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CCoRN, the Constructive Coq Repository at Nijmegan
"... We present CCoRN, the Constructive Coq Repository at Nijmegen. It consists of a library of constructive algebra and analysis, formalized in the theorem prover Coq. In this paper we explain the structure, the contents and the use of the library. Moreover we discuss the motivation and the (possible) ..."
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We present CCoRN, the Constructive Coq Repository at Nijmegen. It consists of a library of constructive algebra and analysis, formalized in the theorem prover Coq. In this paper we explain the structure, the contents and the use of the library. Moreover we discuss the motivation and the (possible) applications of such a library.
The algebraic hierarchy of the FTA Project
 Journal of Symbolic Computation, Special Issue on the Integration of Automated Reasoning and Computer Algebra Systems
, 2002
"... Abstract. We describe a framework for algebraic expressions for the proof assistant Coq. This framework has been developed as part of the FTA project in Nijmegen, in which a complete proof of the fundamental theorem of algebra has been formalized in Coq. The algebraic framework that is described her ..."
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Abstract. We describe a framework for algebraic expressions for the proof assistant Coq. This framework has been developed as part of the FTA project in Nijmegen, in which a complete proof of the fundamental theorem of algebra has been formalized in Coq. The algebraic framework that is described here is both abstract and structured. We apply a combination of record types, coercive subtyping and implicit arguments. The algebraic framework contains a full development of the real and complex numbers and of the rings of polynomials over these fields. The framework is constructive. It does not use anything apart from the Coq logic. The framework has been successfully used to formalize nontrivial mathematics as part of the FTA project.
A Computational Approach to Reflective MetaReasoning about Languages with Bindings
 In MERLIN ’05: Proceedings of the 3rd ACM SIGPLAN workshop on Mechanized
, 2005
"... We present a foundation for a computational metatheory of languages with bindings implemented in a computeraided formal reasoning environment. Our theory provides the ability to reason abstractly about operators, languages, openended languages, classes of languages, etc. The theory is based on th ..."
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Cited by 12 (2 self)
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We present a foundation for a computational metatheory of languages with bindings implemented in a computeraided formal reasoning environment. Our theory provides the ability to reason abstractly about operators, languages, openended languages, classes of languages, etc. The theory is based on the ideas of higherorder abstract syntax, with an appropriate induction principle parameterized over the language (i.e. a set of operators) being used. In our approach, both the bound and free variables are treated uniformly and this uniform treatment extends naturally to variablelength bindings. The implementation is reflective, namely there is a natural mapping between the metalanguage of the theoremprover and the object language of our theory. The object language substitution operation is mapped to the metalanguage substitution and does not need to be defined recursively. Our approach does not require designing a custom type theory; in this paper we describe the implementation of this foundational theory within a generalpurpose type theory. This work is fully implemented in the MetaPRL theorem prover, using the preexisting NuPRLlike MartinL ofstyle computational type theory. Based on this implementation, we lay out an outline for a framework for programming language experimentation and exploration as well as a general reflective reasoning framework. This paper also includes a short survey of the existing approaches to syntactic reflection. 1
A Constructive Formalization of the Fundamental Theorem of Calculus
"... We have finished a constructive formalization in the theorem prover Coq of the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse processes. In this formalization, we have closely followed Bishop's work ([4]). In this paper, we describe the formalizat ..."
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We have finished a constructive formalization in the theorem prover Coq of the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse processes. In this formalization, we have closely followed Bishop's work ([4]). In this paper, we describe the formalization in some detail, focusing on how some of Bishop's original proofs had to be refined, adapted or redone from scratch.
Estimating the cost of a standard library for a mathematical proof checker
, 2001
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Hierarchical Reflection
"... Abstract. The technique of reflection is a way to automate proof construction in type theoretical proof assistants. Reflection is based on the definition of a type of syntactic expressions that gets interpreted in the domain of discourse. By allowing the interpretation function to be partial or even ..."
