Proof for mathd_algebra_513, eq_four

[aleph-prover-test]

Proven lemmas: 2/2

The current goal consists of proving two separate theorems in Lean. The first theorem, mathd_algebra_513, asks you to solve a 2×2 linear system over ℝ: from 3·a + 2·b = 5 and a + b = 2, prove the conjunction a = 1 ∧ b = 1. The second theorem, eq_four, is a basic equality-chaining statement over ℕ: for all a b c d, if a = b, a = d, and a = c, then c = b.

Each theorem was decomposed into straightforward subgoals: for mathd_algebra_513, the conjunction a = 1 ∧ b = 1 is split into two parts (prove a = 1, then prove b = 1); for eq_four, the task is to derive c = b from the given equalities. Progress so far is complete: 2 out of 2 theorems have working proofs.

For mathd_algebra_513, the proof strategy is to use linear arithmetic (linarith) to solve the system directly, handling each component of the conjunction in turn. Informally, this corresponds to standard elimination/substitution (the system has a unique solution), but Lean’s linarith automates the algebra. For eq_four, the proof is pure equality reasoning: from a = c you get c = a, then transitivity with a = b yields c = b; the hypothesis a = d is present but not needed. Nothing remains to be proved at this stage.