## A General Formulation of Simultaneous Inductive-Recursive Definitions in Type Theory (1998)

Venue: | Journal of Symbolic Logic |

Citations: | 66 - 9 self |

### BibTeX

@ARTICLE{Dybjer98ageneral,

author = {Peter Dybjer},

title = {A General Formulation of Simultaneous Inductive-Recursive Definitions in Type Theory},

journal = {Journal of Symbolic Logic},

year = {1998},

volume = {65},

pages = {2000}

}

### Years of Citing Articles

### OpenURL

### Abstract

The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löf'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 Martin-Löf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous induction-recursion. 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 ...

### Citations

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431 | Constructive Analysis
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Citation Context ...nduction is the principal notion. It is, to quote Martin-Lof [27, p73], "intended to be a full scale system for formalising intuitionistic mathematics as developed, for example, in the book by Bi=-=shop [10]". It is a-=-lso a typed functional programming language not unlike ML [33] or Miranda [9]. A "set" in the theory is defined inductively by listing its constructors with their types in much the same way ... |

348 |
Intuitionistic type theory
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- 1984
(Show Context)
Citation Context ...priate notion corresponding to "strict positivity" becomes more complex in the context of dependent types. Instead of formulating such a general condition for inductive definitions of sets M=-=artin-Lof [31, 27, 28, 29]-=- gave rules for a collection of specific set formers. However, this collection may be extended when there is a need for it provided the informal semantic principles of the theory are respected. The po... |

279 |
P.: Constructive mathematics and computer programming
- Martin-Löf
- 1982
(Show Context)
Citation Context ...of the type set), so we have recursively defined families of sets. Note that we have reversed the priority of the following concepts as compared to the standard formulations of Martin-Lof type theory =-=[28, 29, 34]-=-: ffl Elimination and equality rules are special instances of the recursive schemata, whereas in the standard formulation the recursive schemata are derived from the elimination and equality rules. ff... |

271 |
Programming in Martin–Löf ’s Type Theory. An introduction
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- 1990
(Show Context)
Citation Context ...[23], Setzer [41], and Rathjen, Griffor, and Palmgren [39]. We conclude this introduction with a few words about the notation. We employ the "logical framework " formulation of Martin-Lof ty=-=pe theory [30, 34]-=-. The core of this theory is a typed fij-calculus with dependent types. There is a base type set, the type of sets, and for each object A : set, there is the type El(A) (often written just A) of the e... |

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185 |
An introduction to inductive definitions
- Aczel
- 1977
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Citation Context ..., we shall directly construct the collections of sets Set and elements El[A] (for each A 2 Set) in a manner similar to Allen [5]. These collections will be inductively defined using Aczel's rule sets =-=[1]-=-. Like Aczel, we use `object' to refer to an element of a -structure and `collection' to refer to a subset of the elements of a -structure. Furthermore, since substitution in the -calculus is interpre... |

173 | Inductive definitions in the system coq - rules and properties
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- 1993
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Citation Context ...sis of Gim'enez' [22] implementation of inductive definitions in a proof editor for Martin-Lof type theory, and as Coquand and Paulin's formulation of inductive types in the calculus of constructions =-=[16, 38]-=- is the basis for the Coq-system [18]. To illustrate the schema, we show how the rules for the first universe and the fresh-lists can be derived by instantiation. Later sections contain further exampl... |

168 |
An intuitionistic theory of types: Predicative part
- Martin-Löf
- 1975
(Show Context)
Citation Context ...by constructing a realisability model in the style of Allen. 1 1 Introduction Martin-Lof type theory is a foundational framework in which induction is the principal notion. It is, to quote Martin-Lof =-=[27, p73], "in-=-tended to be a full scale system for formalising intuitionistic mathematics as developed, for example, in the book by Bishop [10]". It is also a typed functional programming language not unlike M... |

124 | The Type Theoretic Interpretation of Constructive Set Theory
- Aczel
- 1978
(Show Context)
Citation Context ... of universes (here in conjunction with generalised inductive definitions) is Aczel's universe of iterative sets, in which a constructive version of Zermelo-Fraenkel set theory CZF can be interpreted =-=[3]-=-. But in the standard formulations of type theory universes are needed also for the more basic purpose of defining families of sets by structural recursion. For example, the predicate Z (as used 2 in ... |

109 |
An algorithm for testing conversion in type theory
- Coquand
(Show Context)
Citation Context ...like to mention in particular the computability predicates used by Martin-Lof [31, 27] and C. Coquand [12] for proving normalisation of type theory, and the logical relations introduced by T. Coquand =-=[13]-=- for proving soundness and completeness of an algorithm for testing conversion in type theory. These constructions are similar in nature to the collections of propositions and truths in a Frege struct... |

