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A General Formulation of Simultaneous InductiveRecursive Definitions in Type Theory
 Journal of Symbolic Logic
, 1998
"... The first example of a simultaneous inductiverecursive definition in intuitionistic type theory is MartinLöf's universe à la Tarski. A set U0 of codes for small sets is generated inductively at the same time as a function T0 , which maps a code to the corresponding small set, is defined by recursi ..."
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Cited by 65 (10 self)
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The first example of a simultaneous inductiverecursive definition in intuitionistic type theory is MartinLöf's universe à la Tarski. A set U0 of codes for small sets is generated inductively at the same time as a function T0 , which maps a code to the corresponding small set, is defined by recursion on the way the elements of U0 are generated. In this paper we argue that there is an underlying general notion of simultaneous inductiverecursive definition which is implicit in MartinLöf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous inductionrecursion. This enables us to give a unified treatment of several interesting constructions including various universe constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a realisability model ...
Normalization by evaluation for MartinLöf type theory with one universe
 IN 23RD CONFERENCE ON THE MATHEMATICAL FOUNDATIONS OF PROGRAMMING SEMANTICS, MFPS XXIII, ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE
, 2007
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Optimized Encodings of Fragments of Type Theory in First Order Logic
 JLC: Journal of Logic and Computation
, 1994
"... The paper presents sound and complete translations of several fragments of MartinLof's monomorphic type theory to first order predicate calculus. The translations are optimised for the purpose of automated theorem proving in the mentioned fragments. The implementation of the theorem prover Gand ..."
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Cited by 7 (4 self)
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The paper presents sound and complete translations of several fragments of MartinLof's monomorphic type theory to first order predicate calculus. The translations are optimised for the purpose of automated theorem proving in the mentioned fragments. The implementation of the theorem prover Gandalf and several experimental results are described. 1 Introduction The subject of this paper is the problem of automated theorem proving in MartinLof's monomorphic type theory [19, 8], which is the underlying logic of the interactive proof development system ALF [2, 14]. In the scope of our paper the task of automated theorem proving in type theory is understood as demonstrating that a certain type is inhabited by constructing a term of that type. The problem of inhabitedness of a type A is understood in the following way: given a set of judgements \Gamma (these may be constant declarations, explicit definitions and defining equalities), find a term a such that a2A is derivable from \Gam...
TYPE THEORY AND HOMOTOPY
"... The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Geometry, Algebra, and Logic, which has recently come to light in the form of an interpretation of the constructive type theory of Per MartinLöf into homotopy ..."
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Cited by 4 (0 self)
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The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Geometry, Algebra, and Logic, which has recently come to light in the form of an interpretation of the constructive type theory of Per MartinLöf into homotopy
Propositional Functions and Families of Types
 In Workshop on Programming Logic
, 1989
"... Introduction In order to capture some of the programmers errors, several computer languages, like Pascal and ML, are equipped with a type system. Using the CurryHoward interpretation of propositions as types [3, 8], or as we shall say here, propositions as sets, a type system can be made strong en ..."
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Introduction In order to capture some of the programmers errors, several computer languages, like Pascal and ML, are equipped with a type system. Using the CurryHoward interpretation of propositions as types [3, 8], or as we shall say here, propositions as sets, a type system can be made strong enough to be used to specify the task a program is supposed to do. This is one of the basis for MartinLof's suggestion in [11] to use his formulation of type theory for programming; his ideas are exploited in [14] and there are several computer implementations of type theory [4, 16]. Similar ideas are also behind Coquand and Huet's calculus of constructions [2]. The idea of propositions as sets is closely related to the intuitionistic explanations of the logical constants given by Heyting [7]. In MartinLof's type theory, the interpretation of propositions as sets is fundamental since the notions of proposition and set are identical. So a logical constant is definitionally equal to th
Relative Predicativity and Dependent Recursion in Secondorder Set Theory and Higherorder Theories
"... This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditional ω. Namely, the equivalence between normal transfinite recursion scheme and new ..."
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This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditional ω. Namely, the equivalence between normal transfinite recursion scheme and new dependent transfinite recursion scheme, which does hold in the context of subsystems of second order number theory, does not hold in the contexts of subsystems of second order set theory where the universe V of sets is treated as the given totality (nor in the context of those of n+3th order number or set theories, where the class of all n+2th order objects is treated as the given totality). 1
A note on the theory SID< ω of stratified induction
"... We introduce and analyse a theory of finitely stratified general inductive definitions over the natural numbers, SID< ω, and establish its prooftheoretic ordinal, ϕε0 (0). The definition of SID< ω bears some similarities with D. Leivant’s ramified theories for finitary inductive definitions. ..."
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We introduce and analyse a theory of finitely stratified general inductive definitions over the natural numbers, SID< ω, and establish its prooftheoretic ordinal, ϕε0 (0). The definition of SID< ω bears some similarities with D. Leivant’s ramified theories for finitary inductive definitions.