A Full Formalization of SLD-Resolution in the Calculus of Inductive Constructions

This paper presents a full formalization of the semantics of definite programs, in the calculus of inductive constructions. First, we describe a formalization of the proof of first-order terms unification: this proof is obtained from a similar proof dealing with quasi-terms, thus showing how to relate an inductive set with a subset defined by a predicate. Then, SLD-resolution is explicitely defined: the renaming process used in SLD-derivations is made explicit, thus introducing complications, usually overlooked, during the proofs of classical results. Last, switching and lifting lemmas and soundness and completeness theorems are formalized. For this, we present two lemmas, usually omitted, which are needed. This development also contains a formalization of basic results on operators and their fixpoints in a general setting. All the proofs of the results, presented here, have been checked with the proof assistant Coq.