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Inductive Sets and Families in MartinLöf's Type Theory and Their SetTheoretic Semantics
 Logical Frameworks
, 1991
"... MartinLof's type theory is presented in several steps. The kernel is a dependently typed calculus. Then there are schemata for inductive sets and families of sets and for primitive recursive functions and families of functions. Finally, there are set formers (generic polymorphism) and universes. ..."
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Cited by 76 (13 self)
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MartinLof's type theory is presented in several steps. The kernel is a dependently typed calculus. Then there are schemata for inductive sets and families of sets and for primitive recursive functions and families of functions. Finally, there are set formers (generic polymorphism) and universes. At each step syntax, inference rules, and settheoretic semantics are given. 1 Introduction Usually MartinLof's type theory is presented as a closed system with rules for a finite collection of set formers. But it is also often pointed out that the system is in principle open to extension: we may introduce new sets when there is a need for them. The principle is that a set is by definition inductively generated  it is defined by its introduction rules, which are rules for generating its elements. The elimination rule is determined by the introduction rules and expresses definition by primitive recursion on the way the elements of the set are generated. (In this paper I shall use the term ...
Inductive Families
 Formal Aspects of Computing
, 1997
"... A general formulation of inductive and recursive definitions in MartinLof's type theory is presented. It extends Backhouse's `DoItYourself Type Theory' to include inductive definitions of families of sets and definitions of functions by recursion on the way elements of such sets are generated. Th ..."
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Cited by 65 (13 self)
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A general formulation of inductive and recursive definitions in MartinLof's type theory is presented. It extends Backhouse's `DoItYourself Type Theory' to include inductive definitions of families of sets and definitions of functions by recursion on the way elements of such sets are generated. The formulation is in natural deduction and is intended to be a natural generalization to type theory of MartinLof's theory of iterated inductive definitions in predicate logic. Formal criteria are given for correct formation and introduction rules of a new set former capturing definition by strictly positive, iterated, generalized induction. Moreover, there is an inversion principle for deriving elimination and equality rules from the formation and introduction rules. Finally, there is an alternative schematic presentation of definition by recursion. The resulting theory is a flexible and powerful language for programming and constructive mathematics. We hint at the wealth of possible applic...
Universes for Generic Programs and Proofs in Dependent Type Theory
 Nordic Journal of Computing
, 2003
"... We show how to write generic programs and proofs in MartinL of type theory. To this end we consider several extensions of MartinL of's logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introduction, el ..."
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Cited by 42 (2 self)
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We show how to write generic programs and proofs in MartinL of type theory. To this end we consider several extensions of MartinL of's logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introduction, elimination, and equality rules. These extensions are modeled on Dybjer and Setzer's finitely axiomatized theories of inductiverecursive definitions, which also have a universe of codes for sets, and generic formation, introduction, elimination, and equality rules.
Representing Inductively Defined Sets by Wellorderings in MartinLöf's Type Theory
, 1996
"... We prove that every strictly positive endofunctor on the category of sets generated by MartinLof's extensional type theory has an initial algebra. This representation of inductively defined sets uses essentially the wellorderings introduced by MartinLof in "Constructive Mathematics and Computer Pr ..."
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Cited by 15 (0 self)
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We prove that every strictly positive endofunctor on the category of sets generated by MartinLof's extensional type theory has an initial algebra. This representation of inductively defined sets uses essentially the wellorderings introduced by MartinLof in "Constructive Mathematics and Computer Programming". 1 Background MartinLof [10] introduced a general set former for wellorderings in intuitionistic type theory. It has formation rule Aset (x : A) B(x)set W x:A B(x)set introduction rule a : A (x : B(a)) b(x) : W x:A B(x) sup(a; b) : W x:A B(x) : elimination rule c : W x:A B(x) (x : A; y : B(x) !W x:A B(x); z : Q t:B(x) C(y(t))) d(x; y; z) : C(sup(a; b)) T (c; d) : C(c) and equality rule a : A (x : B(a)) b(x) : W x:A B(x) (x : A; y : B(x) !W x:A B(x); z : Q t:B(x) C(y(t))) d(x; y; z) : C(sup(a; b)) T (sup(a; b); d) = d(a; b; t:T (b(t); d) : C(c) The elimination rule can be viewed either as a rule of transfinite induction or as a rule of definition by transfinite re...