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Abstract. The technique of reflection is a way to automate proof construction in type theoretical proof assistants. Reflection is based on the definition of a type of syntactic expressions that gets interpreted in the domain of discourse. By allowing the interpretation function to be partial or even a relation one gets a more general method known as ``partial reflection''. In this paper we show how one can take advantage of the partiality of the interpretation to uniformly define a family of tactics for equational reasoning that will work in different algebraic structures. The tactics then follow the hierarchy of those algebraic structures in a natural way.
Formalizing Real Calculus in Coq
, 2002
"... We have finished a constructive formalization in the theorem prover Coq of the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse processes. This formalization is built upon the library of constructive algebra created in the FTA (Fundamental Theorem of Alg ..."
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We have finished a constructive formalization in the theorem prover Coq of the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse processes. This formalization is built upon the library of constructive algebra created in the FTA (Fundamental Theorem of Algebra) project, which is extended with results about the real numbers, namely about (power) series. Two important issues that arose in this formalization and which will be discussed in this paper are partial functions (different ways of dealing with this concept and the advantages of each different approach) and the high level tactics that were developed in parallel with the formalization (which automate several routine procedures involving results about realvalued functions).
Explicit Convertibility Proofs in Pure Type Systems
 LOGICAL FRAMEWORKS AND METALANGUAGES: THEORY AND PRACTICE
, 2013
"... We define type theory with explicit conversions. When type checking a term in normal type theory, the system searches for convertibility paths between types. The results of these searches are not stored in the term, and need to be reconstructed every time again. In our system, this information is al ..."
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We define type theory with explicit conversions. When type checking a term in normal type theory, the system searches for convertibility paths between types. The results of these searches are not stored in the term, and need to be reconstructed every time again. In our system, this information is also represented in the term. The system we define has the property that the type derivation of a term has exactly the same structure as the term itself. This has the consequence that there exists a natural LF encoding of such a system in which the encoded type is a dependent parameter of the type of the encoded term. For every Pure Type System we define a system in our style. We show that such a system is always equivalent to the normal system without explicit conversions (even for nonfunctional systems), in the sense that the typability relation can be lifted. This proof has been fully formalised in the Coq system, building on a formalisation by Vincent Siles. In our system, explicit conversions are not allowed to be removed when checking for convertibility. This means that all terms in convertibility proofs are well typed, even in the sense of our system.
www.elsevier.com/locate/entcs A Logical Framework with Explicit Conversions
"... The type theory λP corresponds to the logical framework LF. In this paper we present λH, a variant of λP where convertibility is not implemented by means of the customary conversion rule, but instead type conversions are made explicit in the terms. This means that the time to type check a λH term is ..."
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The type theory λP corresponds to the logical framework LF. In this paper we present λH, a variant of λP where convertibility is not implemented by means of the customary conversion rule, but instead type conversions are made explicit in the terms. This means that the time to type check a λH term is proportional to the size of the term itself. We define an erasure map from λH to λP, and show that through this map the type theory λH corresponds exactly to λP: any λH judgment will be erased to a λP judgment, and conversely each λP judgment can be lifted to a λH judgment. We also show a version of subject reduction: if two λH terms are provably convertible then their types are also provably convertible. Keywords:
A Decision Procedure for Equational Reasoning in Commutative Algebraic Structures
"... Abstract. We present a decision procedure for equational reasoning in abelian groups, commutative rings and fields that checks whether a given equality can be proven from the axioms of these structures. This has been implemented as a tactic in Coq; here we give a mathematical description of the deci ..."
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Abstract. We present a decision procedure for equational reasoning in abelian groups, commutative rings and fields that checks whether a given equality can be proven from the axioms of these structures. This has been implemented as a tactic in Coq; here we give a mathematical description of the decision procedure that abstracts from Coq specifics, making the work in principle adaptable to other theorem provers. Within Coq we prove that this decision procedure is correct. On the metalevel we analyse its completeness, showing that it is complete for groups and rings in the sense that the tactic succeeds in finding a proof of an equality if and only if that equality is provable from the group/ring axioms without any hypotheses. Finally we characterize in what way our method is incomplete for fields.