## A Proof of Strong Normalization For the Theory of Constructions Using a Kripke-Like Interpretation (1990)

Venue: | In Workshop on Logical Frameworks--Preliminary Proceedings |

Citations: | 20 - 0 self |

### BibTeX

@INPROCEEDINGS{Coquand90aproof,

author = {Thierry Coquand and Jean Gallier and Le Chesnay Cedex},

title = {A Proof of Strong Normalization For the Theory of Constructions Using a Kripke-Like Interpretation},

booktitle = {In Workshop on Logical Frameworks--Preliminary Proceedings},

year = {1990}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We give a proof that all terms that type-check in the theory of constructions are strongly normalizing (under fi-reduction). The main novelty of this proof is that it uses a "Kripke-like" interpretation of the types and kinds, and that it does not use infinite contexts. We explore some consequences of strong normalization, consistency and decidability of typechecking. We also show that our proof yields another proof of strong normalization for LF (under fi-reduction), using the reducibility method. 1 Introduction We give a proof that all terms that type-check in the theory of constructions are strongly normalizing (under fi-reduction). The main novelty of this proof is that it uses a "Kripkelike " interpretation of the types and kinds, and that it does not use infinite contexts. The idea used for avoiding infinite contexts comes from Coquand's thesis [Coq85]. Our proof yields as a corollary another proof of strong normalization (under fi-reduction) of well-formed terms of LF . In f...

### Citations

696 | A Framework for Defining Logics
- Harper, Honsel, et al.
- 1993
(Show Context)
Citation Context ...istence of the canonical sets and functions, which itself depends on the existence of the measure c(K), where K a kind. This is not surprising in view of another proof by Harper, Honsell, and Plotkin =-=[HHP89]-=-, in which a mapping from LF into the simply-typed -calculus is used. It should be noted that their proof applies to fi and j reduction, but we do not know presently how to extend our approach to j-re... |

471 |
The calculus of constructions
- Coquand, Huet
- 1988
(Show Context)
Citation Context ...t. Thus, in order to avoid ambiguities, we formulate in the next section a version of the theory of constructions equivalent (but not identical in syntax) to the version presented by Coquand and Huet =-=[CH88]-=-. We refer to this version of the theory of constructions as CC. 2 Syntax of the Theory of Constructions CC We find it pedagogically convenient to first describe a theory of constructions whose syntax... |

285 |
Interpre'tation fonctionelle et e'limination des coupures duns l'arithme'tique d'ordre supe'rieure
- Girard
- 1972
(Show Context)
Citation Context ...fs and type families are disjoint. 6 5 Strong Normalization in CC The proof of strong normalization for well-typed terms of CC is obtained by generalizing the proof given by Girard for the system F ! =-=[Gir72]-=-, as presented in Gallier [Gal90]. However, there are some significant technical complications. In F ! , we have an ascending hierarchy, kinds, type families, and terms, where kinds do not depend on t... |

60 |
Metamathematical investigations of a calculus of constructions. Rapport de recherche de l’INRIA
- Coquand
- 1989
(Show Context)
Citation Context ...ample, it is logical to define "the values" before defining [[\Gamma . A]]ae\Delta, the interpretation of types, since this latter definition requires the former (the term "values"=-= is used by Coquand [Coq87], but in G-=-irard's terminology and ours, these are the sets C A;\Delta of "candidates of reducibility"). However, we believe that it more intuitive and easier for understanding such proofs, if [[\Gamma... |

44 | Kripke-style models for typed lambda calculus
- Mitchell, Moggi
- 1991
(Show Context)
Citation Context ...e thoroughly the nature of Kripke models for the theory of constructions. Among the sources of inspiration for this research, Moggi and Mitchell's work on Kripke models for the simply-typed -calculus =-=[MM87] should be-=- mentioned. We also became recently aware of work by Aarne Ranta [Ran90], in which the notion of "contexts as (finite approximation of) worlds" is used. One of the motivations for this work ... |

43 |
Modular proof of strong normalization for the calculus of constructions
- Geuvers, Nederhof
- 1991
(Show Context)
Citation Context ...o, 1989 [Luo90]. There is apparently a proof of strong normalization for an extension of CC with universes, given in Luo's thesis. We do not have this document yet. 7. Geuvers and Nederhof, June 1989 =-=[GN89]-=-. The authors present what they call a modular proof of strong normalization, by reducing strong normalization in CC to strong normalization in Girard's F ! . This is accomplished by defining a mappin... |

