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Real number calculations and theorem proving
 Proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics, TPHOLs 2005, volume 3603 of Lecture Notes in Computer Science
, 2005
"... Abstract. Wouldn’t it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework. First, we formally establish upper and lower bounds for trigonometric and transcendental functions. ..."
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Abstract. Wouldn’t it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework. First, we formally establish upper and lower bounds for trigonometric and transcendental functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations can be performed in an algebraic setting. This pragmatic approach has been implemented as a strategy in PVS. The strategy provides a safe way to perform explicit calculations over real numbers in formal proofs. 1
Reflecting proofs in firstorder logic with equality
 In Proceedings of EUROSPEECH’97
, 2005
"... Abstract. Our general goal is to provide better automation in interactive proof assistants such as Coq. We present an interpreter of proof traces in firstorder multisorted logic with equality. Thanks to the reflection ability of Coq, this interpreter is both implemented and formally proved sound — ..."
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Cited by 10 (3 self)
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Abstract. Our general goal is to provide better automation in interactive proof assistants such as Coq. We present an interpreter of proof traces in firstorder multisorted logic with equality. Thanks to the reflection ability of Coq, this interpreter is both implemented and formally proved sound — with respect to a reflective interpretation of formulae as Coq properties — inside Coq’s type theory. Our generic framework allows to interpret proofs traces computed by any automated theorem prover, as long as they are precise enough: we illustrate that on traces produced by the CiME tool when solving unifiability problems by ordered completion. We discuss some benchmark results obtained on the TPTP library. The aim of this paper is twofold: first we want to validate a reflective approach for proofs in interactive proof assistants, and second show how to provide a better automation for such assistants. Both aspects can be achieved by using external provers designed to automatically solve some problems of interest: these provers can “feed ” the assistant with large proofs, and help to compare the direct and the reflective approaches, and they can also release the user from (parts of) the proof.
A verified runtime for a verified theorem prover
"... rely on the correctness of runtime systems for programming languages like ML, OCaml or Common Lisp. These runtime systems are complex and critical to the integrity of the theorem provers. In this paper, we present a new Lisp runtime which has been formally verified and can run the Milawa theorem pro ..."
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Cited by 10 (5 self)
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rely on the correctness of runtime systems for programming languages like ML, OCaml or Common Lisp. These runtime systems are complex and critical to the integrity of the theorem provers. In this paper, we present a new Lisp runtime which has been formally verified and can run the Milawa theorem prover. Our runtime consists of 7,500 lines of machine code and is able to complete a 4 gigabyte Milawa proof effort. When our runtime is used to carry out Milawa proofs, less unverified code must be trusted than with any other theorem prover. Our runtime includes a justintime compiler, a copying garbage collector, a parser and a printer, all of which are HOL4verified down to the concrete x86 code. We make heavy use of our previously developed tools for machinecode verification. This work demonstrates that our approach to machinecode verification scales to nontrivial applications. 1
Biform theories in Chiron
 Towards Mechanized Mathematical Assistants, volume 4573 of Lecture Notes in Computer Science
, 2007
"... Abstract. An axiomatic theory represents mathematical knowledge declaratively as a set of axioms. An algorithmic theory represents mathematical knowledge procedurally as a set of algorithms. A biform theory is simultaneously an axiomatic theory and an algorithmic theory. It represents mathematical k ..."
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Cited by 8 (5 self)
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Abstract. An axiomatic theory represents mathematical knowledge declaratively as a set of axioms. An algorithmic theory represents mathematical knowledge procedurally as a set of algorithms. A biform theory is simultaneously an axiomatic theory and an algorithmic theory. It represents mathematical knowledge both declaratively and procedurally. Since the algorithms of algorithmic theories manipulate the syntax of expressions, biform theories—as well as algorithmic theories—are difficult to formalize in a traditional logic without the means to reason about syntax. Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. It includes elements of type theory, a scheme for handling undefinedness, and a facility for reasoning about the syntax of expressions. It is an exceptionally wellsuited logic for formalizing biform theories. This paper defines the notion of a biform theory, gives an overview of Chiron, and illustrates how biform theories can be formalized in Chiron. 1
Complex quantifier elimination in HOL
 TPHOLs 2001: Supplemental Proceedings
, 2001
"... Abstract. Building on a simple construction of the complex numbers and a proof of the Fundamental Theorem of Algebra, we implement, as a HOL derived inference rule, a decision method for the first order algebraic theory of C based on quantifier elimination. Although capable of solving some mildly in ..."
