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12
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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Cited by 18 (9 self)
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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.
A certified, corecursive implementation of exact real numbers
 Theoretical Computer Science
, 2006
"... We implement exact real numbers in the logical framework Coq using streams, i.e., infinite sequences, of digits, and characterize constructive real numbers through a minimal axiomatization. We prove that our construction inhabits the axiomatization, working formally with coinductive types and corecu ..."
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Cited by 12 (0 self)
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We implement exact real numbers in the logical framework Coq using streams, i.e., infinite sequences, of digits, and characterize constructive real numbers through a minimal axiomatization. We prove that our construction inhabits the axiomatization, working formally with coinductive types and corecursive proofs. Thus we obtain reliable, corecursive algorithms for computing on real numbers.
Real number calculations and theorem proving
 Proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics, TPHOLs 2005, volume 3603 of Lecture Notes in Computer Science
, 2005
"... Abstract. Wouldn’t it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework. First, we formally establish upper and lower bounds for trigonometric and transcendental functions. ..."
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Cited by 11 (4 self)
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Abstract. Wouldn’t it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework. First, we formally establish upper and lower bounds for trigonometric and transcendental functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations can be performed in an algebraic setting. This pragmatic approach has been implemented as a strategy in PVS. The strategy provides a safe way to perform explicit calculations over real numbers in formal proofs. 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 formalization i ..."
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Cited by 8 (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. 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.
Verified Real Number Calculations: A Library for Interval Arithmetic
, 2007
"... Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally est ..."
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Cited by 6 (1 self)
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Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally establish upper and lower bounds for elementary functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations take place in an algebraic setting. In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and taylor series expansions. This pragmatic approach has been developed, and formally verified, in a theorem prover. The formal development also includes a set of customizable strategies to automate proofs involving explicit calculations over real numbers. Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theoremprover interaction.
Formal Proof of a Wave Equation Resolution Scheme: the Method Error ⋆
"... Abstract. Popular finite difference numerical schemes for the resolution of the onedimensional acoustic wave equation are wellknown to be convergent. We present a comprehensive formalization of the simplest scheme and formally prove its convergence in Coq. The main difficulties lie in the proper d ..."
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Cited by 3 (2 self)
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Abstract. Popular finite difference numerical schemes for the resolution of the onedimensional acoustic wave equation are wellknown to be convergent. We present a comprehensive formalization of the simplest scheme and formally prove its convergence in Coq. The main difficulties lie in the proper definition of asymptotic behaviors and the implicit way they are handled in the mathematical penandpaper proofs. To our knowledge, this is the first time this kind of mathematical proof is machinechecked. Key words: partial differential equation, acoustic wave equation, numerical scheme, Coq formal proofs 1
Bridging the gap between formal specification and bitlevel floatingpoint arithmetic
"... Floatingpoint arithmetic is defined by the IEEE754 standard and has often been
formalized. We propose a new Coq formalization based on the bitlevel representation of the standard and we prove strong links between this new formalization and
a previous highlevel one. In this process, we have defin ..."
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Cited by 2 (0 self)
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Floatingpoint arithmetic is defined by the IEEE754 standard and has often been
formalized. We propose a new Coq formalization based on the bitlevel representation of the standard and we prove strong links between this new formalization and
a previous highlevel one. In this process, we have defined functions for any rounding mode described by the standard. Our library can now be applied to certify
both software and hardware. Developing results in those two dramatically different
directions, like no other formal development so far, guarantees that nothing was
forgotten or poorly specified in our formalization. It also lets us compare our work
with the existing bitlevel formalizations developed with other proof assistants.
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).