## Induction-recursion and initial algebras (2003)

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Venue: | Annals of Pure and Applied Logic |

Citations: | 29 - 11 self |

### BibTeX

@INPROCEEDINGS{Dybjer03induction-recursionand,

author = {Peter Dybjer and Anton Setzer},

title = {Induction-recursion and initial algebras},

booktitle = {Annals of Pure and Applied Logic},

year = {2003},

pages = {2003}

}

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### Abstract

1 Introduction Induction-recursion is a powerful definition method in intuitionistic type theory in the sense of Scott ("Constructive Validity") [31] and Martin-L"of [17, 18, 19]. The first occurrence of formal induction-recursion is Martin-L"of's definition of a universe `a la Tarski [19], which consists of a set U

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Introduction to Higher Order Categorical Logic
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- 1986
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Citation Context ...d categorical models, such as the correspondence between the typed lambda calculus and Cartesian closed categories, between (impredicative) intuitionistic type theory in the sense of Lambek and Scott =-=[15]-=- and toposes, etc. Note however, that we here only treat the categorical semantics of induction-recursion and not of the logical framework. The reader is referred to the literature on categorical sema... |

347 |
Intuitionistic Type Theory
- Martin–Löf
- 1984
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Citation Context ... of at least Rathjen’s theory KPM. 1 Introduction Induction-recursion is a powerful definition method in intuitionistic type theory in the sense of Scott (“Constructive Validity”) [31] and Martin-Löf =-=[17, 18, 19]-=-. The first occurrence of formal induction-recursion is Martin-Löf’s definition of a universe à la Tarski [19], which consists of a set U0 of codes for small sets together with a decoding function T0 ... |

276 |
Constructive mathematics and computer programming
- Martin-Löf
- 1982
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Citation Context ... of at least Rathjen’s theory KPM. 1 Introduction Induction-recursion is a powerful definition method in intuitionistic type theory in the sense of Scott (“Constructive Validity”) [31] and Martin-Löf =-=[17, 18, 19]-=-. The first occurrence of formal induction-recursion is Martin-Löf’s definition of a universe à la Tarski [19], which consists of a set U0 of codes for small sets together with a decoding function T0 ... |

271 | Programming in Martin–Löf ’s Type Theory. An introduction - Nordström, Petersson, et al. - 1990 |

172 | Inductive definitions in the system Coq: Rules and properties
- Paulin-Mohring
- 1993
(Show Context)
Citation Context ... for definitions of this kind was identified by Dybjer [8, 11], who presents an external schema for their syntactic form. This schema extends earlier schemata for inductive definitions in type theory =-=[6, 7, 9, 26]-=-. Dybjer and Setzer [12] give a finite axiomatization of inductionrecursion as a very general reflection principle. They also show the consistency of their axiomatization by building a model in classi... |

163 |
An intuitionistic Theory of types: predicative part
- Martin–Löf
- 1973
(Show Context)
Citation Context ... of at least Rathjen’s theory KPM. 1 Introduction Induction-recursion is a powerful definition method in intuitionistic type theory in the sense of Scott (“Constructive Validity”) [31] and Martin-Löf =-=[17, 18, 19]-=-. The first occurrence of formal induction-recursion is Martin-Löf’s definition of a universe à la Tarski [19], which consists of a set U0 of codes for small sets together with a decoding function T0 ... |

78 | Inductive sets and families in Martin-Löf’s type theory and their settheoretic semantics
- Dybjer
- 1991
(Show Context)
Citation Context ... for definitions of this kind was identified by Dybjer [8, 11], who presents an external schema for their syntactic form. This schema extends earlier schemata for inductive definitions in type theory =-=[6, 7, 9, 26]-=-. Dybjer and Setzer [12] give a finite axiomatization of inductionrecursion as a very general reflection principle. They also show the consistency of their axiomatization by building a model in classi... |

69 | Inductive families
- Dybjer
- 1994
(Show Context)
Citation Context ... for definitions of this kind was identified by Dybjer [8, 11], who presents an external schema for their syntactic form. This schema extends earlier schemata for inductive definitions in type theory =-=[6, 7, 9, 26]-=-. Dybjer and Setzer [12] give a finite axiomatization of inductionrecursion as a very general reflection principle. They also show the consistency of their axiomatization by building a model in classi... |

67 | A theory of types - Martin-Lof - 1971 |

65 | A General Formulation of Simultaneous Inductive-Recursive Definitions in Type Theory
- Dybjer
- 2000
(Show Context)
Citation Context ...f for the fact that computability has been defined. In order to obtain an explicit inductiverecursive definition one has to formalize the metalanguage. It is an example of indexed induction-recursion =-=[11, 13]-=-, since we are defining computability predicates and thus by Curry-Howard indexed families of sets. More examples of formal induction-recursion occur in recent work on large universes in type theory. ... |

