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Arbitrary Precision Real Arithmetic: Design and Algorithms
, 1996
"... this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms fo ..."
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this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms for rational operations and transcendental functions for this representation. Finally we describe briefly the prototype written in Caml. 2. Computable real numbers
Program Extraction from Nested Definitions
"... Abstract. Minlog is a proof assistant which automatically extracts computational content in an extension of Gödel’s T from formalized proofs. We report on extending Minlog to deal with predicates defined using a particular combination of induction and coinduction, via socalled nested definitions. I ..."
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Abstract. Minlog is a proof assistant which automatically extracts computational content in an extension of Gödel’s T from formalized proofs. We report on extending Minlog to deal with predicates defined using a particular combination of induction and coinduction, via socalled nested definitions. In order to increase the efficiency of the extracted programs, we have also implemented a feature to translate terms into Haskell programs. To illustrate our theory and implementation, a formalisation of a theory of uniformly continuous functions due to Berger is presented. 1
and Helmut Schwichtenberg1
"... Abstract. Minlog is a proof assistant which automatically extracts computational content in an extension of Gödel’s T from formalized proofs. We report on extending Minlog to deal with predicates defined using a particular combination of induction and coinduction, via socalled nested definitions. ..."
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Abstract. Minlog is a proof assistant which automatically extracts computational content in an extension of Gödel’s T from formalized proofs. We report on extending Minlog to deal with predicates defined using a particular combination of induction and coinduction, via socalled nested definitions. In order to increase the efficiency of the extracted programs, we have also implemented a feature to translate terms into Haskell programs. To illustrate our theory and implementation, a formalisation of a theory of uniformly continuous functions due to Berger is presented. 1
PROGRAM EXTRACTION IN EXACT REAL ARITHMETIC KENJI MIYAMOTO AND HELMUT SCHWICHTENBERG
"... Abstract. The importance of an abstract approach to a computation theory over general data types has been stressed by John Tucker in many of his papers. Ulrich Berger and Monika Seisenberger recently reals. They considered a proof involving coinduction of the proposition that any two reals in [−1, 1 ..."
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Abstract. The importance of an abstract approach to a computation theory over general data types has been stressed by John Tucker in many of his papers. Ulrich Berger and Monika Seisenberger recently reals. They considered a proof involving coinduction of the proposition that any two reals in [−1, 1] have their average in the same interval, and informally extract a Haskell program from this proof, which works with stream representations of reals. Here we formalize the proof, and machineextract its computational content using the Minlog proof assistant. This required an extension of this system to also take coinduction into account.
Arbitrary precision real arithmetic: design and algorithms Valerie MenissierMorain
"... We describe here a representation of computable real numbers and a set of algorithms for the elementary functions associated to this representation. A real number is represented as a sequence of nite Badic numbers and for each classical function (rational, algebraic or transcendental), we describe ..."
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We describe here a representation of computable real numbers and a set of algorithms for the elementary functions associated to this representation. A real number is represented as a sequence of nite Badic numbers and for each classical function (rational, algebraic or transcendental), we describe how to produce a sequence representing the result of the application of this function to its arguments, according to the sequences representing these arguments. For each algorithm we prove that the resulting sequence is a valid representation of the exact real result. This arithmetic is the rst abritrary precision real arithmetic with mathematically proved algorithms. Resume Nous proposons une representation des nombres reels calculables ainsi que des algorithmes pour les fonctions elementaires usuelles pour cette representation. Un nombre reel est represente par une suite de nombres Badiques nis et pour chaque fonction classique (rationnelle, algebrique ou transcendante), nous montrons comment produire une suite representant le resultat a partir de suites representant les parametres. Pour chacun de ces algorithmes nous demontrons que la suite qui en resulte represente bien le resultat reel exact. Cette arithmetique est la premiere arithmetique reelle en precision arbitraire dotee d'un jeu complet d'algorithmes