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Pattern Matching with Dependent Types
 Electronic Proceedings of the Third Annual BRA Workshop on Logical Frameworks (B̊astad
, 1992
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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 ar ..."
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Cited by 70 (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...
Internal Type Theory
 Lecture Notes in Computer Science
, 1996
"... . We introduce categories with families as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also present categories with families as a generalized algebraic theory. Then we define categories with f ..."
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Cited by 40 (7 self)
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. We introduce categories with families as a new notion of model for a basic framework of dependent types. This notion is close to ordinary syntax and yet has a clean categorical description. We also present categories with families as a generalized algebraic theory. Then we define categories with families formally in MartinLof's intensional intuitionistic type theory. Finally, we discuss the coherence problem for these internal categories with families. 1 Introduction In a previous paper [8] I introduced a general notion of simultaneous inductiverecursive definition in intuitionistic type theory. This notion subsumes various reflection principles and seems to pave the way for a natural development of what could be called "internal type theory", that is, the construction of models of (fragments of) type theory in type theory, and more generally, the formalization of the metatheory of type theory in type theory. The present paper is a first investigation of such an internal type theor...
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 ..."
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Cited by 33 (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...
Inductionrecursion and initial algebras
 Annals of Pure and Applied Logic
, 2003
"... 1 Introduction Inductionrecursion is a powerful definition method in intuitionistic type theory in the sense of Scott ("Constructive Validity") [31] and MartinL"of [17, 18, 19]. The first occurrence of formal inductionrecursion is MartinL"of's definition ..."
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Cited by 29 (11 self)
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1 Introduction Inductionrecursion is a powerful definition method in intuitionistic type theory in the sense of Scott (&quot;Constructive Validity&quot;) [31] and MartinL&quot;of [17, 18, 19]. The first occurrence of formal inductionrecursion is MartinL&quot;of's definition of a universe `a la Tarski [19], which consists of a set U
Dependently Typed Records for Representing Mathematical Structure
 Theorem Proving in Higher Order Logics, TPHOLs 2000
, 2000
"... this paper appears in Theorem Proving in Higher Order Logics, TPHOLs 2000, c ..."
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Cited by 15 (0 self)
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this paper appears in Theorem Proving in Higher Order Logics, TPHOLs 2000, c
Integrated Verification in Type Theory (Lecture Notes)
, 1996
"... Contents 1 Introduction 2 2 Type Theory as a Programming Language 3 2.1 Hello World in Type Theory . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Hiding and argument synthesis . . . . . . . . . . . . . . . . . . . . . 4 2.3 Using dependent types in programming . . . . . . . . . . . . . . . . 4 ..."
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Cited by 7 (0 self)
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Contents 1 Introduction 2 2 Type Theory as a Programming Language 3 2.1 Hello World in Type Theory . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Hiding and argument synthesis . . . . . . . . . . . . . . . . . . . . . 4 2.3 Using dependent types in programming . . . . . . . . . . . . . . . . 4 2.4 Higherorder sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Logic for free 8 3.1 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Predicate logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.4 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.5 Inductively defined relations . . . . . . . . . . . . . . . . . . . . . . . 13 4 ALF's Type Theory 14 4.1 Judgements of Type Theory . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 Conventions
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. ...