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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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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
On the Desirability of Mechanizing Calculational Proofs
"... Dijkstra argues that calculational proofs are preferable to traditional pictorial and/or verbal proofs. First, due to the calculational proof format, incorrect proofs are less likely. Second, syntactic considerations (letting the "symbols do the work") have led to an impressive array of techniques f ..."
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Dijkstra argues that calculational proofs are preferable to traditional pictorial and/or verbal proofs. First, due to the calculational proof format, incorrect proofs are less likely. Second, syntactic considerations (letting the "symbols do the work") have led to an impressive array of techniques for elegant proof construction. However, calculational proofs are not formal and are not awless. Why not make them formal and check them mechanically?
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