Abstract not found

An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause.

Martin-Löf's Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of well-founded relations is presented. Using primitive recursion over higher types, induction and recursion are formally derived for a large class of well-founded relations. Included are < on natural numbers, and relations formed by inverse images, addition, multiplication, and exponentiation of other relations. The constructions are given in full detail to allow their use in theorem provers for Type Theory, such as Nuprl. The theory is compared with work in the field of ordinal recursion over higher types.

The distinctive merit of the declarative reading of logic programs is the validity ofallthelaws of reasoning supplied by the predicate calculus with equality. Surprisingly many of these laws are still valid for the procedural reading � they can therefore be used safely for algebraic manipulation, program transformation and optimisation of executable logic programs. This paper lists a number of common laws, and proves their validity for the standard (depth- rst search) procedural reading of Prolog. They also hold for alternative search strategies, e.g. breadth- rst search. Our proofs of the laws are based on the standard algebra of functional programming, after the strategies have been given a rather simple implementation in Haskell. 1

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