Proof for mathd_algebra_513, eq_four

[aleph-prover-test]

Proven lemmas: 2/2

The current goal contains two separate theorems to prove in Lean. The first, mathd_algebra_513, asks you to solve a 2×2 linear system over ℝ: from 3·a + 2·b = 5 and a + b = 2, conclude a = 1 and b = 1 (i.e., prove a = 1 ∧ b = 1). The second, eq_four, is a basic equality-chaining fact over ℕ: for all a b c d, if a = b, a = d, and a = c, then c = b.

Both statements have been decomposed naturally into their component conclusions: mathd_algebra_513 is split into proving a = 1 and proving b = 1 (a conjunction), while eq_four is handled by introducing the variables and hypotheses and then composing equalities. Progress-wise, all sub-problems are already solved: 2 theorems out of 2 total are completed.

For mathd_algebra_513, the proof uses a standard linear-arithmetic strategy: after splitting the conjunction, linarith solves each equality directly from the two linear hypotheses, avoiding manual substitution. For eq_four, the proof uses symmetry and transitivity of equality: from a = c you get c = a, then chain with a = b to obtain c = b; the hypothesis a = d is present but not needed. No remaining work is left for this blueprint, and the main “interesting” tactic is that linarith can solve the linear system in one line once the goal is split.