Patching Proofs for Reuse (Extended Abstract)

We investigate the application of machine learning paradigms [2, 4, 3] in automated reasoning for improving a theorem prover by reusing previously computed proofs [7]. Assume that we have already computed a proof P of a conjecture φ := ¡∀u plus(sum(x), sum(u)) ≡ sum(append(x, u))¢ → plus(sum(add(n, x)), sum(y)) ≡ sum(append(add(n, x), y)) from a set of axioms AX . The schematic conjecture Φ := H→ C := ¡∀u F (G(x), G(u)) ≡ G(H(x, u))¢→ F (G(D(n, x)), G(y)) ≡ G(H(D(n, x), y)) is obtained from φ via the generalization {plus 7→ F, sum 7→ G, append 7→ H, add 7→ D} of function symbols plus, sum, ... to function variables F,G, ... In the same way a schematic catch, i.e. a set of schematic axioms AX 0 = {(1), (2), (3)} is obtained from AX where e.g. (1) stems from the axiom sum(add(n, x)) ≡ plus(n, sum(x)). The generalization of P finally yields a schematic proof P 0 of Φ in which the schematic conclusion C is modified in a backward chaining style: G(D(n, x)) ≡ F (n,G(x)) (1) H(D(n, x), y) ≡ D(n,H(x, y)) (2) F (F (x, y), z) ≡ F (x, F (y, z)) (3) F (G(D(n, x)), G(y)) ≡ G(H(D(n, x), y)) C F (F (n,G(x)), G(y)) ≡ G(H(D(n, x), y)) Replace (1) F (F (n,G(x)), G(y)) ≡ G(D(n,H(x, y))) Replace (2) F (F (n,G(x)), G(y)) ≡ F (n,G(H(x, y))) Replace (1) F (F (n,G(x)), G(y)) ≡ F (n,F (G(x), G(y))) Replace (H) F (n, F (G(x), G(y))) ≡ F (n,F (G(x), G(y))) Replace (3) true Reflexivity