x<\rail@tmpa\rail@x=\rail@tmpa\fi
-\rail@eline
-\rail@jx=\rail@x
-\rail@jy=\rail@y
-\advance\rail@x by \rail@joinsz
-\ifnum\rail@x>\rail@rx\rail@rx=\rail@x\fi
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tr]
-\advance\rail@jx by \rail@joinhsz
-\rail@sety{#1}
-\rail@tmpa=\rail@jy
-\advance\rail@tmpa by -\rail@y
-\advance\rail@tmpa by -\rail@boxsp
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\rail@jy=\rail@y
-\advance\rail@jy by \rail@boxsp
-\advance\rail@jy by \rail@joinhsz
-\advance\rail@jx by -\rail@joinhsz
-\rail@joval[br]
-\advance\rail@jy by -\rail@joinhsz
-\rail@tmpa=\rail@jx
-\advance\rail@tmpa by -\rail@sx
-\advance\rail@tmpa by -\rail@joinhsz
-\rail@jput{\line(-1,0){\number\rail@tmpa}}
-\rail@jx=\rail@sx
-\advance\rail@jx by \rail@joinhsz
-\advance\rail@jy by -\rail@joinhsz
-\rail@joval[tl]
-\advance\rail@jx by -\rail@joinhsz
-\rail@tmpa=\rail@boxsp
-\advance\rail@tmpa by -\rail@joinsz
-\rail@jput{\line(0,-1){\number\rail@tmpa}}
-\advance\rail@jy by -\rail@tmpa
-\advance\rail@jx by \rail@joinhsz
-\rail@joval[bl]
-\rail@x=\rail@jx
-\rail@ex=\rail@x
-}
-
-% default setup for Isabelle
-\newenvironment{railoutput}%
-{\begin{list}{}{\rail@param}\def\rail@expand{\relax}\makeatletter}{\makeatother\end{list}}
-
-\def\rail@termfont{\isabellestyle{tt}}
-\def\rail@nontfont{\isabellestyle{it}}
-\def\rail@namefont{\isabellestyle{it}}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/root.bib b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/root.bib
deleted file mode 100644
index 6f429f0..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/root.bib
+++ /dev/null
@@ -1,71 +0,0 @@
-@InProceedings{adt2019,
-    author    = {Karsten Diekhoff and
-                 Michael Kirsten and
-                 Jonas Kr{\"{a}}mer},
-    editor    = {Sa{\v{s}}a Peke{\v{c}} and
-                 {Kristen Brent} Venable},
-    title     = {Formal Property-Oriented Design of
-                 Voting Rules Using Composable Modules},
-    booktitle = {6th International Conference on Algorithmic
-                 Decision Theory ({ADT} 2019)},
-    series    = {Lecture Notes in Artificial Intelligence},
-    part      = {Short Papers},
-    year      = {2019},
-    abstract  = {Voting rules aggregate multiple individual preferences in order to
-                 make a collective decision. Commonly, these mechanisms are expected
-                 to respect a multitude of different notions of fairness and reliability,
-                 which must be carefully balanced to avoid inconsistencies.
-
-                 We present an approach for the sound and flexible design of voting rules
-                 from composable modules. Formal composition rules guarantee social
-                 choice properties from properties of the individual components. The
-                 approach can be applied to many voting rules from the literature.},
-    publisher = {Springer},
-    volume    = {11834},
-    pages     = {164--166},
-    doi       = {10.1007/978-3-030-31489-7}
-}
-
-@InProceedings{lopstr2019,
-  author        = {Karsten Diekhoff and
-                   Michael Kirsten and
-                   Jonas Kr{\"{a}}mer},
-  title         = {Verified Construction of Fair Voting Rules},
-  booktitle     = {29th International Symposium on Logic-Based Program
-                   Synthesis and Transformation ({LOPSTR} 2019),
-                   Revised Selected Papers},
-  series        = {Lecture Notes in Computer Science},
-  editor        = {Maurizio Gabbrielli},
-  year          = {2020},
-  abstract      = {Voting rules aggregate multiple individual preferences in order to
-                   make collective decisions. Commonly, these mechanisms are expected
-                   to respect a multitude of different fairness and reliability
-                   properties, e.g., to ensure that each voter's ballot accounts for
-                   the same proportion of the elected alternatives, or that a voter
-                   cannot change the election outcome in her favor by insincerely
-                   filling out her ballot. However, no voting rule is fair in all
-                   respects, and trade-off attempts between such properties often
-                   bring out inconsistencies, which makes the construction of arguably
-                   practical and fair voting rules non-trivial and error-prone.
- 
-                   In this paper, we present a formal and systematic approach for the
-                   flexible and verified construction of voting rules from composable
-                   core modules to respect such properties by construction. Formal
-                   composition rules guarantee resulting properties from properties of
-                   the individual components, which are of generic nature to be reused
-                   for various voting rules. We provide a prototypical logic-based
-                   implementation with proofs for a selected set of structures and
-                   composition rules within the theorem prover Isabelle/HOL. The
-                   approach can be readily extended in order to support many voting
-                   rules from the literature by extending the set of basic modules and
-                   composition rules. We exemplarily construct the well-known voting
-                   rule sequential majority comparison (SMC) from simple generic
-                   modules, and automatically produce a formal proof that SMC
-                   satisfies the fairness property monotonicity. Monotonicity is a
-                   well-known social-choice property that is easily violated by voting
-                   rules in practice.},
-  publisher     = {Springer},
-  doi           = {10.1007/978-3-030-45260-5_6},
-  volume        = {12042},
-  pages         = {90--104}
-}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/root.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/root.tex
deleted file mode 100644
index 2bf1d6e..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/root.tex
+++ /dev/null
@@ -1,56 +0,0 @@
-\documentclass[11pt,a4paper,notitlepage]{report}
-\input{settings}
-\usepackage{isabelle,isabellesym}
-\usepackage{amssymb} % amsfonts, amssymb
-\usepackage{pdfsetup}
-
-\urlstyle{rm}
-\isabellestyle{it}
-
-% for uniform font size
-%\renewcommand{\isastyle}{\isastyleminor}
-
-% This is required due to some unusual behaviour by Isabelle.
