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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 ...
A finite axiomatization of inductiverecursive definitions
 Typed Lambda Calculi and Applications, volume 1581 of Lecture Notes in Computer Science
, 1999
"... Inductionrecursion is a schema which formalizes the principles for introducing new sets in MartinLöf’s type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an in ..."
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Cited by 42 (14 self)
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Inductionrecursion is a schema which formalizes the principles for introducing new sets in MartinLöf’s type theory. It states that we may inductively define a set while simultaneously defining a function from this set into an arbitrary type by structural recursion. This extends the notion of an inductively defined set substantially and allows us to introduce universes and higher order universes (but not a Mahlo universe). In this article we give a finite axiomatization of inductiverecursive definitions. We prove consistency by constructing a settheoretic model which makes use of one Mahlo cardinal. 1
Constructions, Inductive Types and Strong Normalization
, 1993
"... This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and typechecking, based on the equalityasjudgement presentation. We present a settheoretic notio ..."
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Cited by 31 (2 self)
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This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and typechecking, based on the equalityasjudgement presentation. We present a settheoretic notion of model, CCstructures, and use this to give a new strong normalization proof based on a modification of the realizability interpretation. An extension of the core calculus by inductive types is investigated and we show, using the example of infinite trees, how the realizability semantics and the strong normalization argument can be extended to nonalgebraic inductive types. We emphasize that our interpretation is sound for large eliminations, e.g. allows the definition of sets by recursion. Finally we apply the extended calculus to a nontrivial problem: the formalization of the strong normalization argument for Girard's System F. This formal proof has been developed and checked using the...
Presupposition and Abduction in Type Theory
 Siekmann Edinburgh Conference on Computational Logic and Natural Language Processing
, 1995
"... This paper is about reasoning with presuppositions in natural language. Presupposition accommodation, as predicted by the linguistic theory of presuppositions as anaphoric expressions, is reconstructed logically as abductive inference in a framework that supports both anaphoric links and a contextd ..."
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Cited by 12 (0 self)
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This paper is about reasoning with presuppositions in natural language. Presupposition accommodation, as predicted by the linguistic theory of presuppositions as anaphoric expressions, is reconstructed logically as abductive inference in a framework that supports both anaphoric links and a contextdependent notion of propositionhood. Abductive inference arises as a sideeffect of the use of the formalism and of characteristics of the communication situation. The proposal is illustrated by some examples and compared to related approaches. Keywords: presupposition resolution, abduction, logical frameworks, semanticspragmatics interface, contextdependence 1 Introduction In this working note we will study the relation between the presupposition theory of [van der Sandt, 1992] and the type theory of MartinLof (MLTT) and its implications for inference processes at the semanticspragmatics interface. The main claim will be that the anaphoric theory of presuppositions of [van der Sandt, ...
On the Syntax of Dependent Types and the Coherence Problem (working draft)
, 1994
"... We discuss different ways to represent the syntax of dependent types using MartinLof type theory as a metalanguage. In particular, we show how to give an intrinsic syntax in which meaningful contexts, types in a context, and terms of a certain type in a context, are generated directly without first ..."
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We discuss different ways to represent the syntax of dependent types using MartinLof type theory as a metalanguage. In particular, we show how to give an intrinsic syntax in which meaningful contexts, types in a context, and terms of a certain type in a context, are generated directly without first introducing raw terms, types, and contexts. In the first representation we define inductively the normal contexts, types, and terms of the pure theory of dependent types. Simultaneously substitution and lifting are defined recursively. Equality in the object language is here syntactic equality and is represented by the equality of the metalanguage. The second representation is a calculus of explicit substitutions. This is a pure inductive definition and proofs of equalities are generated simultaneously. As for Curien's explicit syntax there are term constructors corresponding to applications of the type equality rules, and coherence conditions related to those appearing in category theory. ...