J. Cederquist. A pointfree approach to Constructive Analysis in Type Theory. PhD thesis, Goteborg University, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....w.r.t. choice principles. Our conclusion is that total setoids are better suited for the formalisation of mathematics. 1 Introduction Proof development systems such as Agda [9] Coq [5] and Lego [16] rely on powerful type systems and have been successfully used in the formalisation of mathematics [3, 7, 19, 20]. Nevertheless, their underlying type theoriesMartin L#f s intensional type theory [18] and the Calculus of Inductive Constructions [23]fail to support extensional concepts such as quotients and subsets, which play a fundamental role in mathematics. While signicant eoeorts have been devoted to ....

J. Cederquist. A point-free approach to constructive analysis in type theory. PhD thesis, Chalmers Tekniska H#gskola, 1997.

....setoids is unsuitable, and more generally that total setoids seem better suited for formalising mathematics. 1 Introduction Proof development systems such as Agda [19] Coq [10] and Lego [36] rely on powerful type systems and have been successfully used in the formalisation of mathematics [4, 7, 8, 12, 14 16, 40, 42]. Nevertheless, their underlying type theories Martin L of s intensional type theory [39] and the Calculus of Inductive Constructions [46] fail to support extensional concepts such as quotients and subsets, which play a fundamental role in mathematics. While signi cant e orts have been devoted to ....

J. Cederquist. A point-free approach to constructive analysis in type theory. PhD thesis, Chalmers Tekniska Hogskola, 1997.

.... analysis in Lego [Pol94] More recently Harrison [Har96] presented a significant part of classical analysis, constructing the real numbers by means of a technique closely related to Cantor s classical method in the context of the system Isabelle HOL [GM93] Later Cederquist, Coquand and Negri [Ced97] approached constructive analysis using a point free topology, and proved constructively Hahn Banach theorem in an intensional Martin Lof s type theory and certifying such a proof by Half, a successor of the Alf [Mag95] system. The main difference between our approach and the previous ones ....

J.Cederquist: "A pointfree approach to Constructive Analysis in Type Theory" PhD Thesis, Goteborg University, 1997

....language. The main contributions are: ffl use of explicit substitution to simplify and make fully precise the metatheory of our system 1 ; ffl a simplification of the constraint solving mechanism w.r.t. to the work of [9] that has been actually tested in large examples in our implementation [3, 11], ffl use of the notion of parametrised constants to a package mechanism, that has also been tested in our implementation, and brings some module structure without complicating the meta theory of the system. 2 The type checking problem is: given a correct type A is the term e a correct term of ....

J. Cederquist. A Pointfree approach to Constructive Analysis in Type Theory. PhD thesis, Chalmers University of Technology, 1997. 8

.... explicit substitution to simplify and make fully precise the meta theory of our system 1 and to allow the addition of meta variables; ffl a simplification of the constraint solving mechanism with respect to the work of [9] that has been actually tested in large examples in our implementation [3, 11], ffl use of the notion of parametrised constants to a package mechanism, that has also been tested in our implementation, and brings some module structure without complicating the meta theory of the system. 2 The type checking problem is: given a correct type A is the term e a correct term of ....

J. Cederquist. A Pointfree approach to Constructive Analysis in Type Theory. PhD thesis, Chalmers University of Technology, 1997. 8

....can be expressed formally by a diagram which extends the initial algebra diagram used for categorical semantics of inductively defined sets. The resulting theory has been implemented in the Half system, a proof assistant for Martin Lof s type theory developed by Coquand and Synek, see Cederquist [4]. Plan of the paper. In Section 2 we present Martin Lof s Logical Framework. In Section 3 we recall how to use initial algebras for giving categorical semantics of inductive types in the simply typed lambda calculus. In Section 4 we discuss the step from induction to induction recursion and how ....

J. Cederquist. Pointfree Approach to Constructive Analysis in Type Theory. PhD thesis, Department of Computing Science, Chalmers University of Technology and University of Goteborg, 1997.

....form the objects of the type Set. Subsets of a set S are propositional functions, that is, they are objects of the function type S Set. If U is a subset of S and a 2 S, we usually write a ffl U for the judgement U(a) true. Formal topology has been developed in computer systems for type theory [2]; in particular, the completeness proof in [15] has been checked in the ALF system [12] Some of the results in this paper first appeared in Sara Sadocco s tesi di laurea [14] 2 Definition of formal space A formal topology, as defined by Sambin [16] is a commutative idempotent monoid hS; ....

Jan Cederquist. A pointfree approach to constructive analysis in type theory. Ph.d. thesis, Chalmers University of Technology and University of Goteborg, Sweden, 1997.

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J. Cederquist. A pointfree approach to Constructive Analysis in Type Theory. PhD thesis, Goteborg University, 1997.

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