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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 Constructive Algebraic Hierarchy in Coq
"... We describe a framework of algebraic structures in the proof assistant Coq. We have developed this framework as part of the FTA project in Nijmegen, in which a constructive proof of the Fundamental Theorem of Algebra has been formalized in Coq. The algebraic hierarchy that is described here is both ..."
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We describe a framework of algebraic structures in the proof assistant Coq. We have developed this framework as part of the FTA project in Nijmegen, in which a constructive proof of the Fundamental Theorem of Algebra has been formalized in Coq. The algebraic hierarchy that is described here is both abstract and way, dening e.g. a ring as a tuple consisting of a group, a binary operation and a constant that together satisfy the properties of a ring. In this way, a ring automatically inherits the group properties of the additive subgroup. The algebraic hierarchy is formalized in Coq by applying a combination of labeled record types and coercions. In the labeled record types of Coq, one can use dependent types: the type of one label may depend on another label. This allows to give a type to a dependenttyped tuple like hA; f; ai, where A is a set, f an operation on A and a an element of A. Coercions are
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.
Changing Data Structures in Type Theory: a study of natural numbers
 Types for Proofs and Programs, Intl. Workshop (TYPES 2000), LNCS 2277
, 2000
"... In typetheory based proof systems that provide inductive structures, computation tools are automatically associated to inductive de nitions. Choosing a particular representation for a given concept has a strong inuence on proof structure. We propose a method to make the change from one represe ..."
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In typetheory based proof systems that provide inductive structures, computation tools are automatically associated to inductive de nitions. Choosing a particular representation for a given concept has a strong inuence on proof structure. We propose a method to make the change from one representation to another easier, by systematically translating proofs from one context to another. We show how this method works by using it on natural numbers, for which a unary representation (based on Peano axioms) and a binary representation are available. This method leads to an automatic translation tool that we have implemented in Coq and successfully applied to several arithmetical theorems.
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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Cited by 2 (0 self)
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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).
Towards Automation of Real Analysis in Coq
"... 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. This formalization was done closely following the work of Bishop [1]; the real numbers were first axiomatically characterized as a complete ordered field with the archimedian property; later, this axiomatization was proved by Geuvers and Niqui [4] to be appropriate (in the sense that the construction of real numbers as Cauchy sequences of rationals satisfies the axioms) and categorical (as any two models of these axioms are isomorphic). Using this work as a basis, partial functions are defined as a Coq record type consisting of a predicate and a total function on the set of real numbers that satisfy that predicate (see [3]). The usual operations (composition, addition, multiplication, division) are then defined as yielding partial functions from partial functions. We can then define continuity, differentiability and integration, and prove the usual properties of these: preservation of continuity and differentiability through algebraic operations and functional