### Abstract

We present a coinductive framework for defining and reasoning about the infinitary analogues of equational logic and term rewriting in a uniform way. We define Equation found, the infinitary extension of a given equational theory =R, and →^{∞}, the standard notion of infinitary rewriting associated to a reduction relation →R, as follows: (Formula Presented) Equation found Here μ and ν are the least and greatest fixed-point operators, respectively, and (Formula Presented) Equation found The setup captures rewrite sequences of arbitrary ordinal length, but it has neither the need for ordinals nor for metric convergence. This makes the framework especially suitable for formalizations in theorem provers.

Original language | English |
---|---|

Article number | 3 |

Pages (from-to) | 1-44 |

Number of pages | 44 |

Journal | Logical Methods in Computer Science |

Volume | 14 |

Issue number | 1 |

DOIs | |

Publication status | Published - 10 Jan 2018 |

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### Keywords

- Coinduction
- Infinitary equational reasoning
- Infinitary rewriting

### Cite this

*Logical Methods in Computer Science*,

*14*(1), 1-44. [3]. https://doi.org/10.23638/LMCS-14(1:3)2018

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*Logical Methods in Computer Science*, vol. 14, no. 1, 3, pp. 1-44. https://doi.org/10.23638/LMCS-14(1:3)2018

**Coinductive foundations of infinitary rewriting and infinitary equational logic.** / Endrullis, Jörg; Hansen, Helle Hvid; Hendriks, Dimitri; Polonsky, Andrew; Silva, Alexandra.

Research output: Contribution to Journal › Article › Academic › peer-review

TY - JOUR

T1 - Coinductive foundations of infinitary rewriting and infinitary equational logic

AU - Endrullis, Jörg

AU - Hansen, Helle Hvid

AU - Hendriks, Dimitri

AU - Polonsky, Andrew

AU - Silva, Alexandra

PY - 2018/1/10

Y1 - 2018/1/10

N2 - We present a coinductive framework for defining and reasoning about the infinitary analogues of equational logic and term rewriting in a uniform way. We define Equation found, the infinitary extension of a given equational theory =R, and →∞, the standard notion of infinitary rewriting associated to a reduction relation →R, as follows: (Formula Presented) Equation found Here μ and ν are the least and greatest fixed-point operators, respectively, and (Formula Presented) Equation found The setup captures rewrite sequences of arbitrary ordinal length, but it has neither the need for ordinals nor for metric convergence. This makes the framework especially suitable for formalizations in theorem provers.

AB - We present a coinductive framework for defining and reasoning about the infinitary analogues of equational logic and term rewriting in a uniform way. We define Equation found, the infinitary extension of a given equational theory =R, and →∞, the standard notion of infinitary rewriting associated to a reduction relation →R, as follows: (Formula Presented) Equation found Here μ and ν are the least and greatest fixed-point operators, respectively, and (Formula Presented) Equation found The setup captures rewrite sequences of arbitrary ordinal length, but it has neither the need for ordinals nor for metric convergence. This makes the framework especially suitable for formalizations in theorem provers.

KW - Coinduction

KW - Infinitary equational reasoning

KW - Infinitary rewriting

UR - http://www.scopus.com/inward/record.url?scp=85055789501&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055789501&partnerID=8YFLogxK

U2 - 10.23638/LMCS-14(1:3)2018

DO - 10.23638/LMCS-14(1:3)2018

M3 - Article

VL - 14

SP - 1

EP - 44

JO - Logical Methods in Computer Science

JF - Logical Methods in Computer Science

SN - 1860-5974

IS - 1

M1 - 3

ER -