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Semantic Foundations for Embedding HOL in Nuprl
 ALGEBRAIC METHODOLOGY AND SOFTWARE TECHNOLOGY
, 1996
"... We give a new semantics for Nuprl's constructive type theory that justifies a useful embedding of the logic of the HOL theorem prover inside Nuprl. The embedding gives Nuprl effective access to most of the large body of formalized mathematics that the HOL community has amassed over the las ..."
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Cited by 30 (2 self)
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We give a new semantics for Nuprl's constructive type theory that justifies a useful embedding of the logic of the HOL theorem prover inside Nuprl. The embedding gives Nuprl effective access to most of the large body of formalized mathematics that the HOL community has amassed over the last decade. The new semantics is dramatically simpler than the old, and gives a novel and general way of adding settheoretic equivalence classes to untyped functional programming languages.
Towards Selfverification of HOL Light
 In International Joint Conference on Automated Reasoning
, 2006
"... Abstract. The HOL Light prover is based on a logical kernel consisting of about 400 lines of mostly functional OCaml, whose complete formal verification seems to be quite feasible. We would like to formally verify (i) that the abstract HOL logic is indeed correct, and (ii) that the OCaml code does c ..."
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Abstract. The HOL Light prover is based on a logical kernel consisting of about 400 lines of mostly functional OCaml, whose complete formal verification seems to be quite feasible. We would like to formally verify (i) that the abstract HOL logic is indeed correct, and (ii) that the OCaml code does correctly implement this logic. We have performed a full verification of an imperfect but quite detailed model of the basic HOL Light core, without definitional mechanisms, and this verification is entirely conducted with respect to a settheoretic semantics within HOL Light itself. We will duly explain why the obvious logical and pragmatic difficulties do not vitiate this approach, even though it looks impossible or useless at first sight. Extension to include definitional mechanisms seems straightforward enough, and the results so far allay most of our practical worries. 1 Introduction: quis custodiet ipsos custodes? Mathematical proofs are subjected to peer review before publication, but there
SourceLevel Proof Reconstruction for Interactive Theorem Proving
"... Abstract. Interactive proof assistants should verify the proofs they receive from automatic theorem provers. Normally this proof reconstruction takes place internally, forming part of the integration between the two tools. We have implemented sourcelevel proof reconstruction: resolution proofs are ..."
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Cited by 29 (2 self)
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Abstract. Interactive proof assistants should verify the proofs they receive from automatic theorem provers. Normally this proof reconstruction takes place internally, forming part of the integration between the two tools. We have implemented sourcelevel proof reconstruction: resolution proofs are automatically translated to Isabelle proof scripts. Users can insert this text into their proof development or (if they wish) examine it manually. Each step of a proof is justified by calling Hurd’s Metis prover, which we have ported to Isabelle. A recurrent issue in this project is the treatment of Isabelle’s axiomatic type classes. 1
Importing mathematics from hol into Nuprl
 Theorem Proving in Higher Order Logics (TPHOLs 1996), volume 1125 of LNCS
, 1996
"... Abstract. Nuprl and HOL are both tacticbased interactive theorem provers for higherorder logic, and both have been used in many substantial applications over the last decade. However, the HOL community has accumulated a much larger collection of formalized mathematics of the kind useful for hardwa ..."
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Cited by 27 (2 self)
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Abstract. Nuprl and HOL are both tacticbased interactive theorem provers for higherorder logic, and both have been used in many substantial applications over the last decade. However, the HOL community has accumulated a much larger collection of formalized mathematics of the kind useful for hardware and software veri cation. This collection would be of great bene t in applying Nuprl to veri cation problems of real practical interest. This paper describes a connection we have implemented between HOL and Nuprl that gives Nuprl e ective access to mathematics formalized in HOL. In designing this connection, we had to overcome a number of problems related to di erences in the logics, logical infrastructures and stylistic conventions of Nuprl and HOL. 1
A mechanically verified, sound and complete theorem prover for first order logic
 In Theorem Proving in Higher Order Logics, 18th International Conference, TPHOLs 2005, volume 3603 of Lecture Notes in Computer Science
, 2005
"... Abstract. We present a system of first order logic, together with soundness and completeness proofs wrt. standard first order semantics. Proofs are mechanised in Isabelle/HOL. Our definitions are computable, allowing us to derive an algorithm to test for first order validity. This algorithm may be e ..."
