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34
Typing Algorithm in Type Theory with Inheritance
 Proc of POPL'97
, 1997
"... We propose and study a new typing algorithm for dependent type theory. This new algorithm typechecks more terms by using inheritance between classes. This inheritance mechanism turns out to be powerful: it supports multiple inheritance, classes with parameters and uses new abstract classes FUNCLASS ..."
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We propose and study a new typing algorithm for dependent type theory. This new algorithm typechecks more terms by using inheritance between classes. This inheritance mechanism turns out to be powerful: it supports multiple inheritance, classes with parameters and uses new abstract classes FUNCLASS and SORTCLASS (respectively classes of functions and sorts). We also defines classes as records, particularily suitable for the formal development of mathematical theories. This mechanism, implemented in the proof checker Coq, can be adapted to all typed calculus. 1 Introduction In the last years, proof checkers based on type theory appeared as convincing systems to formalize mathematics (especially constructive mathematics) and to prove correctness of software and hardware. In a proof checker, one can interactively build definitions, statements and proofs. The system is then able to check automatically whether the definitions are wellformed and the proofs are correct. Modern systems ar...
General recursion via coinductive types
 Logical Methods in Computer Science
"... Vol. 1 (2:1) 2005, pp. 1–28 ..."
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Proving Equalities in a Commutative Ring Done Right in Coq
 Theorem Proving in Higher Order Logics (TPHOLs 2005), LNCS 3603
, 2005
"... We present a new implementation of a reflexive tactic which solves equalities in a ring structure inside the Coq system. The e#ciency is improved to a point that we can now prove equalities that were previously beyond reach. A special care has been taken to implement e#cient algorithms while kee ..."
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We present a new implementation of a reflexive tactic which solves equalities in a ring structure inside the Coq system. The e#ciency is improved to a point that we can now prove equalities that were previously beyond reach. A special care has been taken to implement e#cient algorithms while keeping the complexity of the correctness proofs low.
Packaging mathematical structures
 THEOREM PROVING IN HIGHER ORDER LOGICS 5674
, 2009
"... This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system. This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated ..."
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Cited by 19 (5 self)
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This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system. This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated structure inference. Our methodology is robust enough to handle a hierarchy comprising a broad variety of algebraic structures, from types with a choice operator to algebraically closed fields. Interfaces for the structures enjoy the convenience of a classical setting, without requiring any axiom. Finally, we present two applications of our proof techniques: a key lemma for characterising the discrete logarithm, and a matrix decomposition problem.
Extending Sledgehammer with SMT Solvers
"... Abstract. Sledgehammer is a component of Isabelle/HOL that employs firstorder automatic theorem provers (ATPs) to discharge goals arising in interactive proofs. It heuristically selects relevant facts and, if an ATP is successful, produces a snippet that replays the proof in Isabelle. We extended Sl ..."
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Cited by 18 (7 self)
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Abstract. Sledgehammer is a component of Isabelle/HOL that employs firstorder automatic theorem provers (ATPs) to discharge goals arising in interactive proofs. It heuristically selects relevant facts and, if an ATP is successful, produces a snippet that replays the proof in Isabelle. We extended Sledgehammer to invoke satisfiability modulo theories (SMT) solvers as well, exploiting its relevance filter and parallel architecture. Isabelle users are now pleasantly surprised by SMT proofs for problems beyond the ATPs ’ reach. Remarkably, the best SMT solver performs better than the best ATP on most of our benchmarks. 1
A modular formalisation of finite group theory
 In TPHOLs
, 2007
"... Abstract. In this paper, we present a formalisation of elementary group theory done in Coq. This work is the first milestone of a longterm effort to formalise FeitThompson theorem. As our further developments will heavily rely on this initial base, we took special care to articulate it in the most ..."
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Abstract. In this paper, we present a formalisation of elementary group theory done in Coq. This work is the first milestone of a longterm effort to formalise FeitThompson theorem. As our further developments will heavily rely on this initial base, we took special care to articulate it in the most compositional way. 1
A MachineChecked Formalization of the Random Oracle Model
 in &quot;Proceedings of TYPES’04&quot;, Lecture Notes in Computer Science
, 2005
"... Abstract. Most approaches to the formal analysis of cryptography protocols make the perfect cryptographic assumption, which entails for example that there is no way to obtain knowledge about the plaintext pertaining to a ciphertext without knowing the key. Ideally, one would prefer to abandon the pe ..."
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Abstract. Most approaches to the formal analysis of cryptography protocols make the perfect cryptographic assumption, which entails for example that there is no way to obtain knowledge about the plaintext pertaining to a ciphertext without knowing the key. Ideally, one would prefer to abandon the perfect cryptography hypothesis and reason about the computational cost of breaking a cryptographic scheme by achieving such goals as gaining information about the plaintext pertaining to a ciphertext without knowing the key. Such a view is permitted by nonstandard computational models such as the Generic Model and the Random Oracle Model. Using the proof assistant Coq, we provide a machinechecked account of the Generic Model and the Random Oracle Model. We exploit this framework to prove the security of the ElGamal cryptosystem against adaptive chosen ciphertexts attacks. 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.
Proving Formally the Implementation of an Efficient gcd Algorithm for Polynomials
 in &quot;3rd International Joint Conference on Automated Reasoning (IJCAR)&quot;, U. FURBACH, N. SHANKAR (editors). , Lecture Notes in Artificial Intelligence
"... Abstract. We describe here a formal proof in the Coq system of the structure theorem for subresultants, which allows to prove formally the correctness of our implementation of the subresultants algorithm. Up to our knowledge it is the first mechanized proof of this result. 1 ..."
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Abstract. We describe here a formal proof in the Coq system of the structure theorem for subresultants, which allows to prove formally the correctness of our implementation of the subresultants algorithm. Up to our knowledge it is the first mechanized proof of this result. 1
A minimalist twolevel foundation for constructive mathematics
, 811
"... We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other lev ..."
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We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model. This twolevel theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the “proofsasprograms ” paradigm and acts as a programming language.