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Translating Specifications in VDMSL to PVS
 Theorem Proving in Higher Order Logics: 9th International Conference, TPHOLs '96, volume 1125 of Lecture Notes in Computer Science
, 1996
"... . This paper presents a method for translating a subset of VDMSL to higher order logic, more specifically the PVS specification language. This method has been used in an experiment where we have taken three existing, relatively large specifications written in VDMSL, handtranslated these to PVS an ..."
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. This paper presents a method for translating a subset of VDMSL to higher order logic, more specifically the PVS specification language. This method has been used in an experiment where we have taken three existing, relatively large specifications written in VDMSL, handtranslated these to PVS and then tried to type check the results. This is not as simple as it may sound since the specifications make extensive use of subtypes, via type invariants and pre and postconditions, and therefore type checking necessarily involves some theorem proving. In trying to prove some of these type checking conditions, a worrying number of errors were identified in the specifications. 1 Introduction In a research project entitled "Towards industrially applicable proof support for VDMSL", we aim at developing tool support for proving theorems about specifications written in the VDM Specification Language (VDMSL) [6]. We would like to base our work on available theorem proving technology. The goal...
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
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.
Count(q) versus the PigeonHole Principle
, 1996
"... For each p 2 there exists a model M of I \Delta 0 (ff) which satisfies the Count(p) principle. Furthermore, if p contains all prime factors of q there exist n; r 2 M and a bijective map f 2 dom(M ) mapping f1; 2; :::; ng onto f1; 2; :::; n+ q r g. A corollary is a complete classificati ..."
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Cited by 3 (1 self)
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For each p 2 there exists a model M of I \Delta 0 (ff) which satisfies the Count(p) principle. Furthermore, if p contains all prime factors of q there exist n; r 2 M and a bijective map f 2 dom(M ) mapping f1; 2; :::; ng onto f1; 2; :::; n+ q r g. A corollary is a complete classification of the Count(q) versus Count(p) problem. Another corollary shows that the pigeonhole principle for injective maps does not follow from any of the Count(q) principles. This solves an open question [Ajtai 94]. 1 Introduction The most fundamental questions in the theory of the complexity of calculations are concerned with complexity classes in which `counting' is only possible in a quite restricted sense. Thus it is not surprising that many elementary counting principles are unprovable in systems of Bounded Arithmetic. These are axiom systems where the induction axiom schema is restricted to predicates of low syntactic complexity. For a good basic reference see [Krajicek 95]. The status of...
Formalizing FixedPoint Theory in PVS
 Universitat Ulm
, 1996
"... We describe an encoding of major parts of domain theory in the PVS extension of the simplytyped calculus; these encodings consist of: ffl Formalizations of basic structures like partial orders and complete partial orders (domains). ffl Various domain constructions. ffl Notions related to monoto ..."
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We describe an encoding of major parts of domain theory in the PVS extension of the simplytyped calculus; these encodings consist of: ffl Formalizations of basic structures like partial orders and complete partial orders (domains). ffl Various domain constructions. ffl Notions related to monotonic functions and continuous functions. ffl KnasterTarski fixedpoint theorems for monotonic and continuous functions; the proof of this theorem requires Zorn's lemma which has been derived from Hilbert's choice operator. ffl Scott's fixedpoint induction for admissible predicates and various variations of fixedpoint induction like Park's lemma. Altogether, these encodings form a conservative extension of the underlying PVS logic, since all developments are purely definitional. Most of our proofs are straightforward transcriptions of textbook knowledge. The purpose of this work, however, was not to merely reproduce textbook knowledge. To the contrary, our main motivation derived from ou...
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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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...
Nonprimitive Recursive Function Definitions
 Proceedings of the 8th International Workshop on Higher Order Logic Theorem Proving and Its Applications (LNCS 971
, 1995
"... This paper presents an approach to the problem of introducing nonprimitive recursive function definitions in higher order logic. A recursive specification is translated into a domain theory version, where the recursive calls are treated as potentially nonterminating. Once we have proved termin ..."
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This paper presents an approach to the problem of introducing nonprimitive recursive function definitions in higher order logic. A recursive specification is translated into a domain theory version, where the recursive calls are treated as potentially nonterminating. Once we have proved termination, the original specification can be derived easily.
Bootstrapping the Primitive Recursive Functions by 47 Colors
, 1994
"... I construct a concrete colouring of the 3 element subsets of N. ..."
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I construct a concrete colouring of the 3 element subsets of N.