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The Rooster and the Butterflies
[chapter]
2013
Lecture Notes in Computer Science
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This paper describes a machine-checked proof of the Jordan-Hölder theorem for finite groups. This purpose of this description is to discuss the representation of the elementary concepts of finite group theory inside type theory. The design choices underlying these representations were crucial to the successful formalization of a complete proof of the Odd Order Theorem with the Coq system.
doi:10.1007/978-3-642-39320-4_1
fatcat:jz33gwq7pze3zpr3u632kky3l4
Two simulations about DPLL(T)
[article]
2012
arXiv
pre-print
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In this paper we relate different formulations of the DPLL(T) procedure. The first formulation is based on a system of rewrite rules, which we denote DPLL(T). The second formulation is an inference system of, which we denote LKDPLL(T). The third formulation is the application of a standard proof-search mechanism in a sequent calculus LKp(T) introduced here. We formalise an encoding from DPLL(T) to LKDPLL(T) that was, to our knowledge, never explicitly given and, in the case where DPLL(T) is
arXiv:1204.5159v1
fatcat:ykoithbqtnfpznultvkm3udsbi
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... nded with backjumping and Lemma learning, never even implicitly given. We also formalise an encoding from LKDPLL(T) to LKp(T), building on Ivan Gazeau's previous work: we extend his work in that we handle the "-modulo-Theory" aspect of SAT-modulo-theory, by extending the sequent calculus to allow calls to a theory solver (seen as a blackbox). We also extend his work in that we handle advanced features of DPLL such as backjumping and Lemma learning, etc. Finally, we re fine the approach by starting to formalise quantitative aspects of the simulations: the complexity is preserved (number of steps to build complete proofs). Other aspects remain to be formalised (non-determinism of the search / width of search space).
Calcul Formel et Preuves Formelles
2018
Les cours du CIRM
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Texte mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/
doi:10.5802/ccirm.27
fatcat:6wo2edsbv5hfzjwsjo6s3kl5oi
Packaging Mathematical Structures
[chapter]
2009
Lecture Notes in Computer Science
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Notation eqType :=
inria-00368403, version 2 -3 Jul 2009
François Garillot, Georges Gonthier, Assia Mahboubi, Laurence Rideau
It is nevertheless impractical to use the Coq Module construct ...
doi:10.1007/978-3-642-03359-9_23
fatcat:vkan4oitzberffz5sd6b3ukkju
An induction principle over real numbers
2016
Archive for Mathematical Logic
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We give a constructive proof of the open induction principle on real numbers, using bar induction and enumerative open sets. We comment the algorithmic content of this result.
doi:10.1007/s00153-016-0513-8
fatcat:7n33mbyqqbflpa2vappslmulka
Canonical Structures for the Working Coq User
[chapter]
2013
Lecture Notes in Computer Science
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This paper provides a gentle introduction to the art of programming type inference with the mechanism of Canonical Structures. Programmable type inference has been one of the key ingredients for the successful formalization of the Odd Order Theorem using the Coq proof assistant. The paper concludes comparing the language of Canonical Structures to the one of Type Classes and Unification Hints.
doi:10.1007/978-3-642-39634-2_5
fatcat:inn2peddmnbdfev5yk5f267hb4
Formally Verified Approximations of Definite Integrals
2018
Journal of automated reasoning
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Finding an elementary form for an antiderivative is often a difficult task, so numerical integration has become a common tool when it comes to making sense of a definite integral. Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy. Yet numerical integration is still missing from the toolbox when performing formal proofs in analysis. This paper presents an efficient method for
doi:10.1007/s10817-018-9463-7
fatcat:g3jumo65prehlovv7mwnoqjsse
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... matically computing and proving bounds on some definite integrals inside the Coq formal system. Our approach is not based on traditional quadrature methods such as Newton-Cotes formulas. Instead, it relies on computing and evaluating antiderivatives of rigorous polynomial approximations, combined with an adaptive domain splitting. Our approach also handles improper integrals, provided that a factor of the integrand belongs to a catalog of identified integrable functions. This work has been integrated to the CoqInterval library.
Formally Verified Approximations of Definite Integrals
[chapter]
2016
Lecture Notes in Computer Science
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Finding an elementary form for an antiderivative is often a difficult task, so numerical integration has become a common tool when it comes to making sense of a definite integral. Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy. Yet numerical integration is still missing from the toolbox when performing formal proofs in analysis. This paper presents an efficient method for
doi:10.1007/978-3-319-43144-4_17
fatcat:gbqhtd2kvvbxzlbd6i7g7wqoqa
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... matically computing and proving bounds on some definite integrals inside the Coq formal system. Our approach is not based on traditional quadrature methods such as Newton-Cotes formulas. Instead, it relies on computing and evaluating antiderivatives of rigorous polynomial approximations, combined with an adaptive domain splitting. This work has been integrated to the CoqInterval library.
