G. Dowek, G. Huet and B. Werner. On the Definition of the j-long Normal Form in Type Systems of the Cube. Submitted to publication. See also http://pauillac.inria.fr/~werner/, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on well typed terms of the same type [9] Geuvers paper also proves that typed fij equality is decidable by orienting the j equation as a contraction and showing that the resulting rewrite relation, in conjunction with the usual fi redexes, is strongly normalising and confluent. A related paper [7] (which we draw upon) also uses strong normalisation and confluence of fi and j contractions to prove the existence of fij long normal forms for C. After collecting some defintions in section 2, presenting C in section 3 and defining our restricted rewrite relation in section 4, we prove: ....

....construction of long fij normal forms Lemma 3.7 There is a well founded order on the terms of C such that If u is a subterm of t, then u t If t is a fi normal form, then there is also a fi normal form T such that there is a judgement Gamma t : T and T t. Proof The proof follows [7] in using an alternate syntax where a type for a term occurs as a subterm of the term. 2 4 A Rewrite Relation Recall that if t and A are pre terms, then j A (t) is notational shorthand for the pre term x:A:tx where x 6 2FV(t) is a correctly sorted variable. The theory of j expansions as ....

G. Dowek. On the defintion of the j-long normal form in type systems of the cube. Informal proceedings of the 1993 Workshop on Types for Proofs and Programs, 1993.

....x; y; In the meta theoretic study hereafter, we will consider a variant of the usual presentation of Pure Type Systems: the terms will carry more type information than usual in the cases of abstraction and application. This approach can be seen as related to the labeled terms used in [4]. However, here, we will use these labels to restrict the usual formulation of fi reduction, see [1, 3] In section 7, we will verify that, provided the strong normalization property holds, our definition of PTS s is equivalent to the usual ones, and hence strong normalization itself is inherited ....

G. Dowek, G. Huet and B. Werner. On the Definition of the j-long Normal Form in Type Systems of the Cube. Submitted to publication. See also http://pauillac.inria.fr/~werner/, 1996.

....judgement forms. In [11] typed and untyped equality are shown to coincide on well type terms of the same type, Gamma t 1 = fij t 2 : T iff Gamma t i : T t 1 = fij t 2 Since fi with j contraction is strongly normalising and confluent on well typed terms, typed equality is decidable. In [7] j contractions are used to prove the existence of fij long normal forms for all systems of the cube. In summary, there are many reasons for using j expansions in type theory, e.g. the ease with which they generalise to other type constructors and their good modularity properties. This paper ....

G. Dowek. On the definition of j-long normal forms in the type systems of the cube. Informal Proceedings of the 1993 Workshop on Types for Proofs and Programs, 1993.

....by x; y; In the meta theoretic study hereafter, we will consider a variant of the usual presentation of Pure Type Systems: the terms will carry more type information than usual in the cases of abstraction and application. This approach can be seen as related to the labeled terms used in [4]. However, here, we will use these labels to restrict the usual formulation of fi reduction, see [1, 3] In section 7, we will verify that, provided the strong normalization property holds, our definition of PTS s is equivalent to the usual ones, and hence strong normalization itself is inherited ....

G. Dowek, G. Huet and B. Werner. On the Definition of the j-long Normal Form in Type Systems of the Cube. Submitted to publication. See also http://pauillac.inria.fr/~werner/, 1996.

....fij long normal form with respect to ( Sigma; hx 1 :A 1 ; x i Gamma1 :A i Gamma1 i) The key property of fij long normal forms is that they provide canonical terms for the equivalence classes under fij equality. The proof of this property is non trivial, and can be adapted from results in [DHW93] and [Gar93a] 3.5 Theorem Let Gamma LF Sigma A : B. The fij long normal form of A with respect to ( Sigma; Gamma) exists and is unique. 3.3 Adequate Representations We now give the definition of adequate representation, which characterises when the consequence relation of a logic has ....

G. Dowek, G. Huet and B. Werner. On the Definition of the j-long normal form in Type Systems of the Cube, Proceedings of the 1993 Workshop on Types for Proofs and Programs, Nijmejen, 1993.

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G. Dowek, G. Huet and B. Werner. On the Definition of the j-long Normal Form in Type Systems of the Cube. Submitted to publication. See also http://pauillac.inria.fr/~werner/, 1996.

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