86 | Pattern matching with dependent types
- Coquand
- 1992
(Show Context)
Citation Context ...es. But it is also possible to formulate definition by structural recursion by an external schema (so that for example the addition function is obtained as an instance) as in Martin-Lof [27], Coquand =-=[14]-=-, and Dybjer [19]. When considering simultaneous inductive-recursive definitions it is essential to adopt the latter approach (using an external schema). The reason is that the elimination rule expres... |

78 | Inductive sets and families in Martin-Löf’s type theory and their settheoretic semantics
- Dybjer
- 1991
(Show Context)
Citation Context ...mulated by Backhouse [7] and covered the case of inductively defined sets (possibly depending on parameters). This schema was generalised to the case of inductively defined families of sets by Dybjer =-=[19, 20]-=-. Inductively defined families subsume inductively defined predicates, and this schema can be viewed as the type-theoretic generalisation of the natural deduction schema for inductively defined predic... |

69 | Inductive families
- Dybjer
- 1994
(Show Context)
Citation Context ...remises of either kind and they may appear in arbitrary order. If we remove the possibility that fi; ; p, and q depend on previous recursive premises, then we essentially recover the schema in Dybjer =-=[20]-=-, because then f cannot appear in the introduction rules for P . Moreover, since a non-recursive premise then cannot depend on a recursive one, we can without loss of generality assume that all non-re... |

67 | A theory of types
- Martin-Lof
- 1971
(Show Context)
Citation Context ...priate notion corresponding to "strict positivity" becomes more complex in the context of dependent types. Instead of formulating such a general condition for inductive definitions of sets M=-=artin-Lof [31, 27, 28, 29]-=- gave rules for a collection of specific set formers. However, this collection may be extended when there is a need for it provided the informal semantic principles of the theory are respected. The po... |

60 |
Hauptsatz for the intuitionistic theory of iterated inductive definitions
- Martin-Löf
- 1971
(Show Context)
Citation Context ...erable to treat this special case rather than to give a necessarily much more complicated general formulation which would include (\Sigma 2 A)B(x), A+B, Nn and N as special cases. See Martin-Lof 1971 =-=[26]-=- for a general formulation of inductive definitions in the language of ordinary first order predicate logic. The first such general schema was formulated by Backhouse [7] and covered the case of induc... |

56 | A Non-Type-Theoretic Semantics for Type-Theoretic Language - Allen - 1987 |

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38 | Internal type theory
- Dybjer
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Citation Context ...e structure, and can be given a classical explanation in an analogous way. By defining constructions of these kinds by simultaneous inductionrecursion we have paved the way for "internal type the=-=ory" [21]-=-, that is, to locally reflect the metatheory of type theory in itself. In particular, we hope to extend the technique of reduction-free normalization [15, 11, 17] developed for the simply typed case t... |

33 | Constructions, Inductive Types and Strong Normalization
- Altenkirch
- 1993
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Citation Context ...rs. Independent families of objects can always be found, but the definability of case forces us to use "recursive" rather than "iterative" encodings, see the discussion by Parigot =-=[37] and Altenkirch [6]-=-. Secondly, for each recursively defined function constant, we construct an element satisfying the recursion equations in question. This can be done in a standard way using fixed points and case analy... |

33 | On universes in type theory
- Palmgren
- 1998
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Citation Context ...universe operator as well as under all the usual set formers in type theory. Recently, even larger universes and universe operators have been proposed by Rathjen, Griffor, and Palmgren [39], Palmgren =-=[36]-=-, and Setzer [42]. Martin-Lof type theory with generalised inductive definitions and universes have great proof-theoretic strength, see Griffor and Rathjen [23], Setzer [41], and Rathjen, Griffor, and... |

31 |
Programming with proofs: A second order type theory
- Parigot
- 1988
(Show Context)
Citation Context ...a list of constructors. Independent families of objects can always be found, but the definability of case forces us to use "recursive" rather than "iterative" encodings, see the di=-=scussion by Parigot [37]-=- and Altenkirch [6]. Secondly, for each recursively defined function constant, we construct an element satisfying the recursion equations in question. This can be done in a standard way using fixed po... |

