## Pure Type Systems with Definitions (1993)

Venue: | In Logical Foundations of Computer Science '94, number 813 in LNCS |

Citations: | 20 - 1 self |

### BibTeX

@INPROCEEDINGS{Severi93puretype,

author = {Paula Severi and Erik Poll},

title = {Pure Type Systems with Definitions},

booktitle = {In Logical Foundations of Computer Science '94, number 813 in LNCS},

year = {1993},

pages = {316--328},

publisher = {Springer Verlag}

}

### OpenURL

### Abstract

In this paper, an extension of Pure Type Systems (PTS's ) to include definitions is presented and the meta-theory of these PTS's with definitions is treated in detail. We prove that all the properties of a PTS are preserved by the extension. 1 Introduction A PTS does not provide the possibility to introduce a definition, i.e. an abbreviation (name) for a larger term which can be used several times in a program or proof. A definition mechanism is essential for practical use, and indeed implementantions of PTS's such as Coq [ea91], Lego [LP92] or Constructor [Hel91] do provide such facility, even though the formal definition of the systems they implement do not. In this chapter, we introduce an extension of the PTS with (non-recursive) definitions. The extension of a PTS with definitions looks very harmless and this may not seem a topic worthy of investigation. Howewer the local definitions complicate matters and it is an open problem whether extending an arbitrary PTS with defintions p...

### Citations

696 | A Framework for Defining Logics - Harper, Honsel, et al. - 1993 |

565 | Term Rewriting Systems
- Klop
- 1992
(Show Context)
Citation Context ... Rosser for fi, ffi and fi ffi-reductions The proof of Church Rosser for fi-reduction is very easy. It follows from the fact that the Combinatorial Reduction System ! T ffi ; ! fi ? is orthogonal(see =-=[Klo90]-=-). Theorem 4.14. (Church Rosser for fi-reduction) Let \Gamma 2 C ffi and a; a 1 ; a 2 2 T ffi such that \Gamma ` a !! fi a 1 and \Gamma ` a !! fi a 2 . Then there exists a term a 3 such that \Gamma ` ... |

518 | Lambda calculi with types
- Barendregt
- 1992
(Show Context)
Citation Context ...ization apply only to the particular type system that they consider and do not extend to other type systems. 2 Pure Type Systems In this chapter the concept of Pure Type System (PTS) is defined as in =-=[Bar92a]-=-. Only the notion of topsort and the system C1 (see definitions 2.11 and 2.14) are not defined in [Bar92a]. Definition 2.1. The set T of pseudoterms and the set C of contexts are defined as follows: T... |

471 | The calculus of constructions - Coquand, Huet - 1988 |

364 | Towards a theory of type structure - Reynolds - 1974 |

285 | Interpre'tation fonctionelle et e'limination des coupures duns l'arithme'tique d'ordre supe'rieure - Girard - 1972 |

90 |
A survey of the project AUTOMATH
- Bruijn
- 1980
(Show Context)
Citation Context ...other (larger) PTS. This enables us to prove that for all strongly normalizing PTS's that we know the extensions with definitions are also strongly normalizing. In the systems of AUTOMATH family (see =-=[dB80]-=-) definitions are considered as part of the formal language. The meta-theory of these systems-including strong normalization - is treated in detail in [Daa80]. Howewer, the proofs of strong normalizat... |

67 | An Analysis of Girard’s Paradox - Coquand - 1986 |

43 |
Modular proof of strong normalization for the calculus of constructions
- Geuvers, Nederhof
- 1991
(Show Context)
Citation Context ... Worse still, proving strong normalization for particular PTS's extended with definitions is already a problem. The strong normalization proofs for particular type systems given in [Coq85], [Luo89a], =-=[GN91]-=-, [Bar92b] cannot be extended in any obvious way. In this chapter, we show how strong normalization of a PTS extended with definitions follows from strong normalization of another (larger) PTS. This e... |

