Simple Type Theory as a Clausal Theory

Deduction modulo is an extension of first-order predicate logic where axioms are replaced by rewrite rules. For instance, the axiom P ⇔ (Q ⇒ R) is replaced by the rule P −→ (Q ⇒ R). These rules define an equivalence relation and, in a proof, a proposition can be replaced by an equivalent one at any time. A presentation of Simple Type Theory in Deduction modulo, calledHOL, has been given in [3]. Polarized deduction modulo [1] is an extension of Deduction modulo where possibly different rewrite rules apply to the negative and positive occurrences of atomic propositions. Like any theory expressed in Deduction modulo, Simple Type Theory can be expressed in Polarized deduction modulo. Each rule just needs to be duplicated in a negative and a positive instance. A rewrite system in Polarized deduction modulo is said to be clausal when negative rules rewrite atomic propositions to clausal propositions and positive rules rewrite atomic propositions to the negation of clausal propositions. This way, clauses rewrite to clauses, which is a useful property in automated theorem proving [2]. We give in this note a presentation of Simple Type Theory as a clausal rewrite system in Polarized deduction modulo, called HOL. This system is does not have the cut elimination property in general but cut elimination holds for sequents well-formed in the language of HOL and, for such sequents, provability in HOL and in HOL are equivalent.