A Formal Quantifier Elimination for Algebraically Closed Fields
[chapter]
2010
Lecture Notes in Computer Science
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We prove formally that the first order theory of algebraically closed fields enjoy quantifier elimination, and hence is decidable. This proof is organized in two modular parts. We first reify the first order theory of rings and prove that quantifier elimination leads to decidability. Then we implement an algorithm which constructs a quantifier free formula from any first order formula in the theory of ring. If the underlying ring is in fact an algebraically closed field, we prove that the two
doi:10.1007/978-3-642-14128-7_17
fatcat:ddgdtdkjlfbv3dq2q7tvfspvdy
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... rmulas have the same semantic. The algorithm producing the quantifier free formula is programmed in continuation passing style, which leads to both a concise program and an elegant proof of semantic correctness.
Is Impredicativity Implicitly Implicit?
2020
Types for Proofs and Programs
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Of all the threats to the consistency of a type system, such as side effects and recursion, impredicativity is arguably the least understood. In this paper, we try to investigate it using a kind of blackbox reverse-engineering approach to map the landscape. We look at it with a particular focus on its interaction with the notion of implicit arguments, also known as erasable arguments. More specifically, we revisit several famous type systems believed to be consistent and which do include some
doi:10.4230/lipics.types.2019.9
dblp:conf/types/MonnierB19
fatcat:elcjyuestje6tkxh37diflg6nm
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... rm of impredicativity, and show that they can be refined to equivalent systems where impredicative quantification can be marked as erasable, in a stricter sense than the kind of proof irrelevance notion used for example for Prop terms in systems like Coq. We hope these observations will lead to a better understanding of why and when impredicativity can be sound. As a first step in this direction, we discuss how these results suggest some extensions of existing systems where constraining impredicativity to erasable quantifications might help preserve consistency.
Coherence for Monoidal Groupoids in HoTT
2020
Types for Proofs and Programs
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We present a proof of coherence for monoidal groupoids in homotopy type theory. An important role in the formulation and in the proof of coherence is played by groupoids with a free monoidal structure; these can be represented by 1-truncated higher inductive types, with constructors freely generating their defining objects, natural isomorphisms and commutative diagrams. All results included in this paper have been formalised in the proof assistant Coq.
doi:10.4230/lipics.types.2019.8
dblp:conf/types/Piceghello19
fatcat:xuutg4xugzckrmfvk47upw6zui
Preface: Selected Extended Papers from Interactive Theorem Proving 2018
2020
Journal of automated reasoning
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Mahboubi Manuel Eberl, Max W. ...
doi:10.1007/s10817-020-09557-w
fatcat:rl6vtxuabbh7nbl4b3avvykhqe
Higher Inductive Type Eliminators Without Paths
2020
Types for Proofs and Programs
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Cubical Agda has support for higher inductive types. Paths are integral to the working of this feature. However, there are other notions of equality. For instance, Cubical Agda comes with an identity type family for which the J rule computes in the usual way when applied to the canonical proof of reflexivity, whereas typical implementations of the J rule for paths do not. This text shows how one can use some of the higher inductive types definable in Cubical Agda with arbitrary notions of
doi:10.4230/lipics.types.2019.10
dblp:conf/types/Danielsson19
fatcat:pgeov2ojdfez7d3ndx5hvppow4
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... ty satisfying certain axioms. The method works for several examples taken from the HoTT book, including the interval, the circle, suspensions, pushouts, the propositional truncation, a general truncation operator, and set quotients.
Formalization of Mathematics in Type Theory (Dagstuhl Seminar 18341)
2019
Dagstuhl Reports
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Mahboubi(INRIA -Nantes, FR) License Creative Commons BY 3.0 Unported license © Cyril Cohen and Assia Mahboubi Joint work of Reynald Affeldt, Cyrill Cohen, Damien Rouhling, Assia Mahboubi, Pierre-Yves ...
of Pittsburgh, US) License Creative Commons BY 3.0 Unported license © Auke Booij and Floris van Doorn URL https://github.com/fpvandoorn/Dagstuhl-tables/
Cyril Cohen(INRIA Sophia Antipolis, FR)and Assia ...
doi:10.4230/dagrep.8.8.130
dblp:journals/dagstuhl-reports/BauerELM18
fatcat:23a6bn6hg5dqhacgv6jsp7ek7u
Modular pre-processing for automated reasoning in dependent type theory
[article]
2022
arXiv
pre-print
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The power of modern automated theorem provers can be put at the service of interactive theorem proving. But this requires in particular bridging the expressivity gap between the logics these provers are respectively based on. This paper presents the implementation of a modular suite of pre-processing transformations, which incrementally bring certain formulas expressed in the Calculus of Inductive Constructions closer to the first-order logic of Satifiability Modulo Theory solvers. These
arXiv:2204.02643v1
fatcat:6fdepv3og5adrmpiyv74siz67i
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... rmations address issues related to the axiomatization of inductive types, to polymorphic definitions or to the different implementations of a same theory signature. This suite is implemented as a plugin for the Coq proof assistant, and integrated to the SMTCoq toolchain.
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