## Coinductive Field of Exact Real Numbers and General Corecursion (2006)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Niqui06coinductivefield,

author = {Milad Niqui},

title = {Coinductive Field of Exact Real Numbers and General Corecursion},

year = {2006}

}

### OpenURL

### Abstract

In this article we present a method to define algebraic structure (field operations) on a representation of real numbers by coinductive streams. The field operations will be given in two algorithms (homographic and quadratic algorithm) that operate on streams of Möbius maps. The algorithms can be seen as coalgebra maps on the coalgebra of streams and hence they will be formalised as general corecursive functions. We use the machinery of Coq proof assistant for coinductive types to present the formalisation.

### Citations

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Citation Context ...s of I. This means that every occurrence of f in the body of f should be direct argument of one of the constructors of I. This condition is due to Giménez [18] and is based on earlier work of Coquand =-=[10]-=-. A precise definition of G can be found in [18, p. 175]. Like other syntactic extensions of coiteration scheme, the guardedness condition of Coq is too restrictive a requirement to allow for formalis... |

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Citation Context ...lus of productivity mh will be a nested function too. It is well-known that applying Bove–Capretta method for formalising nested recursive functions requires the presence of inductive–recursive types =-=[5,13]-=-. In this case the inductive domain predicate will become an inductive–recursive predicate that is defined simultaneously with the nested function. A similar phenomenon happens in our method, in the s... |

64 |
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Citation Context ...tructures. In any case, theoretically this does not change the coalgebraic semantics and the coinductive types can still be interpreted as weakly final coalgebras in any categorical model of CIC (see =-=[1]-=- where a stronger results is proven). Furthermore, the usual coiteration and corecursion schemes can be derived in terms of cofix operator [17]. Therefore in this article we present our method using t... |

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Citation Context ... real numbers is formalised as a coinductive type together with algebraic structure (field operations). However our works is different in many aspects: we use the more general setting of Edalat et al =-=[14]-=- for simultaneously defining algebraic operations of +,×,− and division in one algorithm (quadratic algorithm). As a result our method relies on formalising a non-syntactically productive function 3 A... |

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Citation Context ...arguments. In order to formalise such function in constructive type theory, there is a method of adding an inductive domain predicate introduced in [12] and extensively developed by Bove and Capretta =-=[5]-=-. According to this method we need to define an inductively defined predicate Eh(µ,α) with the intended meaning that µ and α are in the domain of mh which in turn means that the homographic algorithm ... |

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Citation Context ... A final coalgebras with the uniqueness property dropped. 3sNiqui for which we use a method that we call general corecursion. This is related to (but different from) the method presented by Bertot in =-=[4]-=- for formalising Eratosthenes’ sieve. Moreover, our formalisation of the homographic and quadratic algorithm are the first step towards the formalisation of the very powerful normalisation algorithm o... |

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Citation Context ... ever expanding class of specifications. Among these schemes are corecursion [16], dual of course of value recursion [31], T-coiteration for pointed functors [21], λ-coiteration for distributive laws =-=[2]-=- and bialgebraic T-coiteration [6]. But these schemes have one thing in common: they all impose some syntactic criterion for the class of specifications that they are capable of handling; while the pr... |

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Citation Context ...uction rules. Working extensionally, or in a setoid category (with bisimulation as setoid equality), one can formulate the uniqueness property [7, p. 74]. Our works is similar to the formalisation in =-=[8,3]-=- where the stream of real numbers is formalised as a coinductive type together with algebraic structure (field operations). However our works is different in many aspects: we use the more general sett... |

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Citation Context ...tions. Among these schemes are corecursion [16], dual of course of value recursion [31], T-coiteration for pointed functors [21], λ-coiteration for distributive laws [2] and bialgebraic T-coiteration =-=[6]-=-. But these schemes have one thing in common: they all impose some syntactic criterion for the class of specifications that they are capable of handling; while the productivity of the algorithms on re... |

8 |
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Citation Context ...g their specification in a syntax based on Haskell programming language, and would like to find their corresponding coalgebra maps. But this task is not easy mainly due to the problem of productivity =-=[11]-=-. Productive functions are those functions on infinite objects that produce provably infinite output. The basic coiteration scheme which is obtained from the universality of final coalgebra can be use... |

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Citation Context ...recursive functions with non-structurally recursive arguments. In order to formalise such function in constructive type theory, there is a method of adding an inductive domain predicate introduced in =-=[12]-=- and extensively developed by Bove and Capretta [5]. According to this method we need to define an inductively defined predicate Eh(µ,α) with the intended meaning that µ and α are in the domain of mh ... |

3 | Coinduction in Coq
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Citation Context ...uction rules. Working extensionally, or in a setoid category (with bisimulation as setoid equality), one can formulate the uniqueness property [7, p. 74]. Our works is similar to the formalisation in =-=[8,3]-=- where the stream of real numbers is formalised as a coinductive type together with algebraic structure (field operations). However our works is different in many aspects: we use the more general sett... |