64 |
Frege structures and the notions of proposition, truth and set
- Aczel
- 1980
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Citation Context ... but did not provide an explicit discussion of why this is so. It is a non-trivial problem to give classical mathematical meaning to MartinLöf’s computability predicates. One approach is due to Aczel =-=[1]-=- for the closely related construction of a Frege structure. Other approaches have been proposed by Allen [2] and by Löfwall and Sjödin [16]. Although Martin-Löf’s computability predicates nowadays can... |

60 |
Locally Cartesian closed categories and type theory
- Seely
- 1984
(Show Context)
Citation Context ...ntics of induction-recursion and not of the logical framework. The reader is referred to the literature on categorical semantics of dependent types for the latter, see for example Cartmell [4], Seely =-=[32]-=-, Dybjer [10] or Hofmann [14]. The categorical semantics of universes has previously been investigated by Mendler [21]. There he considers various universes which are all inductive-recursive definitio... |

59 |
Generalised algebraic theories and contextual categories
- Cartmell
- 1986
(Show Context)
Citation Context ...orical semantics of induction-recursion and not of the logical framework. The reader is referred to the literature on categorical semantics of dependent types for the latter, see for example Cartmell =-=[4]-=-, Seely [32], Dybjer [10] or Hofmann [14]. The categorical semantics of universes has previously been investigated by Mendler [21]. There he considers various universes which are all inductive-recursi... |

56 |
A Non-Type-Theoretic Semantics for Type-Theoretic Language
- Allen
- 1987
(Show Context)
Citation Context ... mathematical meaning to MartinLöf’s computability predicates. One approach is due to Aczel [1] for the closely related construction of a Frege structure. Other approaches have been proposed by Allen =-=[2]-=- and by Löfwall and Sjödin [16]. Although Martin-Löf’s computability predicates nowadays can be regarded as an informal example of an inductive-recursive definition and therefore as a precursor of the... |

44 | Indexed induction-recursion - Dybjer, Setzer |

43 | A finite axiomatization of inductive-recursive definitions
- Dybjer, Setzer
- 1998
(Show Context)
Citation Context ...es. We shall therefore recall an alternative formalization of induction-recursion which maintains the distinctions in the table above. This formalization was previously presented in Dybjer and Setzer =-=[12]-=-. It expresses induction-recursion as a reflection principle: for any type D and any D-operation d of “arity” φ, there is a set U ′ φ,d which is closed under d and has decoding function T ′ φ,d : U′ φ... |

41 | Syntax and semantics of dependent types
- Hofmann
- 1997
(Show Context)
Citation Context ...and not of the logical framework. The reader is referred to the literature on categorical semantics of dependent types for the latter, see for example Cartmell [4], Seely [32], Dybjer [10] or Hofmann =-=[14]-=-. The categorical semantics of universes has previously been investigated by Mendler [21]. There he considers various universes which are all inductive-recursive definitions with D = set. Our approach... |

38 | Internal type theory
- Dybjer
(Show Context)
Citation Context ...ction-recursion and not of the logical framework. The reader is referred to the literature on categorical semantics of dependent types for the latter, see for example Cartmell [4], Seely [32], Dybjer =-=[10]-=- or Hofmann [14]. The categorical semantics of universes has previously been investigated by Mendler [21]. There he considers various universes which are all inductive-recursive definitions with D = s... |

33 | On universes in type theory
- Palmgren
- 1998
(Show Context)
Citation Context ...en’s superuniverse [23] is an analogue of a hyperinaccessible cardinal. Rathjen, Griffor and Palmgren’s quantifier universes [30] are analogues of Mahlo’s π-numbers; Palmgren’s higher order universes =-=[25]-=- go even further and are generally conjectured to reach the strength of Rathjen’s theory KPM; in Section 6 we will describe a weak version of Setzer’s Mahlo universe [36, 34, 35], which is still induc... |

25 | Proof theoretical strength of Martin-Löf Type Theory with W-type and one universe
- Setzer
- 1993
(Show Context)
Citation Context ...1 ⇒ θn+1 of dependent judgements in the language of type theory with respect to a certain collection of constructors (for a 5full formalization of the language of type theory see for instance Setzer =-=[33]-=-, chapter 2). If R is a collection of rules we introduce the type theory TT(R). We use the notation R ⊢ Γ ⇒ θ to make explicit that the judgement θ is derivable in the context Γ by using the rules of ... |

24 |
Proof-theoretic analysis of KPM
- RATHJEN
- 1991
(Show Context)
Citation Context ...heory KPM; in Section 6 we will describe a weak version of Setzer’s Mahlo universe [36, 34, 35], which is still inductive-recursive, and show that it has at least the strength of Rathjen’s theory KPM =-=[28]-=-. Setzer’s original Mahlo universe is an example of a universe which goes beyond induction-recursion. Induction-recursion as a general unifying principle for definitions of this kind was identified by... |

