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An exercise in coinduction: Moessner’s theorem
"... We present a coinductive proof of Moessner’s theorem. This theorem describes the construction of the stream (1 n, 2 n, 3 n,...) (for n ≥ 1) out of the stream of natural numbers by repeatedly dropping and summing elements. Our formalisation consists of a direct translation of the operational descript ..."
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We present a coinductive proof of Moessner’s theorem. This theorem describes the construction of the stream (1 n, 2 n, 3 n,...) (for n ≥ 1) out of the stream of natural numbers by repeatedly dropping and summing elements. Our formalisation consists of a direct translation of the operational description of Moessner’s procedure into the equivalence of in essence two functional programs. Our proof fully exploits the circularity that is present in Moessner’s procedure and is more elementary than existing proofs. As such, it serves as a nontrivial illustration of the relevance and power of coinduction. 1.
Representing Contractive Functions on Streams
 UNDER CONSIDERATION FOR PUBLICATION IN THE JOURNAL OF FUNCTIONAL PROGRAMMING
, 2011
"... Streams, or infinite lists, have many applications in functional programming, and are naturally defined using recursive equations. But how do we ensure that such equations make sense, i.e. that they actually produce welldefined streams? In this article we present a new approach to this problem, bas ..."
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Cited by 1 (0 self)
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Streams, or infinite lists, have many applications in functional programming, and are naturally defined using recursive equations. But how do we ensure that such equations make sense, i.e. that they actually produce welldefined streams? In this article we present a new approach to this problem, based upon the topological notion of contractive functions on streams. In particular, we give a sound and complete representation theorem for contractive functions on streams, illustrate the use of this theorem as a practical means to produce welldefined streams, and show how the efficiency of the resulting definitions can be improved using another representation of contractive functions.
DOI 10.1007/s1099001290827 A proof of Moessner’s theorem by coinduction
, 2012
"... © The Author(s) 2012. This article is published with open access at Springerlink.com Abstract We present a coinductive proof of Moessner’s theorem. This theorem describes the construction of the stream (1 n, 2 n, 3 n,...)(for n ≥ 1) out of the stream of positive natural numbers by repeatedly droppin ..."
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© The Author(s) 2012. This article is published with open access at Springerlink.com Abstract We present a coinductive proof of Moessner’s theorem. This theorem describes the construction of the stream (1 n, 2 n, 3 n,...)(for n ≥ 1) out of the stream of positive natural numbers by repeatedly dropping and summing elements. Our formalisation consists of a direct translation of the operational description of Moessner’s procedure into the equivalence of—in essence—two functional programs. Our proof fully exploits the circularity that is implicitly present in Moessner’s procedure, and it is more elementary than existing proofs. As such, it serves as a nontrivial illustration of the relevance and power of coinduction.
Representing Contractive Functions on Streams (Extended Version)
 UNDER CONSIDERATION FOR PUBLICATION IN THE JOURNAL OF FUNCTIONAL PROGRAMMING
, 2011
"... Streams, or infinite lists, have many applications in functional programming, and are naturally defined using recursive equations. But how do we ensure that such equations make sense, i.e. that they actually produce welldefined streams? In this article we present a new approach to this problem, bas ..."
Abstract
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Streams, or infinite lists, have many applications in functional programming, and are naturally defined using recursive equations. But how do we ensure that such equations make sense, i.e. that they actually produce welldefined streams? In this article we present a new approach to this problem, based upon the topological notion of contractive functions on streams. In particular, we give a sound and complete representation theorem for contractive functions on streams, illustrate the use of this theorem as a practical means to produce welldefined streams, and show how the efficiency of the resulting definitions can be improved using another representation of contractive functions.