Added Fundamental Theorem of Discrete Calculus Test Case

proofs/disc_calc.dfy

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+lemma {:induction false} fund_theorem_of_disc_calc(s: InfIntSeq, i: nat, j: nat)
+    requires i <= j
+    ensures sum_from_i_to_j(discrete_derivative(s), i, j) == 
+            get_elem_at_index(s, j) - get_elem_at_index(s, i)
+{
+}

testcases/disc_calc.dfy

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+/*
+    This testcase requires students to prove of the "Fundemental Theorem of Discrete Calculus".
+    Infinite sequences of integers are represented using a codatatype 'InfIntSeq'.
+*/
+
+/* ========= InfIntSeq Functions ========= */
+
+// InfIntSeq represents a possibly infinite sequence of integers
+// If the sequence is finite, it is interpreted as being padded with 0s
+codatatype InfIntSeq = End | InfIntSeqCons(head: int, tail: InfIntSeq)
+
+function tail_after_n_elems(s: InfIntSeq, n: nat): InfIntSeq {
+    if n == 0 then s
+    else match s
+        case End => End
+        case InfIntSeqCons(h, t) => tail_after_n_elems(t, n - 1)
+}
+
+function get_elem_at_index(s: InfIntSeq, n: nat): int {
+    match s
+    case End => 0
+    case InfIntSeqCons(h, t) =>
+        if n == 0 then h
+        else get_elem_at_index(t, n - 1)
+}
+
+/* ========= Discrete Derivative and Sum Functions ========= */
+
+/*
+    Let a_i be a sequence of ints (0-indexed).
+
+    da_i = a_(i+1) - a_i (discrete derivative of a_i)
+*/
+function discrete_derivative(s: InfIntSeq): InfIntSeq
+{
+    match s
+    case End => End
+    case InfIntSeqCons(h1, t1) =>
+        match t1
+        case End => InfIntSeqCons(0 - h1, End) // Since we interpret finite InfIntSeq as being padded with 0s
+        case InfIntSeqCons(h2, t2) =>
+            InfIntSeqCons(h2 - h1, discrete_derivative(t1))
+}
+
+function sum_of_first_n(s: InfIntSeq, n: nat): int {
+    if n == 0 then 0
+    else match s
+        case End => 0
+        case InfIntSeqCons(h, t) => h + sum_of_first_n(t, n - 1)
+}
+
+function sum_from_i_to_j(s: InfIntSeq, i: nat, j: nat): int {
+    if i >= j then 0
+    else sum_of_first_n(tail_after_n_elems(s, i), j - i)
+}
+
+/* ========= Test Cases ========= */
+
+/*
+    The fundemental theorem of discrete calculus states that:
+    Sum from k=i to j of (da_k) = a_j - a_i
+
+    Aka sum of the individual changes gives the overall change
+*/
+method tc_fund_theorem_of_disc_calc(s: InfIntSeq, i: nat, j: nat) 
+    requires i <= j
+{
+    assert sum_from_i_to_j(discrete_derivative(s), i, j) == 
+           get_elem_at_index(s, j) - get_elem_at_index(s, i) by {
+        fund_theorem_of_disc_calc(s, i, j);
+    }
+}
+
+/*
+    Nothing to do with the test case, but a cool thing:
+
+    This theorem allows us to compute the sums of sequences in a manner similar to how 
+    we evaluate integerals, by finding a "discrete antiderivative" of the sequence. 
+
+    Here is a recursive algorithm you can use to figure out the discrete antiderivative of i^n:
+
+        First guess i^(n+1). Using the binomial theorem, you will get an expression E involving powers of i from 0 to n.
+
+        Subtract of the discrete antiderivatives of all the lower power terms of i in E from i^(n+1).
+
+        Then divide by the coefficient of i^n in E, and you are done.
+
+    -----
+
+    This gives a mechanical process to discover formulas for the sums of powers of i
+*/