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QArith: Coq formalisation of lazy rational arithmetic
 Types for Proofs and Programs, volume 3085 of LNCS
, 2003
"... Abstract. In this paper we present the Coq formalisation of the QArith library which is an implementation of rational numbers as binary sequences for both lazy and strict computation. We use the representation also known as the SternBrocot representation for rational numbers. This formalisation use ..."
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Abstract. In this paper we present the Coq formalisation of the QArith library which is an implementation of rational numbers as binary sequences for both lazy and strict computation. We use the representation also known as the SternBrocot representation for rational numbers. This formalisation uses advanced machinery of the Coq theorem prover and applies recent developments in formalising general recursive functions. This formalisation highlights the rôle of type theory both as a tool to verify handwritten programs and as a tool to generate verified programs. 1
Program Extraction from Large Proof Developments
, 2003
"... It is well known that mathematical proofs often contain (abstract) algorithms, but although these algorithms can be understood by a human, it still takes a lot of time and effort to implement this algorithm on a computer; moreover, one runs the risk of making mistakes in the process. From a fully... ..."
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It is well known that mathematical proofs often contain (abstract) algorithms, but although these algorithms can be understood by a human, it still takes a lot of time and effort to implement this algorithm on a computer; moreover, one runs the risk of making mistakes in the process. From a fully...
Admissible Digit Sets and a Modified SternBrocot Representation
, 2004
"... We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a nite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sucient conditions that such a "dig ..."
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We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a nite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sucient conditions that such a "digit set" yields an admissible representation of [0; +1]. Furthermore we establish the productivity and correctness of the homographic algorithm for such "admissible" digit sets. In the second part of the paper we discuss representation of positive real numbers based on the SternBrocot tree. We show how we can modify the usual SternBrocot representation to yield a ternary admissible digit set.
On Progress of Investigations in Continued Logarithm Arithmetic ∗
"... Abstract. This is a workinprogress on research of exact real arithmetic using continued fraction paradigm. We introduce a new redundant extension of continued logarithm representation, which offers the lacking real number computability. In a uniform way, we contrast this new representation with ex ..."
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Abstract. This is a workinprogress on research of exact real arithmetic using continued fraction paradigm. We introduce a new redundant extension of continued logarithm representation, which offers the lacking real number computability. In a uniform way, we contrast this new representation with existing alternatives and using few examples we discuss their practical aspects. It follows from this comparison that continued logarithms have a severe competitor in a redundant admissible, continued fraction representation, but they are still worth to continue in their research.
Generalizations of the floor and ceiling functions using the SternBrocot tree
"... We consider a fundamental number theoretic problem where practial applications abound. We decompose any rational number a b in c ratios as evenly as possible while maintaining the sum of numerators and the sum of denominators. The minimum � � a b c and maximum � � a of the ratios b c give rational ..."
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We consider a fundamental number theoretic problem where practial applications abound. We decompose any rational number a b in c ratios as evenly as possible while maintaining the sum of numerators and the sum of denominators. The minimum � � a b c and maximum � � a of the ratios b c give rational estimates of a b from below and from above. The case c = b gives the usual floor and ceiling functions. We furthermore define the difference � � a, whichiszeroiff c ≤ GCD(a, b), quantifying the distance b c to relative primality. A main tool for investigating the properties of � � a b c, � � a b c and � � a b c is the SternBrocot tree, where all positive rational numbers occur in lowest terms and in size order. We prove basic properties such that there is a unique decomposition that gives both � � a b c and � � a. It turns out that b c this decomposition contains at most three distinct ratios. The problem has arisen in a generalization of the 4/3−conjecture in computer science.
Admissible Digit Sets and a Modified Stern–Brocot Representation
"... We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a finite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sufficient conditions that such a “dig ..."
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We examine a special case of admissible representations of the closed interval, namely those which arise via sequences of a finite number of Möbius transformations. We regard certain sets of Möbius transformations as a generalized notion of digits and introduce sufficient conditions that such a “digit set ” yields an admissible representation of [0, +∞]. Furthermore we establish the productivity and correctness of the homographic algorithm for such “admissible” digit sets. In the second part of the paper we discuss representation of positive real numbers based on the Stern–Brocot tree. We show how we can modify the usual Stern–Brocot representation to yield a ternary admissible digit set.