40 |
Une Théorie des Constructions
- Coquand
- 1985
(Show Context)
Citation Context ... this proof is that it uses a "Kripkelike " interpretation of the types and kinds, and that it does not use infinite contexts. The idea used for avoiding infinite contexts comes from Coquand=-='s thesis [Coq85]-=-. Our proof yields as a corollary another proof of strong normalization (under fi-reduction) of well-formed terms of LF . In fact, it is easy to see that this proof does not use the candidates of redu... |

26 |
On Girard's "candidats de reductibilit'es
- Gallier
- 1990
(Show Context)
Citation Context .... 6 5 Strong Normalization in CC The proof of strong normalization for well-typed terms of CC is obtained by generalizing the proof given by Girard for the system F ! [Gir72], as presented in Gallier =-=[Gal90]-=-. However, there are some significant technical complications. In F ! , we have an ascending hierarchy, kinds, type families, and terms, where kinds do not depend on type families or terms, and type f... |

4 |
Strong normalization for terms of the theory of constructions
- Pottinger
- 1987
(Show Context)
Citation Context ...g normalization for the first 19 time. The proof uses infinite contexts, and basically Henkin's technique for adding new witnesses, so that all closed types are inhabited. 4. Pottinger, February 1987 =-=[Pot87]-=-. This paper refers to Coquand 1987, and gives a proof of strong normalization apparently inspired by Coquand's proof. Infinite contexts are also used, as well as an idea due to Seldin. Although we ne... |

3 |
Mathesis: The Mathematical Foundations of Ulysses
- Seldin
- 1987
(Show Context)
Citation Context ...roof. Infinite contexts are also used, as well as an idea due to Seldin. Although we need to examine it more closely, the proof seems correct, but rather difficult to follow. 5. Seldin, November 1987 =-=[Sel87]. This is -=-a report, "Mathesis: the Mathematical Foundations of Ulysses", in which a proof of strong normalization for a variant of the theory of constructions is given. We have not yet had the time to... |

2 |
Constructing possible worlds. Theoria
- Ranta
- 1990
(Show Context)
Citation Context ...he sources of inspiration for this research, Moggi and Mitchell's work on Kripke models for the simply-typed -calculus [MM87] should be mentioned. We also became recently aware of work by Aarne Ranta =-=[Ran90], in which the notio-=-n of "contexts as (finite approximation of) worlds" is used. One of the motivations for this work is to give an intuitionistic treatment of the notion of "possible worlds". Ranta's... |

1 |
an extended calculus of constructions (check title
- ECC
- 1990
(Show Context)
Citation Context ...rs that it also uses infinite contexts. It appears to be more along the line of Martin Lof's proof of normalization for F ! , defined as a Prawitz-style natural deduction system. 6. Zhaohui Luo, 1989 =-=[Luo90]-=-. There is apparently a proof of strong normalization for an extension of CC with universes, given in Luo's thesis. We do not have this document yet. 7. Geuvers and Nederhof, June 1989 [GN89]. The aut... |

1 |
An Intuitionistic Theory of Types
- Lof
- 1972
(Show Context)
Citation Context ...heorem 4.1 (Martin Lof) The Church-Rosser property holds for raw terms of CC (even the economical version). Proof . Such a proof using the so called "Tait/Martin Lof's method" was given by M=-=artin Lof [ML72]-=-. It should be noted that theorem 4.1 is quite handy. It appears that if fi-conversion is defined on raw terms, which is definitely more convenient than using equality judgments, many important proper... |

1 |
Benthem Jutting. Normalization in Coquand's System
- van
- 1986
(Show Context)
Citation Context ...em arises in the definition of [[\Gamma . x: K: B]]ae\Delta, where the argument \Delta 0 . A is omitted. As a consequence, [[\Gamma . A]]ae\Delta is not always well-defined. 2. Jutting, December 1986 =-=[vBJ86]-=-. This is a note attempting to correct Coquand's proof of normalization given in his thesis. The introduction mentions discussions with Coquand, leading to this note. As we see it, the definition of [... |