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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...
Constructive Topology and Combinatorics
 In proceeding of the conference Constructivity in Computer Science, San Antonio, LNCS 613
, 1991
"... We present a method to extract constructive proofs from classical arguments proved by topogical means. Typically, this method will apply to the nonconstructive use of compactness in combinatorics, often in the form of the use of König's lemma (which says that a finitely branching tree that is i ..."
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We present a method to extract constructive proofs from classical arguments proved by topogical means. Typically, this method will apply to the nonconstructive use of compactness in combinatorics, often in the form of the use of König's lemma (which says that a finitely branching tree that is infinite has an infinite branch.) The method consists roughly of working with the corresponding pointfree version of the topological argument, which can be proven constructively using only as primitive the notion of inductive definition. We illustrate here this method on the classical "minimal bad sequence" argument used by NashWilliams in his proof of Kruskal's theorem. The proofs we get by this method are wellsuited for mechanisation in interactive proof systems that allow the user to introduce inductively defined notions, such as NuPrl, or MartinLof set theory.
Inductive Definitions and Type Theory: An Introduction
"... MartinLof's type theory can be described as an intuitionistic theory of iterated inductive definitions developed in a framework of dependent types. It was originally intended to be a fullscale system for the formalization of constructive mathematics, but has also proved to be a powerful fr ..."
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MartinLof's type theory can be described as an intuitionistic theory of iterated inductive definitions developed in a framework of dependent types. It was originally intended to be a fullscale system for the formalization of constructive mathematics, but has also proved to be a powerful framework for programming. The theory integrates an expressive specification language (its type system) and a functional programming language (where all programs terminate). There now exist several proofassistants based on type theory, and many nontrivial examples from programming, computer science, logic, and mathematics have been implemented using these. In this series of lectures we shall describe type theory as a theory of inductive definitions. We emphasize its open nature: much like in a standard functional language such as ML or Haskell the user can add new types whenever there is a need for them. We discuss the syntax and semantics of the theory. Moreover, we present some examples ...