29 | From semantics to rules: A machine assisted analysis - Coquand - 1994 |

27 | The strength of some Martin-Löf type theories
- Griffor, Rathjen
- 1994
(Show Context)
Citation Context ...n, Griffor, and Palmgren [39], Palmgren [36], and Setzer [42]. Martin-Lof type theory with generalised inductive definitions and universes have great proof-theoretic strength, see Griffor and Rathjen =-=[23], Setzer [-=-41], and Rathjen, Griffor, and Palmgren [39]. We conclude this introduction with a few words about the notation. We employ the "logical framework " formulation of Martin-Lof type theory [30,... |

25 | Proof theoretical strength of Martin-Löf Type Theory with W-type and one universe
- Setzer
- 1993
(Show Context)
Citation Context ...nd Palmgren [39], Palmgren [36], and Setzer [42]. Martin-Lof type theory with generalised inductive definitions and universes have great proof-theoretic strength, see Griffor and Rathjen [23], Setzer =-=[41], and Rath-=-jen, Griffor, and Palmgren [39]. We conclude this introduction with a few words about the notation. We employ the "logical framework " formulation of Martin-Lof type theory [30, 34]. The cor... |

24 | Normalization and the Yoneda embedding - Čubrić, Dybjer, et al. - 1998 |

21 |
Inductively defined types, preliminary version
- Coquand, Paulin-Mohring
(Show Context)
Citation Context ...sis of Gim'enez' [22] implementation of inductive definitions in a proof editor for Martin-Lof type theory, and as Coquand and Paulin's formulation of inductive types in the calculus of constructions =-=[16, 38]-=- is the basis for the Coq-system [18]. To illustrate the schema, we show how the rules for the first universe and the fresh-lists can be derived by instantiation. Later sections contain further exampl... |

18 | Inaccessibility in Constructive Set Theory and Type Theory
- Rathjen, Griffor, et al.
- 1998
(Show Context)
Citation Context ...osed under the universe operator as well as under all the usual set formers in type theory. Recently, even larger universes and universe operators have been proposed by Rathjen, Griffor, and Palmgren =-=[39]-=-, Palmgren [36], and Setzer [42]. Martin-Lof type theory with generalised inductive definitions and universes have great proof-theoretic strength, see Griffor and Rathjen [23], Setzer [41], and Rathje... |

18 | The independence of Peano’s fourth axiom from Martin-Löf ’s type theory without Universes
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(Show Context)
Citation Context ...pe theory universes are needed also for the more basic purpose of defining families of sets by structural recursion. For example, the predicate Z (as used 2 in the type-theoretic proof that 0 6= s(n) =-=[29, 43]-=-) with the recursion equations Z(0) = ?; Z(s(n)) = ? is defined in terms of universes and the rule of N-elimination. Martin-Lof [27] introduced an infinite tower of universes U 0 : U 1 : U 2 : \Delta ... |

16 |
Predicative type universes and primitive recursion
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Citation Context ...evidence to the fundamental nature of the schematic natural deduction formulation of inductive definitions in type theory. The idea to consider this generalisation was inspired by Nax Mendler's paper =-=[32]-=- on the categorytheoretic semantics of universes in type theory. Our analysis improves fundamentally on Mendler's, since the category-theoretic machinery can be applied only if the rules for U 0 and T... |

15 |
On the meaning and construction of the rules in Martin-Löf’s theory of types
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(Show Context)
Citation Context ...l cases. See Martin-Lof 1971 [26] for a general formulation of inductive definitions in the language of ordinary first order predicate logic. The first such general schema was formulated by Backhouse =-=[7]-=- and covered the case of inductively defined sets (possibly depending on parameters). This schema was generalised to the case of inductively defined families of sets by Dybjer [19, 20]. Inductively de... |

15 | Extending Martin-Löf Type Theory by one Mahlo-universe
- Setzer
(Show Context)
Citation Context ... as well as under all the usual set formers in type theory. Recently, even larger universes and universe operators have been proposed by Rathjen, Griffor, and Palmgren [39], Palmgren [36], and Setzer =-=[42]-=-. Martin-Lof type theory with generalised inductive definitions and universes have great proof-theoretic strength, see Griffor and Rathjen [23], Setzer [41], and Rathjen, Griffor, and Palmgren [39]. W... |

13 |
The Coq proof assistant version 5.6 user's guide
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Citation Context ...nductive definitions in a proof editor for Martin-Lof type theory, and as Coquand and Paulin's formulation of inductive types in the calculus of constructions [16, 38] is the basis for the Coq-system =-=[18]-=-. To illustrate the schema, we show how the rules for the first universe and the fresh-lists can be derived by instantiation. Later sections contain further examples. It might be helpful to study thes... |