23 | Inductive Families. Formal Aspects of Computing 6(4 - Dybjer |

21 | Inductively defined types, preliminary version - Coquand, Paulin-Mohring |

21 |
Constructive validity
- Scott
- 1970
(Show Context)
Citation Context ...theoretical strength of at least Rathjen’s theory KPM. 1 Introduction Induction-recursion is a powerful definition method in intuitionistic type theory in the sense of Scott (“Constructive Validity”) =-=[31]-=- and Martin-Löf [17, 18, 19]. The first occurrence of formal induction-recursion is Martin-Löf’s definition of a universe à la Tarski [19], which consists of a set U0 of codes for small sets together ... |

18 | Inaccessibility in Constructive Set Theory and Type Theory
- Rathjen, Griffor, et al.
- 1998
(Show Context)
Citation Context ...ple, Martin-Löf’s universes are analogues of inaccessible cardinals; Palmgren’s superuniverse [23] is an analogue of a hyperinaccessible cardinal. Rathjen, Griffor and Palmgren’s quantifier universes =-=[30]-=- are analogues of Mahlo’s π-numbers; Palmgren’s higher order universes [25] go even further and are generally conjectured to reach the strength of Rathjen’s theory KPM; in Section 6 we will describe a... |

16 |
Predicative type universes and primitive recursion
- Mendler
- 1991
(Show Context)
Citation Context ... semantics of dependent types for the latter, see for example Cartmell [4], Seely [32], Dybjer [10] or Hofmann [14]. The categorical semantics of universes has previously been investigated by Mendler =-=[21]-=-. There he considers various universes which are all inductive-recursive definitions with D = set. Our approach goes further since we consider inductive-recursive definitions with arbitrary D and char... |

15 | Extending Martin-Löf Type Theory by one Mahlo-universe
- Setzer
(Show Context)
Citation Context ...gren’s higher order universes [25] go even further and are generally conjectured to reach the strength of Rathjen’s theory KPM; in Section 6 we will describe a weak version of Setzer’s Mahlo universe =-=[36, 34, 35]-=-, which is still inductive-recursive, and show that it has at least the strength of Rathjen’s theory KPM [28]. Setzer’s original Mahlo universe is an example of a universe which goes beyond induction-... |

12 |
Ordinal notations based on a weakly Mahlo cardinal, Archive for Mathematical Logic 29
- Rathjen
- 1990
(Show Context)
Citation Context ...1)) ∈ W ∩ Ω1 ⊑ OT , and from transfinite induction over W follows transfinite induction up to ψΩ1(ωn(M + 1)) for n ∈ ω, which in the limit reaches ψΩ1(ɛM+1). Rathjen determined the strength of KPM in =-=[27, 28, 29]-=-. The ordinal notation systems we used is based on [3], where it is shown that the strength of KPM is at most ψΩ1(ɛM+1) (which can be seen to be sharp as in [29] or by taking the above proof and adapt... |

9 |
Universes and a general notion of simultaneous inductive–recursive de nition in type theory
- Dybjer
- 1992
(Show Context)
Citation Context ...’s original Mahlo universe is an example of a universe which goes beyond induction-recursion. Induction-recursion as a general unifying principle for definitions of this kind was identified by Dybjer =-=[8, 11]-=-, who presents an external schema for their syntactic form. This schema extends earlier schemata for inductive definitions in type theory [6, 7, 9, 26]. Dybjer and Setzer [12] give a finite axiomatiza... |

8 |
On Fixed Point Operators, Inductive Definitions and Universes in Martin-Löf’s Type Theory
- Palmgren
- 1991
(Show Context)
Citation Context ...arge universes in type theory. These are constructive analogues of large cardinals in set theory. For example, Martin-Löf’s universes are analogues of inaccessible cardinals; Palmgren’s superuniverse =-=[23]-=- is an analogue of a hyperinaccessible cardinal. Rathjen, Griffor and Palmgren’s quantifier universes [30] are analogues of Mahlo’s π-numbers; Palmgren’s higher order universes [25] go even further an... |

8 |
Type-theoretic Interpretation of Iterated, Strictly Positive Inductive Definitions,” Archive For Mathematical Logic 32
- Palmgren
- 1992
(Show Context)
Citation Context ... the set of natural numbers or the W-type are special cases of inductive-recursive definitions, so we obtain elimination rules for these sets as well as we do for universes (as introduced by Palmgren =-=[24]-=-). We have to collect the induction hypotheses with respect to an argument of introγ, that is, with respect to an element u of F U γ (Uγ, Tγ). The induction hypothesis consists of the value of the fun... |

