Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, France, May 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....works well, but for fij reduction the situation is much more complex. In particular, fij reduction is confluent only for well typed terms, and subject reduction depends on strengthening, which is difficult to prove directly. These technical problems with fij reduction have been solved by Salvesen [Sal90], Geuvers [Geu92] and later with a different method by Goguen [Gog99] but nevertheless several problems remain. First, canonical forms are not fij normal forms and so conversion to canonical form must be handled separately. Second, the algorithms implicit in the reduction based accounts are not ....

Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, France, May 1990.

....would cause many problems in our efforts, one of which is that equality in Prolog is 50 fij conversion. In many previous studies of LF, such as [20] and [3] however, j was used nontheless. Furthermore, confluence (and strong normalization) of fij conversion in LF have now been proved (see [21]) Thus we can just use fij convertibility ( fij ) as definitional equality and assume the existence of unique canonical forms. There is both precedence and justification to this approach. Another, more difficult approach to the j problem is to adopt a restricted subset of LF, where fi and j ....

A. Salvesen. The church-rosser theorem for lf with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks, Antibes, May 1990.

....but for fij reduction the situation is much more complex. In particular, fij reduction is confluent only for well typed terms, and subject reduction depends on strengthening, which is difficult to prove directly. These technical problems with fij reduction have been addressed in work by Salvesen [Sal90], Geuvers [Geu92] and later with a different method by Goguen [Gog99] but nevertheless several problems remain. First, canonical forms are not fij normal forms and so conversion to canonical form must be handled separately. Second, the algorithms implicit in the reduction based accounts are not ....

Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, France, May 1990.

....type theory that is used as a metarepresentation language is based on fij equivalence. This discrepancy was known to Harper, Honsell and Plotkin when they first presented LF in 1987 [HHP93] A full treatment of the meta theory of LF with fij equivalence was successively devised by various authors [Coq91, Geu93, Sal90] and resulted in non trivial complications. The formulation of the semantics of LLF as a pre canonical system has the advantage of forcing all derivable judgments to mention only terms in j long form, as formally expressed below in Lemma 2.12. Indeed, all the issues concerning j conversion are ....

Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, France, May 1990.

....but for fij reduction the situation is much more complex. In particular, fij reduction is confluent only for well typed terms, and subject reduction depends on strengthening, which is difficult to prove directly. These technical problems with fij reduction have been addressed in work by Salvesen [Sal90], Geuvers [Geu92] and later with a different method by Goguen [Gog99] but nevertheless several problems remain. First, canonical forms are not fij normal forms and so conversion to canonical form must be handled separately. Second, the algorithms implicit in the reduction based accounts are not ....

Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, France, May 1990.

....but for fij reduction the situation is much more complex. In particular, fij reduction is confluent only for well typed terms, and subject reduction depends on strengthening, which is difficult to prove directly. These technical problems with fij reduction have been addressed in work by Salvesen [Sal90], Geuvers [Geu92] and later with a different method by Goguen [Gog99] but nevertheless several problems remain. First, canonical forms are not fij normal forms and so conversion to canonical form must be handled separately. Second, the algorithms implicit in the reduction based accounts are not ....

Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, France, May 1990.

....works well, but for fij reduction the situation is much more complex. In particular, fij reduction is confluent only for well typed terms, and subject reduction depends on strengthening, which is difficult to prove directly. These technical problems with fij reduction have been solved by Salvesen [Sal90], Geuvers [Geu92] and later with a different method by Goguen [Gog99] but nevertheless several problems remain. First, canonical forms are not fij normal forms and so conversion to canonical form must be handled separately. Second, the algorithms implicit in the reduction based accounts are not ....

Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, France, May 1990.

....we consider here is fij conversion. Harper et al. 12] formulate definitional equality only with fi conversion and conjecture that the system resulting from adding the j rule would have the properties we list below. This has recently been proved by Coquand [3] and independently by Salvesen [30]. For practical purposes the formulation including the j rule is superior, since every term has an equivalent canonical form. Thus, for us, j is the least congruence generated by fij conversions in the usual manner. The basic judgments are Gamma Sigma M : A and M j N and analogous judgments ....

Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, May 1990.

....equality we consider here is fij conversion. Harper et al. 8] formulate definitional equality only with fi conversion and conjecture that the system resulting from adding the j rule would have the properties we list below. This has recently been proved by Coquand [1] and independently by Salvesen [22]. For practical purposes the formulation including the j rule is superior, since every term has an equivalent canonical form. Thus, for us, j is the least congruence generated from fij conversions in the usual manner. The basic judgments are Gamma Sigma M : A and M j N and analogous judgments ....

Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, May 1990.

.... it has not been shown that every term in the Calculus of Constructions is equivalent to a unique canonical form we will take this as a working hypothesis as in [5] For the restriction of this system to the LF type theory, this has recently been proved independently by Coquand [2] and Salvesen [25]. It is possible Delta valid Gamma A : Gamma[x:A] valid Sigma(c) A Gamma valid Gamma c : A Gamma(x) A Gamma valid Gamma x : A Gamma valid Gamma Prop : Type Gamma[x:A] B : Gamma fx:AgB : Gamma[x:A] M : B Gamma [x:A] M : fx:Ag B Gamma M : fx:Ag B ....

Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, May 1990.

....for applications in program development and theorem proving. One might consider the problem of including j conversions into the theory, which are useful for some technical reasons; whether they are important in practice is to be seen. For this, we refer to a recent relevant work by Salvesen [Sal89] which considers the ChurchRosser property of LF with j conversion. Another extension of purely theoretical interest might be to extend the predicative levels to larger ordinals, say . This might be interesting when considering the problem of proof theoretic power of the predicative levels ....

A. Salvesen, `The Church-Rosser Theorem for LF with fij reduction', manuscript.

....LF in 1987 [HHP93] but they could not prove some of the necessary properties in the presence of the j expansion rule (in particular the property that we call confluence in Section 2. 3) A full treatment of the meta theory of LF with fij equivalence was successively devised by various authors [Coq91, Geu93, Sal90] and resulted in non trivial complications. The formulation of the semantics of LLF as a pre canonical system has the advantage of forcing all derivable judgments to mention only terms in j long form. Indeed, all the issues concerning j conversion are hardwired into the system and do not require ....

Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, France, May 1990.

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Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, France, May 1990.

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Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, May 1990. 12

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Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, France, May 1990.

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Anne Salvesen. The Church-Rosser theorem for LF with fij-reduction. Unpublished notes to a talk given at the First Workshop on Logical Frameworks in Antibes, France, May 1990.

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