10 |
The strength of Martin–Löf type theory with one universe
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(Show Context)
Citation Context ... 3.1-3.4. This interpretation also applies to definitions with parameters as specified in 3.5 provided these parameters satisfies a particular positivity criterion, see section 6.4.4. We follow Aczel =-=[2, 4]-=- and interpret types as collections of objects of a Frege structure. But instead of first building collections of propositions and truths as in Aczel's work, we shall directly construct the collection... |

8 |
On Fixed Point Operators, Inductive Definitions and Universes in Martin-Löf’s Type Theory
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Citation Context ...set T i (a). A universe `a la Tarski should therefore be understood as a pair (U i ; T i ) consisting of a set U i of codes and a decoding function T i . Further universes were introduced by Palmgren =-=[35]-=-. Firstly, he defined a universe operator, that is, an operator on families of sets which when applied to a universe (U i ; T i ) returns the next universe (U i+1 ; T i+1 ). In this way the external s... |

7 | A realizability interpretation of Martin-Löf’s type theory. Twentyfive years of constructive type theory
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(Show Context)
Citation Context ... structures described above. 20 As further examples of simultaneous induction-recursion, we would like to mention in particular the computability predicates used by Martin-Lof [31, 27] and C. Coquand =-=[12]-=- for proving normalisation of type theory, and the logical relations introduced by T. Coquand [13] for proving soundness and completeness of an algorithm for testing conversion in type theory. These c... |

6 |
Frege Structures and the
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Citation Context ...ses and universe operators can be found in the recent papers by Palmgren [36] and Rathjen, Griffor, and Palmgren [39]. 5 Frege structures The notion of a Frege structure was introduced by Peter Aczel =-=[4]-=-. One purpose was to provide an appropriate setting for -calculus (or abstract realisability) interpretations of Martin-Lof type theory. Another was to provide a model for a foundational framework whe... |

6 |
A normalization proof for an impredicative type system with large eliminations over integers
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Citation Context ...unction defined by recursion necessarily is an element of a set (as in the traditional elimination rules). Instead the value can be an object of an arbitrary type (as in the "large" eliminat=-=ion rules [44, 45]-=-. For example, the value may be a set (an object of the type set), so we have recursively defined families of sets. Note that we have reversed the priority of the following concepts as compared to the... |

4 |
Type Theory and the External Logic of Programs
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Citation Context ... a -structure with set of objects F 0 and an explicit equivalence relation on objectss0 , together with an encoding of the logical constants can be carried out in type theory (see for example Hedberg =-=[24]-=- for a formal development of constructive domain theory inside type theory). It remains to turn the logical schemata into a simultaneous inductive definition of the property P : (a : F 0 )set of propo... |

3 |
Propositional functions and families of types. Notre Dame
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(Show Context)
Citation Context ...unction defined by recursion necessarily is an element of a set (as in the traditional elimination rules). Instead the value can be an object of an arbitrary type (as in the "large" eliminat=-=ion rules [44, 45]-=-. For example, the value may be a set (an object of the type set), so we have recursively defined families of sets. Note that we have reversed the priority of the following concepts as compared to the... |

2 | Amendment to intuitionistic type theory. Notes from a lecture given in Göteborg - Martin-Löf - 1986 |

2 |
Adding proof objects and inductive definition mechanism to Frege structures
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(Show Context)
Citation Context ...t is not clear how to use category-theoretic ideas for obtaining a formal system for simultaneous induction-recursion. The idea to enrich Frege structures with proof objects can also be found in Sato =-=[40]-=-. However, Sato works in a type-free constructive theory and not in type theory. Working in the same framework as Sato, Kameyama [25] has developed an approach to half-positive inductive definitions. ... |

1 |
A command for inductive sets in ILF. Master Thesis, Universidad de la Rep'ublica
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(Show Context)
Citation Context ...and the old schema we use the same notation here as in Dybjer [20]. The present description could form the basis for an implementation in the same way as the old schema [20] is the basis of Gim'enez' =-=[22]-=- implementation of inductive definitions in a proof editor for Martin-Lof type theory, and as Coquand and Paulin's formulation of inductive types in the calculus of constructions [16, 38] is the basis... |

1 |
A type-free theory of half-monotone inductive definitions
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(Show Context)
Citation Context ...rich Frege structures with proof objects can also be found in Sato [40]. However, Sato works in a type-free constructive theory and not in type theory. Working in the same framework as Sato, Kameyama =-=[25]-=- has developed an approach to half-positive inductive definitions. His aims are similar to ours: to formulate a general notion which subsumes the construction of Frege structures and enables the inter... |