40 |
Une Théorie des Constructions
- Coquand
- 1985
(Show Context)
Citation Context ...rmalization or not. Worse still, proving strong normalization for particular PTS's extended with definitions is already a problem. The strong normalization proofs for particular type systems given in =-=[Coq85]-=-, [Luo89a], [GN91], [Bar92b] cannot be extended in any obvious way. In this chapter, we show how strong normalization of a PTS extended with definitions follows from strong normalization of another (l... |

36 |
Proof Development System: User’s Manual
- LEGO
- 1992
(Show Context)
Citation Context ...on (name) for a larger term which can be used several times in a program or proof. A definition mechanism is essential for practical use, and indeed implementantions of PTS's such as Coq [ea91], Lego =-=[LP92]-=- or Constructor [Hel91] do provide such facility, even though the formal definition of the systems they implement do not. In this chapter, we introduce an extension of the PTS with (non-recursive) def... |

36 |
extended calculus of constructions
- ECC
- 1989
(Show Context)
Citation Context ...on or not. Worse still, proving strong normalization for particular PTS's extended with definitions is already a problem. The strong normalization proofs for particular type systems given in [Coq85], =-=[Luo89a]-=-, [GN91], [Bar92b] cannot be extended in any obvious way. In this chapter, we show how strong normalization of a PTS extended with definitions follows from strong normalization of another (larger) PTS... |

34 |
The Language Theory of Automath
- Daalen
- 1980
(Show Context)
Citation Context ...ng. In the systems of AUTOMATH family (see [dB80]) definitions are considered as part of the formal language. The meta-theory of these systems-including strong normalization - is treated in detail in =-=[Daa80]-=-. Howewer, the proofs of strong normalization apply only to the particular type system that they consider and do not extend to other type systems. 2 Pure Type Systems In this chapter the concept of Pu... |

23 |
Benthem Jutting. Typing in pure type systems
- van
- 1993
(Show Context)
Citation Context ... singly sorted specifications) Let S ffi be a functional DPTS. \Gamma 1 ; x:A; \Gamma 2 ` b : B x = 2 FV (\Gamma 2 ) [ FV (b) [ FV (B) 9 = ; ) \Gamma 1 ; \Gamma 2 ` b : B Arbitrary Specifications. In =-=[vBJ93]-=- the set T is partitioned into the sets T V and TS . We follow the main branch of a term. If we find a term of the form \Pix:A:B or a constant then the term will be in TS . If we find neither a produc... |

19 | A modular proof of strong normalisation for the calculus of constructions - Geuvers, Nederhof - 1991 |

10 | A direct proof of the finite developments theorem - Vrijer - 1985 |

6 |
et al. The coq proof assistant version 5.6, users guide
- Dowek
- 1991
(Show Context)
Citation Context ...an abbreviation (name) for a larger term which can be used several times in a program or proof. A definition mechanism is essential for practical use, and indeed implementantions of PTS's such as Coq =-=[ea91]-=-, Lego [LP92] or Constructor [Hel91] do provide such facility, even though the formal definition of the systems they implement do not. In this chapter, we introduce an extension of the PTS with (non-r... |

6 |
Goal directed proof construction in type theory
- Helmink
- 1991
(Show Context)
Citation Context ... term which can be used several times in a program or proof. A definition mechanism is essential for practical use, and indeed implementantions of PTS's such as Coq [ea91], Lego [LP92] or Constructor =-=[Hel91]-=- do provide such facility, even though the formal definition of the systems they implement do not. In this chapter, we introduce an extension of the PTS with (non-recursive) definitions. The extension... |

1 |
Klop Henk Barendregt, Jan Bergstra and Henri Volken. Degrees, reductions and representability in the lambda calculus
- Willem
- 1976
(Show Context)
Citation Context ...iy:A B[x 0 := d]) otherwise (y:A: a)[x 0 := d] = ae (y:A[x 0 := d] a) if x 2 A (y:A a[x 0 := d]) otherwise We define a strategy similar to the perpetual strategy F1 on -terms and for fi-reduction(see =-=[HBV76]-=-). The leftmost ffi -redex of a term could be either a global or a local definition. If the leftmost redex is a variable x such that x=a:A is in the context then the strategy gives the term obtained b... |