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Cited by 18 (0 self)
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Abstract. We present a system of first order logic, together with soundness and completeness proofs wrt. standard first order semantics. Proofs are mechanised in Isabelle/HOL. Our definitions are computable, allowing us to derive an algorithm to test for first order validity. This algorithm may be executed in Isabelle/HOL using the rewrite engine. Alternatively the algorithm has been ported to OCaML. 1
Axiomatic constructor classes in Isabelle/HOLCF
 In In Proc. 18th International Conference on Theorem Proving in Higher Order Logics (TPHOLs ’05), Volume 3603 of Lecture Notes in Computer Science
, 2005
"... Abstract. We have definitionally extended Isabelle/HOLCF to support axiomatic Haskellstyle constructor classes. We have subsequently defined the functor and monad classes, together with their laws, and implemented state and resumption monad transformers as generic constructor class instances. This ..."
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Cited by 18 (7 self)
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Abstract. We have definitionally extended Isabelle/HOLCF to support axiomatic Haskellstyle constructor classes. We have subsequently defined the functor and monad classes, together with their laws, and implemented state and resumption monad transformers as generic constructor class instances. This is a step towards our goal of giving modular denotational semantics for concurrent lazy functional programming languages, such as GHC Haskell. 1
A Proof Planning Framework for Isabelle
, 2005
"... Proof planning is a paradigm for the automation of proof that focuses on encoding intelligence to guide the proof process. The idea is to capture common patterns of reasoning which can be used to derive abstract descriptions of proofs known as proof plans. These can then be executed to provide fully ..."
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Cited by 13 (9 self)
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Proof planning is a paradigm for the automation of proof that focuses on encoding intelligence to guide the proof process. The idea is to capture common patterns of reasoning which can be used to derive abstract descriptions of proofs known as proof plans. These can then be executed to provide fully formal proofs. This thesis concerns the development and analysis of a novel approach to proof planning that focuses on an explicit representation of choices during search. We embody our approach as a proof planner for the generic proof assistant Isabelle and use the Isar language, which is humanreadable and machinecheckable, to represent proof plans. Within this framework we develop an inductive theorem prover as a case study of our approach to proof planning. Our prover uses the difference reduction heuristic known as rippling to automate the step cases of the inductive proofs. The development of a flexible approach to rippling that supports its various modifications and extensions is the second major focus of this thesis. Here, our inductive theorem prover provides a context in which to evaluate rippling experimentally. This work results in an efficient and powerful inductive theorem prover for Isabelle as well as proposals for further improving the efficiency of rippling. We also draw observations in order
A Classical SetTheoretic Model of Polymorphic Extensional Type Theory
, 1997
"... . We give a new semantic foundation for type theories in the lineage of MartinLof's "polymorphic extensional" type theory, and use it to give a model of the constructive type theory of the interactive theorem proving system Nuprl. These type theories are based on an operational seman ..."
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. We give a new semantic foundation for type theories in the lineage of MartinLof's "polymorphic extensional" type theory, and use it to give a model of the constructive type theory of the interactive theorem proving system Nuprl. These type theories are based on an operational semantics of an untyped programming language. We show how to integrate classical settheoretic objects, such as functionsasgraphs and equivalence classes, into this operational framework. The new semantics is dramatically simpler than the previous ones, and enables direct reasoning about classical mathematics. A practical consequence is that it justifies a useful embedding of the logic of the HOL theorem prover that gives Nuprl effective access to most of the large body of formalized mathematics that the HOL community has amassed over the years. 1 Introduction The socalled "polymorphic extensional" type theory of MartinLof (MartinLof, 1982) has two features that set it apart from other constructive type t...
A proofcentric approach to mathematical assistants
 Journal of Applied Logic: Special Issue on Mathematics Assistance Systems
, 2005
"... We present an approach to mathematical assistants which uses readable, executable proof scripts as the central language for interaction. We examine an implementation that combines the Isar language, the Isabelle theorem prover and the IsaPlanner proof planner. We argue that this synergy provides a f ..."
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We present an approach to mathematical assistants which uses readable, executable proof scripts as the central language for interaction. We examine an implementation that combines the Isar language, the Isabelle theorem prover and the IsaPlanner proof planner. We argue that this synergy provides a flexible environment for the exploration, certification, and presentation of mathematical proof.