The Oyster-Clam System

O Y S T ER Hor88] is an interactive proof editor closely based on the Cornell NuPRL system, but implemented in Prolog. The object-level logic is a version of Martin-LL of type theory (a higher order constructive logic including induction) in a sequent-calculus formulation. Proofs are constructed in a top-down fashion by application of the rules of inference. Notational deenitions and libraries of theorems are supported. The tactic language for the system is Prolog. Predicates describing properties of a proof under construction are available to the user, who may also include arbitrary Prolog in tactics. Soundness of the system is ensured by the use of an abstract data type of proofs: partial proofs can only be altered by application of the primitive proof rules. Tactics can be combined using system deened tacticals. Prolog pattern-matching and backtracking in tactics have proved useful in the automation of proof search. Since the object-level logic is constructive, terms of an enlarged-calculus can be computed from complete proofs, and these so-called ex tr act terms can then be executed by application on appropriate inputs. This allows the system to be used as a program synthesis environment, since a theorem can be regarded as a speciication which is realised by its extract term. The system is written in some 2000 lines of Prolog, making it considerably more compact than the original NuPRL system, while the speed of the two systems is comparable.