Universal Algebra in Type Theory
 Theorem Proving in Higher Order Logics, 12th International Conference, TPHOLs '99, volume 1690 of LNCS
, 1999
"... We present a development of Universal Algebra inside Type Theory, formalized using the proof assistant Coq. We define the notion of a signature and of an algebra over a signature. We use setoids, i.e. ... ..."
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We present a development of Universal Algebra inside Type Theory, formalized using the proof assistant Coq. We define the notion of a signature and of an algebra over a signature. We use setoids, i.e. ...
Programming interfaces and basic topology
 Annals of Pure and Applied Logic
, 2005
"... A pattern of interaction that arises again and again in programming, is a “handshake”, in which two agents exchange data. The exchange is thought of as provision of a service. Each interaction is initiated by a specific agent —the client or Angel, and concluded by the other —the server or Demon. We ..."
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A pattern of interaction that arises again and again in programming, is a “handshake”, in which two agents exchange data. The exchange is thought of as provision of a service. Each interaction is initiated by a specific agent —the client or Angel, and concluded by the other —the server or Demon. We present a category in which the objects —called interaction structures in the paper — serve as descriptions of services provided across such handshaken interfaces. The morphisms —called (general) simulations— model components that provide one such service, relying on another. The morphisms are relations between the underlying sets of the interaction structures. The proof that a relation is a simulation can serve (in principle) as an executable program, whose specification is that it provides the service described by its domain, given an implementation of the service described by its codomain.
Specifying Interactions With Dependent Types
 In Workshop on subtyping and dependent types in programming
, 2000
"... this paper we consider how to express specications of interactions in dependent type theory. The results so far are modest, though we hope we have identied some key structures for describing contracts between independent agents, and shown how to dene them in a dependently typed framework. These are ..."
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Cited by 5 (4 self)
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this paper we consider how to express specications of interactions in dependent type theory. The results so far are modest, though we hope we have identied some key structures for describing contracts between independent agents, and shown how to dene them in a dependently typed framework. These are called below transition systems (2.2) and interaction systems (2.3). Both are coalgebras; transition systems for a functor Fam , and interaction systems for its composite with itself, Fam (Fam ). These structures seems to have interesting connections with predicate transformer semantics for imperative programs, as initiated by Dijkstra, and also with the renement calculus of Back and von Wright as described in their book [2]. We restrict attention to situations in which the system and its environment communicate by exchanging messages in strict alternation, as with the moves in a two player game.
Ordinals and Interactive Programs
, 2000
"... The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be ..."
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The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be
Interactive programs and weakly final coalgebras (extended version
 Dependently typed programming, number 04381 in Dagstuhl Seminar Proceedings, 2004. Available via http://drops.dagstuhl.de/opus
"... GR/S30450/01. 2 A. Setzer, P. Hancock 1 Introduction According to MartinL"of [19]: "... I do not think that the search for logically ever more satisfactory high level programming languages can stop short of anything but ..."
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GR/S30450/01. 2 A. Setzer, P. Hancock 1 Introduction According to MartinL"of [19]: "... I do not think that the search for logically ever more satisfactory high level programming languages can stop short of anything but
Understanding Inductive Types in Constructions
, 1993
"... In this paper we extend the Calculus of Constructions with generalized inductive types. The extension is justified by showing that the usual set theoretical model can be effectivized. It is also pointed out that the model given in a published paper for a collection of inductive types in a different ..."
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In this paper we extend the Calculus of Constructions with generalized inductive types. The extension is justified by showing that the usual set theoretical model can be effectivized. It is also pointed out that the model given in a published paper for a collection of inductive types in a different style is wrong. Copyright c fl1993. All rights reserved. Reproduction of all or part of this work is permitted for educational or research purposes on condition that (1) this copyright notice is included, (2) proper attribution to the author or authors is made and (3) no commercial gain is involved. Technical Reports issued by the Department of Computer Science, Manchester University, are available by anonymous ftp from m1.cs.man.ac.uk (130.88.13.4) in the directory /pub/TR. The files are stored as PostScript, in compressed form, with the report number as filename. Alternatively, reports are available by post from The Computer Library, Department of Computer Science, The University, Oxford R...