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LCF Examples in HOL
 The Computer Journal
, 1994
"... The LCF system provides a logic of fixed point theory and is useful to reason about nontermination, recursive definitions and infinitevalued types such as lazy lists. Because of continual presence of bottom elements, it is clumsy for reasoning about finitevalued types and strict functions. The ..."
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Cited by 12 (4 self)
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The LCF system provides a logic of fixed point theory and is useful to reason about nontermination, recursive definitions and infinitevalued types such as lazy lists. Because of continual presence of bottom elements, it is clumsy for reasoning about finitevalued types and strict functions. The HOL system provides set theory and supports reasoning about finitevalued 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 wellfounded induction.
Some domain theory and denotational semantics in Coq
, 2009
"... Abstract. We present a Coq formalization of constructive ωcpos (extending earlier work by PaulinMohring) up to and including the inverselimit construction of solutions to mixedvariance recursive domain equations, and the existence of invariant relations on those solutions. We then define operatio ..."
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Abstract. We present a Coq formalization of constructive ωcpos (extending earlier work by PaulinMohring) up to and including the inverselimit construction of solutions to mixedvariance 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
Formalising a Model of the lambdacalculus in HOLST
, 1994
"... 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 HOLST which is a version of the HOL s ..."
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Cited by 3 (0 self)
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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 HOLST which is a version of the HOL system supporting a ZFlike set theory in addition to higher order logic. This paper presents a case study on the use of HOLST 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...
Formalizing Domains, Ultrametric Spaces and Semantics of Programming Languages
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2010
"... We describe a Coq formalization of constructive ωcpos, ultrametric spaces and ultrametricenriched categories, up to and including the inverselimit construction of solutions to mixedvariance recursive equations in both categories enriched over ωcppos and categories enriched over ultrametric spac ..."
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Cited by 3 (1 self)
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We describe a Coq formalization of constructive ωcpos, ultrametric spaces and ultrametricenriched categories, up to and including the inverselimit construction of solutions to mixedvariance recursive equations in both categories enriched over ωcppos 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 simplytyped 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 recursivelydefined metric space of worlds to give an interpretation of types over a stepcounting operational semantics for a language with recursive types and general references.