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Abstract. Building on a simple construction of the complex numbers and a proof of the Fundamental Theorem of Algebra, we implement, as a HOL derived inference rule, a decision method for the first order algebraic theory of C based on quantifier elimination. Although capable of solving some mildly interesting problems, we also implement a more efficient semidecision procedure for the universal fragment based on Gröbner bases. This is applied to examples including the automatic proof of some simple geometry theorems. The general and universal procedures present an interesting contrast in that the latter can exploit the findingchecking separation to achieve greater efficiency, though this feature is only partly exploited in the present implementation. 1
A Survey on Embedding Programming Logics in a Theorem Prover
 Institute of Information and Computing Sciences Utrecht University
, 2002
"... Theorem provers were also called 'proof checkers' because that is what they were in the beginning. They have grown powerful, however, capable in many cases to automatically produce complicated proofs. In particular, higher order logic based theorem provers such as HOL and PVS became popular because ..."
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Theorem provers were also called 'proof checkers' because that is what they were in the beginning. They have grown powerful, however, capable in many cases to automatically produce complicated proofs. In particular, higher order logic based theorem provers such as HOL and PVS became popular because the logic is well known and very expressive. They are generally considered to be potential platforms to embed a programming logic for the purpose of formal verification. In this paper we investigate a number of most commonly used methods of embedding programming logics in such theorem provers and expose problems we discover. We will also propose an alternative approach: hybrid embedding.
Verifying mixed realinteger quantifier elimination
 IJCAR 2006, LNCS 4130
, 2006
"... We present a formally verified quantifier elimination procedure for the first order theory over linear mixed realinteger arithmetics in higherorder logic based on a work by Weispfenning. To this end we provide two verified quantifier elimination procedures: for Presburger arithmitics and for lin ..."
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We present a formally verified quantifier elimination procedure for the first order theory over linear mixed realinteger arithmetics in higherorder logic based on a work by Weispfenning. To this end we provide two verified quantifier elimination procedures: for Presburger arithmitics and for linear real arithmetics.
Formalizing the LogicAutomaton Connection
"... Abstract. This paper presents a formalization of a library for automata on bit strings in the theorem prover Isabelle/HOL. It forms the basis of a reflectionbased decision procedure for Presburger arithmetic, which is efficiently executable thanks to Isabelle’s code generator. With this work, we th ..."
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Abstract. This paper presents a formalization of a library for automata on bit strings in the theorem prover Isabelle/HOL. It forms the basis of a reflectionbased decision procedure for Presburger arithmetic, which is efficiently executable thanks to Isabelle’s code generator. With this work, we therefore provide a mechanized proof of the wellknown connection between logic and automata theory. 1
Proof synthesis and reflection for linear arithmetic. Submitted
, 2006
"... This article presents detailed implementations of quantifier elimination for both integer and real linear arithmetic for theorem provers. The underlying algorithms are those by Cooper (for Z) and by Ferrante and Rackoff (for R). Both algorithms are realized in two entirely different ways: once in ta ..."
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Cited by 6 (5 self)
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This article presents detailed implementations of quantifier elimination for both integer and real linear arithmetic for theorem provers. The underlying algorithms are those by Cooper (for Z) and by Ferrante and Rackoff (for R). Both algorithms are realized in two entirely different ways: once in tactic style, i.e. by a proofproducing functional program, and once by reflection, i.e. by computations inside the logic rather than in the metalanguage. Both formalizations are highly generic because they make only minimal assumptions w.r.t. the underlying logical system and theorem prover. An implementation in Isabelle/HOL shows that the reflective approach is between one and two orders of magnitude faster. 1
Formalised Cut Admissibility for Display Logic
 In Proc. TPHOLS'02, LNCS 2410, 131147
, 2002
"... We use a deep embedding of the display calculus for relation algebras RA in the logical framework Isabelle/HOL to formalise a machinechecked proof of cutadmissibility for RA. Unlike other "implementations ", we explicitly formalise the structural induction in Isabelle /HOL and believe this to ..."
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Cited by 6 (3 self)
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We use a deep embedding of the display calculus for relation algebras RA in the logical framework Isabelle/HOL to formalise a machinechecked proof of cutadmissibility for RA. Unlike other "implementations ", we explicitly formalise the structural induction in Isabelle /HOL and believe this to be the first full formalisation of cutadmissibility in the presence of explicit structural rules.