7 | A model for a type theory with Mahlo universe, Draft, available from http://www-compsci.swan.ac.uk/∼csetzer
- Setzer
- 1996
(Show Context)
Citation Context ...gren’s higher order universes [25] go even further and are generally conjectured to reach the strength of Rathjen’s theory KPM; in Section 6 we will describe a weak version of Setzer’s Mahlo universe =-=[36, 34, 35]-=-, which is still inductive-recursive, and show that it has at least the strength of Rathjen’s theory KPM [28]. Setzer’s original Mahlo universe is an example of a universe which goes beyond induction-... |

5 | A note on the ordinal analysis of KPM, in - Buchholz - 1993 |

4 | Agda homepage - Coquand |

3 | Proof-theoretical analysis of - Rathjen - 1991 |

2 |
Collapsing functions based on recursively large cardinals: a well-ordering proof for
- Rathjen
- 1994
(Show Context)
Citation Context ...1)) ∈ W ∩ Ω1 ⊑ OT , and from transfinite induction over W follows transfinite induction up to ψΩ1(ωn(M + 1)) for n ∈ ω, which in the limit reaches ψΩ1(ɛM+1). Rathjen determined the strength of KPM in =-=[27, 28, 29]-=-. The ordinal notation systems we used is based on [3], where it is shown that the strength of KPM is at most ψΩ1(ɛM+1) (which can be seen to be sharp as in [29] or by taking the above proof and adapt... |

2 |
A note on the ordinal analysis of KP M
- Buchholz
- 1993
(Show Context)
Citation Context ...ws transfinite induction up to ψΩ1(ωn(M + 1)) for n ∈ ω, which in the limit reaches ψΩ1(ɛM+1). Rathjen determined the strength of KPM in [27, 28, 29]. The ordinal notation systems we used is based on =-=[3]-=-, where it is shown that the strength of KPM is at most ψΩ1(ɛM+1) (which can be seen to be sharp as in [29] or by taking the above proof and adapting it to KPM). Therefore the assertion of the theorem... |

2 | Inductively de ned types, preliminary version - Coquand, Paulin - 1990 |

2 | A nite axiomatization of inductive-recursive denitions - Dybjer, Setzer - 1999 |

2 | Inductive de nitions in the system Coq—rules and properties, in: Typed lambda calculi and applications - Paulin-Mohring - 1993 |

1 |
Strong normalizability in Martin-Lof’s type theory
- Lofwall, Sjodin
- 1991
(Show Context)
Citation Context ...nLöf’s computability predicates. One approach is due to Aczel [1] for the closely related construction of a Frege structure. Other approaches have been proposed by Allen [2] and by Löfwall and Sjödin =-=[16]-=-. Although Martin-Löf’s computability predicates nowadays can be regarded as an informal example of an inductive-recursive definition and therefore as a precursor of the concept of induction-recursion... |

1 |
An intuitionistic theoy of types
- Martin-Löf
- 1998
(Show Context)
Citation Context ...00303/G; Department of Computer Science, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK, Email: a.g.setzer@swan.ac.uk 1proof of normalization of an early version of his type theory =-=[20]-=- he introduces Taitstyle computability predicates for dependent types. Whereas Tait defines a family of computability predicates indexed by the types of the simply typed lambda calculus, Martin-Löf’s ... |

1 |
A type theory for Mahlo universes, Abstract for Logic Colloquium
- Setzer
- 1997
(Show Context)
Citation Context ...gren’s higher order universes [25] go even further and are generally conjectured to reach the strength of Rathjen’s theory KPM; in Section 6 we will describe a weak version of Setzer’s Mahlo universe =-=[36, 34, 35]-=-, which is still inductive-recursive, and show that it has at least the strength of Rathjen’s theory KPM [28]. Setzer’s original Mahlo universe is an example of a universe which goes beyond induction-... |

1 |
The Agda homepage, February 2000. http://www.cs.chalmers.se/~catarina/agda/. 38 T. Coquand and
- Coquand
- 1990
(Show Context)
Citation Context ...mitted it we could still define the empty set as the set N0 with only one constructor of type N0 → N0, see p. 13. 1 In the proof assistant Agda for type theory (developed by C. Coquand and T. Coquand =-=[5]-=-) the logical framework has been modified so that the type set is closed under the dependent product and dependent function space of the logical framework. If we formulated induction-recursion based o... |

1 | Internal type theory, in: TYPES ’95, Types for Proofs and - Dybjer - 1996 |

1 | On xed point operators, inductive de nitions and universes in Martin-Lof’s type theory - Palmgren - 1991 |

1 | Constructive validity, in: Symp. on Automatic Demonstration - Scott - 1970 |

1 | Extending Martin-Lof type theory by one - Setzer - 2000 |