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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 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 ...
Indexed InductionRecursion
, 2001
"... We give two nite axiomatizations of indexed inductiverecursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductiverecursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in ..."
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Cited by 51 (17 self)
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We give two nite axiomatizations of indexed inductiverecursive de nitions in intuitionistic type theory. They extend our previous nite axiomatizations of inductiverecursive de nitions of sets to indexed families of sets and encompass virtually all de nitions of sets which have been used in intuitionistic type theory. The more restricted of the two axiomatization arises naturally by considering indexed inductiverecursive de nitions as initial algebras in slice categories, whereas the other admits a more general and convenient form of an introduction rule.
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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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
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 33 (12 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
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 C ..."
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Cited by 18 (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...
Proof theory of MartinLöf type theory. An overview
 MATHEMATIQUES ET SCIENCES HUMAINES, 42 ANNÉE, N O 165:59 – 99
, 2004
"... We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert’s programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent ..."
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We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert’s programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent constructive theories. Then we show, how, as part of such a programme, the proof theoretic analysis of MartinLöf type theory with Wtype and one microscopic universe containing only two finite sets in carried out. Then we look at the analysis MartinLöf theory with Wtype and a universe closed under the Wtype, and consider the extension of type theory by one Mahlo universe and its prooftheoretic analysis. Finally we repeat the concept of inductiverecursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalisation, which can be used in generic programming, and explain, what is known about its strength.
Theory and Applications of Induction Recursion Case for Support
"... at the University of Strathclyde and asked to form the Mathematically Structured Programming research group there. Research Summary: Prof. Ghani’s research tries to understand the nature and structure of computation. This is quite a bold statement and inevitably, only partial answers will be forthco ..."
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at the University of Strathclyde and asked to form the Mathematically Structured Programming research group there. Research Summary: Prof. Ghani’s research tries to understand the nature and structure of computation. This is quite a bold statement and inevitably, only partial answers will be forthcoming. Nevertheless, this statement shows his commitment to ask deep and fundamental questions so as to produce research which is of the highest calibre and which will stand the test of time rather than become obsolete within a few years. In particular, he has worked extensively in the following areas which form the pillars upon which this proposal is built. – Category Theory: Category theory is a relatively new mathematical discipline which provides an abstract theory of structure and hence is key to Prof. Ghani’s work. He has applied various categorical structures such as monads, comonads, coalgebras, enriched categories and Kan extensions to problems in computation. Of closest relevance to this proposal is his work on containers which provides a theory of concrete data types. – Type Theory: Prof. Ghani uses type theory as an intermediate abstraction between functional programming and its categorical underpinnings. He has worked on features such as type systems, pattern matching and explicit substitutions which make the lambdacalculus closer to “real ” functional languages. He also developed the subject of etaexpansions and showed it to be better behaved than the more traditional theory of etacontractions. He solved the long standing open problem of the decidability of betaetaequality for sum types which had attracted
A Finite Axiomatization of InductiveRecursive Definitions
, 1999
"... Abstract Inductionrecursion is a schema which formalizes the principles for introducing new sets in MartinL"of'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 ext ..."
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Abstract Inductionrecursion is a schema which formalizes the principles for introducing new sets in MartinL&quot;of'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 Introduction In this article we present an elegant, uniform method for introducing large sets in type theory. We draw on experience from proof theory, category theory, and set theory to formulate a compact, completely formal theory of inductiverecursive definitions, and to prove its consistency. Inductionrecursion is a schema for introducing new sets in type theory developed by Dybjer [7]. All the usual sets in MartinL&quot;of's type theory and practically all sets (data types), which are defined in analogy with it, are instances of this schema. Applications of inductionrecursion include not only a variety of typetheoretic analogues of large cardinals (inaccessible cardinals, hyperinaccessible cardinals, etc) but also various powerful notions needed for the typetheoretic formalization of metamathematics (such as reducibility predicates and logical relations for dependent types). Inductionrecursion can also provide novel ways to formalize simple concepts such as the set of lists with distinct elements [7].
Indexed InductionRecursion
"... An indexed inductive definition (IID) is a simultaneous inductive definition of an indexed family of sets. An inductiverecursive definition (IRD) is a simultaneous inductive definition of a set and a recursive definition of a function from that set into another type. An indexed inductiverecursive ..."
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An indexed inductive definition (IID) is a simultaneous inductive definition of an indexed family of sets. An inductiverecursive definition (IRD) is a simultaneous inductive definition of a set and a recursive definition of a function from that set into another type. An indexed inductiverecursive definition (IIRD) is a combination of both. We present a closed theory which allows us to introduce all IIRDs in a natural way without much encoding. By specialising it we also get a closed theory of IID. Our theory of IIRDs includes essentially all definitions of sets which occur in MartinLöf type theory. We show in particular that MartinLöf’s computability predicates for dependent types and Palmgren’s higher order universes are special kinds of IIRD and thereby clarify why they are constructively acceptable notions. We give two axiomatisations. The first and more restricted one formalises a principle for introducing meaningful IIRD by using the dataconstruct in the original version of the proof assistant Agda for MartinLöf type theory. The second one admits a more general form of introduction rule, including the introduction rule for the intensional identity relation, which is not covered by the restricted one. If we add an extensional identity relation to our logical framework, we show that the theories of restricted and general IIRD are equivalent by interpreting them in each other. Finally, we show the consistency of our theories by constructing a model in classical set theory extended by a Mahlo cardinal.