Theorem Proving Modulo

Projet COQ, Projet PARA, PROTHEO ; «Theorem proving modulo» is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of wider interest because it aims at separating deductions and computations. The first contribution of this paper is to provide a «sequent calculus modulo» that gives a clear distinction between the decidable (computation) and the undecidable (deduction). The congruence on propositions is handled via rewrite rules and equational axioms. Usually rewriting applies only to terms. The second contribution of this paper is to allow rewriting atomic propositions into non atomic ones and to give a complete proof search method, called «Extended Narrowing and Resolution» (ENAR), modulo such congruences. The completeness of this method is proved using the sequent calculus modulo. An important application is that this Extended Narrowing and Resolution method subsumes full higher-order resolution when applied to a first-order presentation of higher-order logic. This shows that such a presentation can yield also efficient proof-search methods.