-% It should be included in isabellesym.sty automatically, but it is not.
-\newcommand{\isasymcirclearrowleft}{\isamath{\circlearrowleft}}
-
-\begin{document}
-
-\title{Verified Construction of Fair Voting Rules}
-\author{Michael Kirsten}
-\institute{Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany\\
-\email{kirsten@kit.edu}}
-\maketitle
-
-\begin{abstract}
-Voting rules aggregate multiple individual preferences in order to make a
-collective decision.
-Commonly, these mechanisms are expected to respect a multitude of different
-notions of fairness and reliability, which must be carefully balanced to avoid
-inconsistencies.
-
-This article contains a formalisation of a framework for the construction of
-such fair voting rules using composable modules~\cite{adt2019,lopstr2019}.
-The framework is a formal and systematic approach for the flexible and verified
-construction of voting rules from individual composable modules to respect such
-social-choice properties by construction.
-Formal composition rules guarantee resulting social-choice properties from
-properties of the individual components which are of generic nature to be
-reused for various voting rules.
-We provide proofs for a selected set of structures and composition rules.
-The approach can be readily extended in order to support more voting rules,
-e.g., from the literature by extending the sets of modules and composition
-rules.
-\end{abstract}
-
-\tableofcontents
-
-\parindent 0pt\parskip 0.5ex
-
-\input{session}
-
-% optional bibliography
-\bibliographystyle{abbrv}
-\bibliography{root}
-
-\end{document}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/session.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/session.tex
deleted file mode 100644
index 00be90a..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/session.tex
+++ /dev/null
@@ -1,48 +0,0 @@
-\input{Preference_Relation.tex}
-\input{Result.tex}
-\input{Profile.tex}
-\input{Electoral_Module.tex}
-\input{Evaluation_Function.tex}
-\input{Aggregator.tex}
-\input{Termination_Condition.tex}
-\input{Defer_Equal_Condition.tex}
-\input{Defer_Module.tex}
-\input{Drop_Module.tex}
-\input{Pass_Module.tex}
-\input{Elect_Module.tex}
-\input{Elimination_Module.tex}
-\input{Maximum_Aggregator.tex}
-\input{Plurality_Module.tex}
-\input{Composite_Elimination_Modules.tex}
-\input{Result_Properties.tex}
-\input{Sequential_Composition.tex}
-\input{Parallel_Composition.tex}
-\input{Loop_Composition.tex}
-\input{Aggregator_Properties.tex}
-\input{Indep_Of_Alt.tex}
-\input{Disjoint_Compatibility.tex}
-\input{Aggregator_Facts.tex}
-\input{Composite_Structures.tex}
-\input{Revision_Composition.tex}
-\input{Borda_Rule.tex}
-\input{Condorcet_Properties.tex}
-\input{Condorcet_Consistency.tex}
-\input{Condorcet_Rules.tex}
-\input{Condorcet_Facts.tex}
-\input{Pairwise_Majority_Rule.tex}
-\input{Copeland_Rule.tex}
-\input{Minimax_Rule.tex}
-\input{Blacks_Rule.tex}
-\input{Nanson_Baldwin_Rule.tex}
-\input{Classic_Nanson_Rule.tex}
-\input{Schwartz_Rule.tex}
-\input{Monotonicity_Properties.tex}
-\input{Weak_Monotonicity.tex}
-\input{Result_Facts.tex}
-\input{Result_Rules.tex}
-\input{Monotonicity_Facts.tex}
-\input{Monotonicity_Rules.tex}
-\input{Disjoint_Compatibility_Facts.tex}
-\input{Disjoint_Compatibility_Rules.tex}
-\input{Sequential_Majority_Comparison.tex}
-\input{Homogeneity.tex}
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/session_graph.pdf b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/session_graph.pdf
index f1569a3..6ad93f9 100644
Binary files a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/session_graph.pdf and b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/session_graph.pdf differ
diff --git a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/settings.tex b/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/settings.tex
deleted file mode 100644
index 9a1f927..0000000
--- a/virage/src/test/resources/verifiedVotingRuleConstruction/theories/output/document/settings.tex
+++ /dev/null
@@ -1,30 +0,0 @@
-\usepackage[T1]{fontenc}
-\usepackage{lmodern}
-\usepackage{sectsty}
-\usepackage{xpatch}
-\usepackage{upquote}
-
-\makeatletter
-\xpatchcmd{\@maketitle}
-  {\@title}
-{%\vspace{-2.45\baselineskip}
-\bfseries\@title\vspace{.25\baselineskip}}
-  {}{}
-\makeatother
-
-\makeatletter
-\providecommand{\institute}[1]{%
-    \apptocmd{\@author}{\end{tabular}%
-    \vspace{.5\baselineskip}\par
-    \begin{tabular}[t]{c}
-    #1}{}{}
-}
-\makeatother
-
-\newcommand{\email}[1]{%
-    \texttt{\href{mailto:#1}{\color{black}{#1}}}%
-}
-
-\pretocmd{\tableofcontents}{\sectionfont{\Huge}}{}{}
-%\let\clearpage\relax
-\apptocmd{\tableofcontents}{\sectionfont{\Large}}{}{}