The Algebra of Equality Proofs

Abstract

Proofs of equalities may be built from assumptions using proof rules for reflexivity, symmetry, and transitivity. Reflexivity is an axiom proving x=x for any x; symmetry is a 1-premise rule taking a proof of x=y and returning a proof of y=x; and transitivity is a 2-premise rule taking proofs of x=y and y=z, and returning a proof of x=z. Define an equivalence relation to hold between proofs iff they prove a theorem in common. The main theoretical result of the paper is that if all assumptions are independent, this equivalence relation is axiomatized by the standard axioms of group theory: reflexivity is the unit of the group, symmetry is the inverse, and transitivity is the multiplication. Using a standard completion of the group axioms, we obtain a rewrite system which puts equality proofs into canonical form. Proofs in this canonical form use the fewest possible assumptions, and a proof can be canonized in linear time using a simple strategy. This result is applied to obtain a simple extension of the union-find algorithm for ground equational reasoning which produces minimal proofs. The time complexity of the original union-find operations is preserved, and minimal proofs are produced in worst-case time $O(n^{\textit{log}_2 3})$, where n is the number of expressions being equated. As a second application, the approach is used to achieve significant performance improvements for the CVC cooperating decision procedure.