Abstract not found

The LCF system provides a logic of fixed point theory and is useful to reason about nontermination, recursive definitions and infinite-valued types such as lazy lists. Because of continual presence of bottom elements, it is clumsy for reasoning about finite-valued types and strict functions. The HOL system provides set theory and supports reasoning about finite-valued types and total functions well. In this paper a number of examples are used to demonstrate that an extension of HOL with domain theory combines the benefits of both systems. The examples illustrate reasoning about infinite values and nonterminating functions and show how domain and set theoretic reasoning can be mixed to advantage. An example presents a proof of correctness of a recursive unification algorithm using well-founded induction.

Abstract. We present a Coq formalization of constructive ω-cpos (extending earlier work by Paulin-Mohring) up to and including the inverselimit construction of solutions to mixed-variance recursive domain equations, and the existence of invariant relations on those solutions. We then define operational and denotational semantics for both a simplytyped CBV language with recursion and an untyped CBV language, and establish soundness and adequacy results in each case. 1

Most new theorem provers implement strong and complicated type theories which eliminate some of the limitations of simple type theories such as the HOL logic. A more accessible alternative might be to use a combination of set theory and simple type theory as in HOL-ST which is a version of the HOL system supporting a ZF-like set theory in addition to higher order logic. This paper presents a case study on the use of HOL-ST to build a model of the -calculus by formalising the inverse limit construction of domain theory. This construction is not possible in the HOL system itself, or in simple type theories in general. 1 Introduction The HOL system [GM93] supports a simple and accessible yet very powerful logic, called higher order logic or simple type theory. This is probably a main reason why it has one of the largest user communities of any theorem prover today. However, it is heard every now and then that users cannot quite do what they would like to do, e.g. due to restrictions in t...

We describe a Coq formalization of constructive ω-cpos, ultrametric spaces and ultrametric-enriched categories, up to and including the inverse-limit construction of solutions to mixed-variance recursive equations in both categories enriched over ω-cppos and categories enriched over ultrametric spaces. We show how these mathematical structures may be used in formalizing semantics for three representative programming languages. Specifically, we give operational and denotational semantics for both a simply-typed CBV language with recursion and an untyped CBV language, establishing soundness and adequacy results in each case, and then use a Kripke logical relation over a recursively-defined metric space of worlds to give an interpretation of types over a step-counting operational semantics for a language with recursive types and general references.