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A MachineChecked Proof of the Odd Order Theorem
"... This paper reports on a sixyear collaborative effort that culminated in a complete formalization of a proof of the FeitThompson Odd Order Theorem in the Coq proof assistant. The formalized proof is constructive, and relies on nothing but the axioms and rules of the foundational framework impleme ..."
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This paper reports on a sixyear collaborative effort that culminated in a complete formalization of a proof of the FeitThompson Odd Order Theorem in the Coq proof assistant. The formalized proof is constructive, and relies on nothing but the axioms and rules of the foundational framework implemented by Coq. To support the formalization, we developed a comprehensive set of reusable libraries of formalized mathematics, including results in finite group theory, linear algebra, Galois theory, and the theories of the real and complex algebraic numbers.
Coquet: A Coq library for verifying hardware
 In Proceedings of the 1st International Conference on Certified Programs and Proofs (CPP ’11), J.P.Jouannaudand Z. Shao, Eds. LNCS
"... Abstract. We propose a new library to model and verify hardware circuits in the Coq proof assistant. This library allows one to easily build circuits by following the usual penandpaper diagrams. We define a deepembedding: we use a (dependently typed) datatype that models the architecture of cir ..."
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Abstract. We propose a new library to model and verify hardware circuits in the Coq proof assistant. This library allows one to easily build circuits by following the usual penandpaper diagrams. We define a deepembedding: we use a (dependently typed) datatype that models the architecture of circuits, and a meaning function. We propose tactics that ease the reasoning about the behavior of the circuits, and we demonstrate that our approach is practicable by proving the correctness of various circuits: a textbook divide and conquer adder of parametric size, some higherorder combinators of circuits, and some sequential circuits: a buffer, and a register.
Foundational propertybased testing
, 2015
"... Abstract Integrating propertybased testing with a proof assistant creates an interesting opportunity: reusable or tricky testing code can be formally verified using the proof assistant itself. In this work we introduce a novel methodology for formally verified propertybased testing and implement i ..."
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Abstract Integrating propertybased testing with a proof assistant creates an interesting opportunity: reusable or tricky testing code can be formally verified using the proof assistant itself. In this work we introduce a novel methodology for formally verified propertybased testing and implement it as a foundational verification framework for QuickChick, a port of QuickCheck to Coq. Our framework enables one to verify that the executable testing code is testing the right Coq property. To make verification tractable, we provide a systematic way for reasoning about the set of outcomes a random data generator can produce with nonzero probability, while abstracting away from the actual probabilities. Our framework is firmly grounded in a fully verified implementation of QuickChick itself, using the same underlying verification methodology. We also apply this methodology to a complex case study on testing an informationflow control abstract machine, demonstrating that our verification methodology is modular and scalable and that it requires minimal changes to existing code. 1
Mirrorshard: Proof by computational reflection with verified hints
 CoRR
"... We describe a method for building composable and extensible verification procedures within the Coq proof assistant. Unlike traditional methods that rely on runtime generation and checking of proofs, we use verifiedcorrect procedures with Coq soundness proofs. Though they are internalized in Coq’ ..."
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We describe a method for building composable and extensible verification procedures within the Coq proof assistant. Unlike traditional methods that rely on runtime generation and checking of proofs, we use verifiedcorrect procedures with Coq soundness proofs. Though they are internalized in Coq’s logic, our provers support sound extension by users with hints over new domains, enabling automated reasoning about userdefined abstract predicates. We maintain soundness by developing an architecture for modular packaging, construction, and composition of hint databases, which had previously only been implemented in Coq at the level of its dynamically typed, proofgenerating tactic language. Our provers also include rich handling of unification variables, enabling integration with other tacticbased deduction steps within Coq. We have implemented our techniques in MirrorShard, an opensource framework for reflective verification. We demonstrate its applicability by instantiating it to separation logic in order to reason about imperative program verification. 1.
and
, 2012
"... Under consideration for publication in J. Functional Programming 1 How to make ad hoc proof automation less ad hoc ..."
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Under consideration for publication in J. Functional Programming 1 How to make ad hoc proof automation less ad hoc
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, 2013
"... Under consideration for publication in J. Functional Programming 1 How to make ad hoc proof automation less ad hoc ..."
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Under consideration for publication in J. Functional Programming 1 How to make ad hoc proof automation less ad hoc
Verifying an algorithm computing Discrete Vector Fields for digital imaging
"... In this paper, we present a formalization of an algorithm to construct admissible discrete vector fields in the Coq theorem prover taking advantage of the SSReflect library. Discrete vector fields are a tool which has been welcomed in the homological analysis of digital images since it provides a ..."
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In this paper, we present a formalization of an algorithm to construct admissible discrete vector fields in the Coq theorem prover taking advantage of the SSReflect library. Discrete vector fields are a tool which has been welcomed in the homological analysis of digital images since it provides a procedure to reduce the amount of information but preserving the homological properties. In particular, thanks to discrete vector fields, we are able to compute, inside Coq, homological properties of biomedical images which otherwise are out of the reach of this system.