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]....