## Coinductive Proofs for Basic Real Computation Tie Hou

### BibTeX

@MISC{Pp_coinductiveproofs,

author = {Sa Pp and Wales Uk},

title = {Coinductive Proofs for Basic Real Computation Tie Hou},

year = {}

}

### OpenURL

### Abstract

Abstract. We describe two representations for real numbers, signed digit streams and Cauchy sequences. We give coinductive proofs for the correctness of functions converting between these two representations to show the adequacy of signed digit stream representation. We also show a coinductive proof for the correctness of a corecursive program for the average function with regard to the signed digit stream representation. We implemented this proof in the interactive proof system Minlog. Thus, reliable, corecursive functions for real computation can be guaranteed, which is very helpful in formal software development for real numbers.

### Citations

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(Show Context)
Citation Context ...cept of computability on infinite streams can be explained by means of ’Oracle Turing machine’ (Alan Turing ([15])). More recent accounts of the complexity of stream functions are studied by e.g. Ko (=-=[10]-=-) and Weihrauch ([16]). Hence, in order to show that the signed digit stream representation is adequate, we need to provide computable functions SDTC : SDR → CR and CTSD : CR → SDR, such that for all ... |

152 |
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(Show Context)
Citation Context ... on infinite streams can be explained by means of ’Oracle Turing machine’ (Alan Turing ([15])). More recent accounts of the complexity of stream functions are studied by e.g. Ko ([10]) and Weihrauch (=-=[16]-=-). Hence, in order to show that the signed digit stream representation is adequate, we need to provide computable functions SDTC : SDR → CR and CTSD : CR → SDR, such that for all r ∈ IR, 1. ∀(ds, k) ∈... |

123 |
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Citation Context ...there are computable back-and-forth translations between these two representations. The concept of computability on infinite streams can be explained by means of ’Oracle Turing machine’ (Alan Turing (=-=[15]-=-)). More recent accounts of the complexity of stream functions are studied by e.g. Ko ([10]) and Weihrauch ([16]). Hence, in order to show that the signed digit stream representation is adequate, we n... |

16 | R.: Computing with real numbers: I. The LFT approach to real number computation; II. A domain framework for computational geometry
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(Show Context)
Citation Context ... digits, base 2/3 with binary digits, nested sequences of rational intervals, Cauchy sequences, continued fractions ([8]), base golden-ratio with binary digits, and linear fractional transformations (=-=[6]-=-). Meanwhile, many algorithms have been proposed for real computations using these representations. However, few give formal proofs for the algorithms. More recently, Chirimar and Howe ([4]) represent... |

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(Show Context)
Citation Context ...roofs of correctness for some algorithms were shown. Formalisation of real numbers using corecursive streams as a coinductive type was discussed in [5], [3], [1] in the logical framework Coq. Lenisa (=-=[11]-=-) introduced set-theoretic generalizations of the coinduction proof principle in the view of bisimulation. However, the usual coinduction, based on bisimulation, is not expressive enough for the equal... |

13 |
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(Show Context)
Citation Context .... The definition of SDTC ′ is an instance of a well-known corecursion scheme for defining infinite streams. More general schemes of corecursion are discussed, for example, in recent work of Buchholz (=-=[2]-=-). Lemma 7 (Convert from SD to Cauchy) ∀ds,k,r,q[(ds, k) ∼ r ⇒ (SDTC ′ (k, q, ds),k+1)∼ c q + r] Proof. We use Lemma 1. We define g, f, h by g((ds, k),r) = ( (tail(ds),k − 1),r − 2k−1 · head(ds) ) , f... |

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Citation Context ...d function minimum and maximum. Only informal proofs of correctness for some algorithms were shown. Formalisation of real numbers using corecursive streams as a coinductive type was discussed in [5], =-=[3]-=-, [1] in the logical framework Coq. Lenisa ([11]) introduced set-theoretic generalizations of the coinduction proof principle in the view of bisimulation. However, the usual coinduction, based on bisi... |

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Citation Context ...ormations ([6]). Meanwhile, many algorithms have been proposed for real computations using these representations. However, few give formal proofs for the algorithms. More recently, Chirimar and Howe (=-=[4]-=-) represented real numbers by Cauchy sequences and implemented real analysis in Nuprl based on the type theory. Plume ([13]) gave algorithms for the basic arithmetic operations, transcendental functio... |

11 |
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Citation Context ...n, and function minimum and maximum. Only informal proofs of correctness for some algorithms were shown. Formalisation of real numbers using corecursive streams as a coinductive type was discussed in =-=[5]-=-, [3], [1] in the logical framework Coq. Lenisa ([11]) introduced set-theoretic generalizations of the coinduction proof principle in the view of bisimulation. However, the usual coinduction, based on... |

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Citation Context .... Hence, the adequacy of signed digit stream representation is proved. 5 Average of Signed Digit Streams The average function plays an important role as a tool to get other computable functions, e.g. =-=[7]-=-. In the following we define the average function on real numbers in the interval [-1, 1]. Then we give the coinduction proof of its correctness. In order to calculate the average of two signed digit ... |

5 | Completing the rationals and metric spaces in LEGO - Jones - 1992 |

3 | Coinduction in Coq
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Citation Context ...ction minimum and maximum. Only informal proofs of correctness for some algorithms were shown. Formalisation of real numbers using corecursive streams as a coinductive type was discussed in [5], [3], =-=[1]-=- in the logical framework Coq. Lenisa ([11]) introduced set-theoretic generalizations of the coinduction proof principle in the view of bisimulation. However, the usual coinduction, based on bisimulat... |

3 | Streaming Representation-Changers
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Citation Context ...–230, 2006. c○ Springer-Verlag Berlin Heidelberg 2006222 T. Hou base with negative digits, base 2/3 with binary digits, nested sequences of rational intervals, Cauchy sequences, continued fractions (=-=[8]-=-), base golden-ratio with binary digits, and linear fractional transformations ([6]). Meanwhile, many algorithms have been proposed for real computations using these representations. However, few give... |

3 | Formalising exact arithmetic in type theory - Niqui - 2005 |

2 |
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(Show Context)
Citation Context ... few give formal proofs for the algorithms. More recently, Chirimar and Howe ([4]) represented real numbers by Cauchy sequences and implemented real analysis in Nuprl based on the type theory. Plume (=-=[13]-=-) gave algorithms for the basic arithmetic operations, transcendental functions, integration, and function minimum and maximum. Only informal proofs of correctness for some algorithms were shown. Form... |

2 |
Inverting monotone continuous functions in constructive analysis
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(Show Context)
Citation Context ...ductive assertions is easy to implement in proof assistants like Minlog, Coq and so on. The average function for signed digit streams in this paper has been implemented in the Minlog system. See also =-=[14]-=- for other proof developments in Minlog based on the Cauchy sequence representation of real numbers. 1.1 Contributions The main contributions of this paper are: – (a) We define (in Section 2) a genera... |