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Infinite Objects in Type Theory
"... . We show that infinite objects can be constructively understood without the consideration of partial elements, or greatest fixedpoints, through the explicit consideration of proof objects. We present then a proof system based on these explanations. According to this analysis, the proof expressions ..."
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. We show that infinite objects can be constructively understood without the consideration of partial elements, or greatest fixedpoints, through the explicit consideration of proof objects. We present then a proof system based on these explanations. According to this analysis, the proof expressions should have the same structure as the program expressions of a pure functional lazy language: variable, constructor, application, abstraction, case expressions, and local let expressions. 1 Introduction The usual explanation of infinite objects relies on the use of greatest fixedpoints of monotone operators, whose existence is justified by the impredicative proof of Tarski's fixed point theorem. The proof theory of such infinite objects, based on the so called coinduction principle, originally due to David Park [21] and explained with this name for instance in the paper [18], reflects this explanation. Constructively, to rely on such impredicative methods is somewhat unsatisfactory (see fo...
Analysis of a Guard Condition in Type Theory
, 1997
"... We present a realizability interpretation of coinductive types based on partial equivalence relations (per's). We extract from the per's interpretation sound rules to type recursive definitions. These recursive definitions are needed to introduce "infinite" and "total" objects of coinductive type ..."
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We present a realizability interpretation of coinductive types based on partial equivalence relations (per's). We extract from the per's interpretation sound rules to type recursive definitions. These recursive definitions are needed to introduce "infinite" and "total" objects of coinductive type such as an infinite stream or a nonterminating process. We show that the proposed type system enjoys the basic syntactic properties of subject reduction and strong normalization with respect to a confluent rewriting system first studied by Gimenez. We also compare the proposed type system with those studied by Coquand and Gimenez. In particular, we provide a semantic reconstruction of Gimenez's system which suggests a rule to type nested recursive definitions.
Data Types, Infinity and Equality in System AF2
 In CSL ’93, volume 832 of LNCS
, 1995
"... This work presents an extension of system AF 2 to allow the use of infinite data types. We extend the logic with inductive and coinductive types, and show that the "programming method" is still correct. Unlike previous work in other typesystems, we only use the pure calculus. Propositions about no ..."
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This work presents an extension of system AF 2 to allow the use of infinite data types. We extend the logic with inductive and coinductive types, and show that the "programming method" is still correct. Unlike previous work in other typesystems, we only use the pure calculus. Propositions about normalization and unicity of the representation of data have no equivalent in other systems. Moreover, the class of data types we consider is very large with some unusual ones. 1 Introduction Since the work of Curry, a lot of typesystems have been created (e.g., De Bruijn's Automath [4]; Girard's system F [5]; MartinLof's type theory [10]; CoquandHuet's Calculus of construction [3]; etc). One of their purposes is program extraction via the CurryHoward isomorphism [6], which establishes a correspondence between programs and proofs of specifications. One of these systems is AF 2 (second order functional arithmetic) due to Leivant and Krivine [9, 7, 8]. It uses equations as algorithmic specif...