Proof for EasyLean/Basic.lean

[aleph-prover-test]

Proven lemmas: 2/2

The current goal is to prove two elementary theorems in Lean: first, that the real system 3a + 2b = 5 and a + b = 2 forces a = 1 and b = 1; second, that for natural numbers, from a = b, a = d, and a = c one can conclude c = b.

This was naturally decomposed into two independent sub-problems, one for each theorem. The first is a 2×2 linear algebra exercise over ℝ, and the second is a basic equality-transitivity fact over ℕ.

Progress is complete: 2 out of 2 sub-problems have been solved. For mathd_algebra_513, the proof uses linear reasoning: one can combine the two equations to isolate one variable and then the other, and Lean’s linarith tactic is enough to finish it directly. For eq_four, the proof is a short chain of equalities: from a = c we get c = a, and composing with a = b gives c = b.

At this point, nothing remains open in this batch. The main interesting proof ideas were that the first theorem can be dispatched either by explicit substitution or by a single linear-arithmetic step, while the second shows that one of the assumptions, a = d, is actually unused.