meagtan / theorem-prover

Proof of concept for theorem proving using heuristic search

theorem-prover

Proof of concept for theorem proving using heuristic search

Description

The theorem proving algorithm operates on expressions comprised of literals and universally quantified variables. A statement is proven by a series of transformations that ends in the literal True. Such transformations can be separated into two rough categories: a rule being applied to an expression to yield another expression, and a rule matching a statement and yielding True. The program applies A* search on the resulting graph of expressions connected by such transformations to find the shortest path starting from an inputted statement and concluding in True.

For example, the proof that addition, defined as in the Peano axioms, is associative, is represented by these transformations:

Action	Statement
Starting hypothesis	M + (N + K) = (M + N) + K
Induction on M	0 + (N + K) = (0 + N) + K and M + (N + K) = (M + N) + K implies s M + (N + K) = (s M + N) + K
Application of rule 0 + N = N	N + K = N + K and M + (N + K) = (M + N) + K implies s M + (N + K) = (s M + N) + K
Application of rule s M + N = s (M + N)	N + K = N + K and M + (N + K) = (M + N) + K implies s (M + (N + K)) = s (M + N) + K
Application of rule s M + N = s (M + N)	N + K = N + K and M + (N + K) = (M + N) + K implies s (M + (N + K)) = s ((M + N) + K)
Left conjunct matches rule X = X	True and M + (N + K) = (M + N) + K implies s (M + (N + K)) = s ((M + N) + K)
Right conjunct matches rule X = Y implies s X = s Y	True and True
Statement matches rule True and True	True