## Polishing Up the Tait-Martin-Löf Proof of the Church-Rosser Theorem (1995)

Citations: | 6 - 0 self |

### BibTeX

@MISC{Pollack95polishingup,

author = {Robert Pollack},

title = {Polishing Up the Tait-Martin-Löf Proof of the Church-Rosser Theorem},

year = {1995}

}

### OpenURL

### Abstract

Introduction The Tait--Martin-Lof proof is the best known and simplest proof of confluence (the Church--Rosser theorem) for various lambda calculi. It is explained in detail, for example, in [Bar84, HS86, Rev88]. The desire to clarify this proof has inspired work on concrete representation of binding [dB72, Coq91]. Perhaps the best modern version is given in [Tak95]. Formal proofs are reported in [Hue94, MP93, Pfe92, Sha88] 1 . In this note I outline the innovation given in [Tak95] (and formalized by McKinna [MP93]), and present a further improvement which I believe has not appeared in the literature before. 1.1 Preliminary Definitions Let Rel2 be the class of binary relations, and R; T 2 Rel2 ; we write aRb for (a; b) 2 R . For R 2 Rel2 the transitive reflexive closure of R , wri

### Citations

310 | Lambda Calculus Notation with Nameless Dummies, A Tool for Automatic Formula Manipulation, with Application to the Church-Rosser Theorem - Bruijn - 1972 |

291 | The Lambda Calculus. Its Syntax and Semantics, volume 103 - Barendregt - 1984 |

109 | An algorithm for testing conversion in type theory - Coquand |

69 | The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions
- Pollack
- 1994
(Show Context)
Citation Context ...ussed in section 2, where I present a proof that I believe is new. 3. Showing AE = ! . This is usually considered trivial, but in fact this is where the names of variables can become problematic (see =-=[Pol94]-=-). I will not discuss this in the present paper except to note that AE is anyway a nicer relation to work with than !, and might be taken as more fundamental for coarse reasoning about reduction such ... |

38 |
A Mechanical Proof of the Church-Rosser Theorem
- Shankar
- 1988
(Show Context)
Citation Context ...3 Parallel Reduction has the Diamond Property Lemma 3.1 dp(AE). The relation AE was defined specifically to meet the two problems observed in attempting to show dp(!). Given a AE b & a AE c , Shankar =-=[Sha88] does &quo-=-t;structural induction" (I quote) on a with case analysis on the two reductions. For example if a is an application there are five subcases: 1. a is not a redex 2. a is a redex, and is only contr... |

36 | A proof of the Church-Rosser theorem and its representation in a logical framework
- Pfenning
- 1992
(Show Context)
Citation Context ...term of a, but a subterm of the left subterm of a.) Now by the substitution lemma (I omit some details) we have [a b r =x]a b l AE [d r =x]d l and [a c r =x]a c l AE [d r =x]d l as required. Pfenning =-=[Pfe92] does &quo-=-t;simultaneous induction on the structure of the reductions" a AE b and a AE c. His argument for the case discussed above proceeds the same way as Shankar's except that the two uses of IH diagram... |

28 |
Residual theory in -calculus: A formal development
- Huet
- 1993
(Show Context)
Citation Context ...ly the redexes that are necessary to bring b and c together. This analysis is not necessary for the Church-Rosser theorem, as the definition of confluence says nothing about "minimal reduction&qu=-=ot;. Huet [Hue94]-=- proves a more subtle theorem and uses it to prove confluence as a corollary, but Shankar and Pfenning do not state the full meaning of their proofs. 3.1 A Coarser Proof Takahashi [Tak95] gives a proo... |

13 |
Pure Type Sytems formalized
- McKinna, Pollack
- 1993
(Show Context)
Citation Context ...]. Perhaps the best modern version is given in [Tak95]. Formal proofs are reported in [Hue94, MP93, Pfe92, Sha88] 1 . In this note I outline the innovation given in [Tak95] (and formalized by McKinna =-=[MP93]-=-), and present a further improvement which I believe has not appeared in the literature before. 1.1 Preliminary Definitions Let Rel2 be the class of binary relations, and R; T 2 Rel2 ; we write aRb fo... |

6 | Lambda-Calculus and Combinators - Hindley, Seldin - 2008 |

4 |
Parallel reductions in W -calculus (Revised version
- Takahashi
- 1995
(Show Context)
Citation Context ...ed in detail, for example, in [Bar84, HS86, Rev88]. The desire to clarify this proof has inspired work on concrete representation of binding [dB72, Coq91]. Perhaps the best modern version is given in =-=[Tak95]-=-. Formal proofs are reported in [Hue94, MP93, Pfe92, Sha88] 1 . In this note I outline the innovation given in [Tak95] (and formalized by McKinna [MP93]), and present a further improvement which I bel... |