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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 univer ..."
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Cited by 80 (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 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 re ..."
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Cited by 79 (9 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 ...
Wellfounded Trees and Dependent Polynomial Functors
 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2004
"... We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class ..."
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Cited by 44 (6 self)
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We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed categories.
Setoids in Type Theory
, 2000
"... Formalising mathematics in dependent type theory often requires to use setoids, i.e. types with an explicit equality relation, as a representation of sets. This paper surveys some possible denitions of setoids and assesses their suitability as a basis for developing mathematics. In particular, we ..."
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Cited by 43 (3 self)
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Formalising mathematics in dependent type theory often requires to use setoids, i.e. types with an explicit equality relation, as a representation of sets. This paper surveys some possible denitions of setoids and assesses their suitability as a basis for developing mathematics. In particular, we argue that a commonly advocated approach to partial setoids is unsuitable, and more generally that total setoids seem better suited for formalising mathematics. 1
Proof Theory and Meaning
 Handbook of Philosophical Logic, Vol III
, 1986
"... 2. Intermezzo: classical truth and sequent calculi 477 3. Dummett's argument against a truthconditional view on meaning 478 4. Proof as a key concept in meaning theories 485 5. The meaning of the logical constants and the soundess of predicate logic 489 6. Questions of completeness 494 7. The ..."
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Cited by 39 (0 self)
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2. Intermezzo: classical truth and sequent calculi 477 3. Dummett's argument against a truthconditional view on meaning 478 4. Proof as a key concept in meaning theories 485 5. The meaning of the logical constants and the soundess of predicate logic 489 6. Questions of completeness 494 7. The type theory of MartinLöf 497
On universes in type theory
 191 – 204
, 1998
"... The notion of a universe of types was introduced into constructive type theory by MartinLöf (1975). According to the propositionsastypes principle inherent in ..."
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Cited by 35 (8 self)
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The notion of a universe of types was introduced into constructive type theory by MartinLöf (1975). According to the propositionsastypes principle inherent in
The Strength of Some MartinLöf Type Theories
 Arch. Math. Logic
, 1994
"... One objective of this paper is the determination of the prooftheoretic strength of Martin Lof's type theory with a universe and the type of wellfounded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, n ..."
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Cited by 30 (5 self)
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One objective of this paper is the determination of the prooftheoretic strength of Martin Lof's type theory with a universe and the type of wellfounded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with \Delta 1 2 comprehension and bar induction. As MartinLof intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a strong classical theory. Also the prooftheoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts. Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman. 0 Introduction MartinLof's intuitionistic theory of types was originally introduce...
Collection Principles in Dependent Type Theory
 Types for Proofs and Programs, International Workshop, TYPES 2000
, 2001
"... We introduce logicenriched intuitionistic type theories, that extend intuitionistic dependent type theories with primitive judgements to express logic. By adding type theoretic rules that correspond to the collection axiom schemes of the constructive set theory CZF we obtain a generalisation of ..."
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Cited by 21 (4 self)
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We introduce logicenriched intuitionistic type theories, that extend intuitionistic dependent type theories with primitive judgements to express logic. By adding type theoretic rules that correspond to the collection axiom schemes of the constructive set theory CZF we obtain a generalisation of the type theoretic interpretation of CZF. Suitable logicenriched type theories allow also the study of reinterpretations of logic. We end the paper with an application to the doublenegation interpretation.