## Closure Under Alpha-Conversion (1993)

Venue: | In The Informal Proceeding of the 1993 Workshop on Types for Proofs and Programs |

Citations: | 22 - 3 self |

### BibTeX

@INPROCEEDINGS{Pollack93closureunder,

author = {Randy Pollack},

title = {Closure Under Alpha-Conversion},

booktitle = {In The Informal Proceeding of the 1993 Workshop on Types for Proofs and Programs},

year = {1993},

pages = {313--332},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper appears in Types for Proofs and Programs: International Workshop TYPES'93, Nijmegen, May 1993, Selected Papers, LNCS 806. abstraction, compute a type for its body in an extended context; to compute a type for an application, compute types for its left and right components, and check that they match appropriately. Lets use the algorithm to compute a type for a = [x:ø ][x:oe]x. FAILURE: no rule applies because x 2 Dom (x:ø )

### Citations

709 | A framework for defining logics - Harper, Honsell, et al. - 1993 |

537 | Lambda calculi with types
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(Show Context)
Citation Context ...ions of Computer Science, University of Edinburgh, The King's Buildings, Edinburgh, EH9 3JZ, Scotland rap@dcs.ed.ac.uk 1 Introduction Consider an informal presentation of simply typed -calculus as in =-=[Bar92]. Le-=-aving out some of the details, let oe, �� range over simple types, x, y range over a class of term variables, and M , N , range over the Church-style terms. A statementshas the form M : oe, where ... |

397 | Explicit substitutions
- Abadi, Cardelli, et al.
- 1991
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Citation Context ...93] goes much further, not only using explicit names, but also explicit substitutions, i.e. making the notion of substitution a part of the formal system (as originally proposed for nameless terms in =-=[ACCL91]-=-). Unfortunately the system of [Tas93], in its current formulation, is not closed under alpha-conversion; e.g. it fails to derive judgement (7) in section 3. Finally, there is a recent proposal [Gor93... |

311 | Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser Theorem
- Bruijn
- 1972
(Show Context)
Citation Context ... we ignore the distinction between alphaconvertible terms, and treat then as equal. It is well known that one solution to the problems of alpha-conversion is the use of de Bruijn "nameless variab=-=les" [dB72]-=-. Although nameless variables have their partisans for use in metatheoretic study, even those partisans admit that the explanation of substitution of a term for a given variable is painful in such a p... |

109 |
An algorithm for testing conversion in type theory
- Coquand
(Show Context)
Citation Context ...syntax for pretty printing, and this translation itself must be formalized. I know of two recent proposals that take names seriously, but avoid the need for alpha-conversion. One proposal, by Coquand =-=[Coq91]-=-, follows a style in logic to distinguish between free variables (parameters) and bound variables (variables). This idea has been used to formalize a large theory of Pure Type Systems, including reduc... |

86 | ECC an Extended Calculus of Constructions - Luo - 1989 |

69 | The Theory of LEGO: A Proof Checker for the Extended Calculus of Constructions
- Pollack
- 1994
(Show Context)
Citation Context ...ubject of any premiss. Such a proof does not extend to the case of Pure Type Systems to be considered in section 3. Here I give a proof that does extend, and is better than the proof given in [MP93]. =-=[Pol94]-=- describes these three different ways to prove such an equivalence in excrutiating detail. We introduce the machinary of renaming. A renaming (ranged over by OE) is a function from parameters to param... |

44 | Modular proof of strong normalization for the calculus of construction - Geuvers, Nederhof - 1991 |

38 |
proof development system: User’s manual
- LEGO
- 1992
(Show Context)
Citation Context ...sion. 1.1 The Constructive Engine The Constructive Engine [Hue89] is an abstract machine for type checking the Calculus of Constructions. It is the basis for the proofcheckers Coq [DFH + 93] and LEGO =-=[LP92]-=-. Among its interesting aspects are: 1. the non-deterministic rules of the underlying type theory are converted into a deterministic, syntax-directed program 2. this syntax-directed program implements... |

33 |
A mechanisation of name-carrying syntax up to alpha-conversion
- Gordon
- 1994
(Show Context)
Citation Context ...CCL91]). Unfortunately the system of [Tas93], in its current formulation, is not closed under alpha-conversion; e.g. it fails to derive judgement (7) in section 3. Finally, there is a recent proposal =-=[Gor93]-=- of a formalization mixing nameless terms and named variables in such a way that named terms are equal up to alpha-conversion. 1.1 The Constructive Engine The Constructive Engine [Hue89] is an abstrac... |

29 | Residual theory in -calculus: A formal development - Huet - 1993 |

29 | Checking Algorithms for Pure Type Systems - Jutting, S, et al. - 1993 |

28 | Type Dependence and Constructive Mathematics - Berardi - 1989 |

25 |
The constructive engine
- Huet
- 1989
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Citation Context ...cent proposal [Gor93] of a formalization mixing nameless terms and named variables in such a way that named terms are equal up to alpha-conversion. 1.1 The Constructive Engine The Constructive Engine =-=[Hue89]-=- is an abstract machine for type checking the Calculus of Constructions. It is the basis for the proofcheckers Coq [DFH + 93] and LEGO [LP92]. Among its interesting aspects are: 1. the non-determinist... |

22 | Implicit syntax
- Pollack
- 1990
(Show Context)
Citation Context ...isses of the rule, whose inputs are computed from the given inputs to the conclusion), and return a result computed from the results of the premisses". We have called such systems translation sys=-=tems [Pol90]. There -=-is one difficulty remaining, with the rule for lambda terms. Following the ce-app example, it should be \Gamma; x:oe ` M )M [x=0] : �� \Gamma ` [x:oe]M ) [oe]M : oe ! �� x 62 M but it is not c... |

17 | Formulation of Martin-Löf’s theory of types with explicit substitutions
- Tasistro
- 1993
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Citation Context ... conversion and typing [MP93]. This formalization does distinguish between alpha-convertible terms, and the typing judgement is indeed closed under alpha-conversion. The other proposal, by Martin-Lof =-=[Tas93]-=- goes much further, not only using explicit names, but also explicit substitutions, i.e. making the notion of substitution a part of the formal system (as originally proposed for nameless terms in [AC... |

13 |
Pure Type Sytems formalized
- McKinna, Pollack
- 1993
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Citation Context ...o distinguish between free variables (parameters) and bound variables (variables). This idea has been used to formalize a large theory of Pure Type Systems, including reduction, conversion and typing =-=[MP93]-=-. This formalization does distinguish between alpha-convertible terms, and the typing judgement is indeed closed under alpha-conversion. The other proposal, by Martin-Lof [Tas93] goes much further, no... |

12 | Generalising automath by means of a lambda-typed lambda calculus - Bruijn - 1984 |

7 | A formalization of the strong normalization proof for system f in lego - Altenkirch - 1993 |

5 | Benthem Jutting. Typing in Pure Type Sytems - van - 1993 |

4 | Introduction to Generalised Type Sytems - Barendregt - 1991 |

4 | Typechecking in Pure Type Sytems - Pollack - 1992 |