Karl Crary. Admissibility of fixpoint induction over partial types. In CADE1998, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....e2 e3 K = if0 (EJe1K ) EJe2K ) EJe3K ) T Jif0 D e1 e2 e3 : K = CJK 3.6 Fixpoints Thus far, all of our translated terms live within a strongly normalizing type theory. If we wish to translate the fixpoint combinator, we give up strong normalization and require a partial type theory instead [10]. However, if we only admit fixpoint operators at run time, all specialization time computations remain strongly normalizing, the extraction function is terminating, and type checking remains decidable. The translation of the run time constructs is straightforward. EJfix D x:e : K = fix x : ....

Karl Crary. Admissibility of fixpoint induction over partial types. In CADE1998, 1998.

....last results in cumbersome theory dependencies. Last but not least, the PVS version does not provide tactics for proving admissibility automatically. There is also work on extending type theory with partial functions that employs notions of domain theory (Constable Smith, 1987; Audebaud, 1991; Crary, 1998). This work is still largely concerned with overcoming theoretical problems arising from the use of type theory. On the other hand, the problems with formalizing and automating continuity mentioned above point out deficiencies in the type systems of both PVS and Isabelle HOL: the former s lack of ....

Crary, Karl. (1998). Admissibility of fixpoint induction over partial types. Pages 270--285 of: Kirchner, C, & Kirchner, H. (eds), Automated deduction --- CADE-15. Lect. Notes in Comp. Sci., vol. 1421. Springer-Verlag.

....principle: t in (T T ) fix (t) in T However, the fixpoint principle is not valid for all types (see Smith [42] for an example of such a type) Types for which the fixpoint principle is valid are called admissible. Admissibility is indicated within the logic by the type T admiss . Crary [15, 17] shows that a wide class of types are admissible, including all types used in conventional programming languages. 3 Intensional Reasoning To lay the groundwork that makes intensional reasoning possible in our type theory, we begin by being more specific about the structure of terms in Section ....

Karl Crary. Admissibility of fixpoint induction over partial types. In Fifteenth International Conference on Automated Deduction, volume 1421 of Lecture Notes in Computer Science, pages 270--285, Lindau, Germany, July 1998. Springer-Verlag. Extended version published as CMU technical report CMU-CS98 -164.

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Crary, K. (1998a) Admissibility of fixpoint induction over partial types. Technical report, Department of Computer Science, Cornell University.

....4, beginning with a summary of Smith s admissibility class and then widening the class using predicate admissibility and monotonicity. Concluding remarks appear in Section 5. Most proofs have been omitted in this paper due to space limitations; those proofs appear in the companion technical report [10]. Type Formation Introduction Elimination universe i U i type formation (for i 1) operators disjoint union T1 T2 inj 1 (e) case(e; x1 :e1 ; x2 :e2 ) inj 2 (e) function space Pix:T1:T2 x:e e1e2 product space Sigma x:T1:T2 he1 ; e2 i 1 (e) 2 (e) natural numbers N 0; 1; 2; assorted ....

....in A) type (a = a 0 2 A) t = t 0 2 T iff T type (t# , t 0 #) t# ) t = t 0 2 T ) 2 (a in A) iff (a in A) type a# Fig. 3. Type Definitions Part of the definition (for types other than universes) appears in Figure 3; the full definition appears in the companion technical report [10]. This equality relation is constructed to respect evaluation: if t 2 T and t 7 t 0 then t = t 0 2 T . 2.3 The Fixpoint Principle The central issue of this paper is the fixpoint principle: f 2 T T ) fix (f) 2 T The fixpoint principle allows us to type recursively defined objects, such as ....

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K. Crary. Admissibility of fixpoint induction over partial types. Technical Report TR98-1674, Department of Computer Science, Cornell University, Apr. 1998.

....in LCF. This condition is required because fixpoint induction can be derived from the recursive typing rule [51] However, all the types used in the embedding in this paper are admissible, so I ignore the admissibility condition in this paper. Additional details appear in Smith [51] and Crary [16, 17]. 3 This terminology can be somewhat confusing. A total type is one that contains only convergent expressions. The partial function type T1 T 2 contains functions that return possibly divergent elements, but those functions themselves converge, so a partial function type is a total type. ....

Karl Crary. Admissibility of fixpoint induction over partial types. Technical report, Department of Computer Science, Cornell University, 1998.

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