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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 13 (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
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
Coinductive Pearl: Modular FirstOrder Logic Completeness
"... Codatatypes are regrettably absent from many programming languages and proof assistants. We make a case for their usefulness by revisiting a classic result: the completeness theorem for firstorder logic established through a Gentzen system. Codatatypes help capture the essence of the proof, which e ..."
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Codatatypes are regrettably absent from many programming languages and proof assistants. We make a case for their usefulness by revisiting a classic result: the completeness theorem for firstorder logic established through a Gentzen system. Codatatypes help capture the essence of the proof, which establishes an abstract property of derivation trees independently of the concrete syntax or inference rules. This separation of concerns simplifies the presentation, especially for readers acquainted with lazy data structures. The proof is formalized in Isabelle/HOL and demonstrates the recently introduced definitional package for codatatypes and its integration with Isabelle’s Haskell code generator.
Mechanizing the Metatheory of Sledgehammer
"... Abstract. This paper presents an Isabelle/HOL formalization of recent research in automated reasoning: efficient encodings of sorts in unsorted firstorder logic, as implemented in Isabelle’s Sledgehammer proof tool. The formalization provides the generalpurpose machinery to reason about formulas a ..."
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Abstract. This paper presents an Isabelle/HOL formalization of recent research in automated reasoning: efficient encodings of sorts in unsorted firstorder logic, as implemented in Isabelle’s Sledgehammer proof tool. The formalization provides the generalpurpose machinery to reason about formulas and models, emulating the theory of institutions. It also establishes classical metatheorems such as completeness, compactness, and downward Löwenheim–Skolem. Quantifiers are represented using a nominallike approach designed for interpreting syntax in semantic domains. 1
Towards SelfVerification of Isabelle’s Sledgehammer
"... Abstract. This paper presents an Isabelle/HOL formalisation of recent research in automated reasoning: efficient encodings of sorts in unsorted firstorder logic, as implemented in the Sledgehammer proof tool. The formalisation provides the machinery to reason about models as well as classical metat ..."
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Abstract. This paper presents an Isabelle/HOL formalisation of recent research in automated reasoning: efficient encodings of sorts in unsorted firstorder logic, as implemented in the Sledgehammer proof tool. The formalisation provides the machinery to reason about models as well as classical metatheorems, emulating the theory of institutions. Quantifiers are represented using an approach that avoids some of the tedium and restrictions associated with better known binder representations. Sledgehammer itself has been useful for discharging the proof obligations arising from its own metatheory. 1
Essential Incompleteness of Arithmetic Verified by Coq
, 2005
"... Abstract. A constructive proof of the GödelRosser incompleteness theorem [9] has been completed using the Coq proof assistant. Some theory of classical firstorder logic over an arbitrary language is formalized. A development of primitive recursive functions is given, and all primitive recursive fu ..."
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Abstract. A constructive proof of the GödelRosser incompleteness theorem [9] has been completed using the Coq proof assistant. Some theory of classical firstorder logic over an arbitrary language is formalized. A development of primitive recursive functions is given, and all primitive recursive functions are proved to be representable in a weak axiom system. Formulas and proofs are encoded as natural numbers, and functions operating on these codes are proved to be primitive recursive. The weak axiom system is proved to be essentially incomplete. In particular, Peano arithmetic is proved to be consistent in Coq’s type theory and therefore is incomplete. 0
Essential Incompleteness of Arithmetic Verified by Coq
, 2006
"... Abstract. A constructive proof of the GödelRosser incompleteness theorem [9] has been completed using the Coq proof assistant. Some theory of classical firstorder logic over an arbitrary language is formalized. A development of primitive recursive functions is given, and all primitive recursive fu ..."
Abstract
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Abstract. A constructive proof of the GödelRosser incompleteness theorem [9] has been completed using the Coq proof assistant. Some theory of classical firstorder logic over an arbitrary language is formalized. A development of primitive recursive functions is given, and all primitive recursive functions are proved to be representable in a weak axiom system. Formulas and proofs are encoded as natural numbers, and functions operating on these codes are proved to be primitive recursive. The weak axiom system is proved to be essentially incomplete. In particular, Peano arithmetic is proved to be consistent in Coq’s type theory and therefore is incomplete. 0