## A finite axiomatization of inductive-recursive definitions (1999)

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Venue: | Typed Lambda Calculi and Applications, volume 1581 of Lecture Notes in Computer Science |

Citations: | 43 - 14 self |

### BibTeX

@INPROCEEDINGS{Dybjer99afinite,

author = {Peter Dybjer and Anton Setzer},

title = {A finite axiomatization of inductive-recursive definitions},

booktitle = {Typed Lambda Calculi and Applications, volume 1581 of Lecture Notes in Computer Science},

year = {1999},

pages = {129--146},

publisher = {Springer}

}

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### Abstract

Induction-recursion is a schema which formalizes the principles for introducing new sets in Martin-Löf’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 inductive-recursive definitions. We prove consistency by constructing a set-theoretic model which makes use of one Mahlo cardinal. 1

### Citations

347 | Intuitionistic Type Theory - Martin-Löf - 1984 |

266 |
Programming in MartinLöf’s Type Theory, an introduction
- Nordstrom, Petersson, et al.
- 1990
(Show Context)
Citation Context ...et constructors by giving appropriate codes. In Section 6 we build a set-theoretic model. In Section 7 we mention some related work. 2 An Extension of the Logical Framework The Logical Framework (see =-=[15]-=-) has the following forms of judgements: Γ context, and A : type, A = B : type, a : A, a = b : A, depending on contexts Γ (written as Γ ⇒ A : type, etc.). We have set : type and if A : set, then A : t... |

166 | Inductive definitions in the system Coq. Rules and properties
- Paulin-Mohring
- 1993
(Show Context)
Citation Context ...y has mainly used external schemata in the style of Martin-Löf’s intuitionistic theory of iterated inductive definitions in predicate logic [11]. See for example Backhouse [3], Dybjer [9], and Paulin =-=[18]-=-. A schema for inductive-recursive definitions was introduced by Dybjer [7]. Categorical semantics of inductive types and of universes. The categorical semantics of inductively defined dependent types... |

145 |
Intuitionistic type theory. Bibliopolis
- Martin-Löf
- 1984
(Show Context)
Citation Context ...ion 5). In the case of inductive-recursive definitions however, a later premise may also depend on an earlier inductive premise. We consider the key example, the ordinary first universe U à la Tarski =-=[12]-=-, which is defined inductively, while simultaneously defining the decoding function T : U → set recursively. Consider one of its constructors, ̂ Σ : (x : U) → (y : T(x) → U) → U with the defining equa... |

77 | Inductive sets and families in Martin-Löf’s Type Theory and their set-theoretic semantics: An inversion principle for Martin-Löf’s type theory
- Dybjer
- 1991
(Show Context)
Citation Context ...inaccessible above M, however all types will be interpreted as elements of VΛ, where Λ is the first (non-regular) fixed point of λα.ℵα above M. We will develop the semantics following the approach in =-=[8]-=-. Let λ0 := ℵM+1, λn+1 := ℵλn, and Λ := supn∈ω λn. If a, a1, . . . , an, c are sets, and b is a function with domain a, let Πx∈ab(x) := {f | f function ∧ dom(f) = a ∧ ∀x ∈ a.f(x) ∈ b(x)} , λx ∈ a.b(x)... |

67 | Inductive families
- Dybjer
- 1994
(Show Context)
Citation Context ...ises. We cannot make use of inductive premises, because they only give information about the set we are currently defining. To capture inductive definitions of sets in the presence of dependent types =-=[8, 9]-=-, we thus only need to change the notion of a strictly positive functor Φ above by replacing the non-inductive case by: 1 In [7] the terminology “non-recursive premise” was used, but “non-inductive pr... |

66 | An intuitionistic theory of types - Martin-Löf - 1998 |

65 | A General Formulation of Simultaneous Inductive-Recursive Definitions in Type Theory
- Dybjer
- 2000
(Show Context)
Citation Context ...ulate a compact, completely formal theory of inductive-recursive definitions, and to prove its consistency. Induction-recursion is a schema for introducing new sets in type theory developed by Dybjer =-=[7]-=-. All the usual sets in Martin-Löf’s type theory and practically all sets (data types), which are defined in analogy with it, are instances of this schema. Applications of induction-recursion include ... |

59 |
Hauptsatz for the intuitionistic theory of iterated inductive definitions
- Martin-Löf
- 1971
(Show Context)
Citation Context ...lization of inductive definitions in Martin-Löf’s type theory has mainly used external schemata in the style of Martin-Löf’s intuitionistic theory of iterated inductive definitions in predicate logic =-=[11]-=-. See for example Backhouse [3], Dybjer [9], and Paulin [18]. A schema for inductive-recursive definitions was introduced by Dybjer [7]. Categorical semantics of inductive types and of universes. The ... |

56 |
A Non- Type-Theoretic Semantics for Type-Theoretic Language
- Allen
- 1987
(Show Context)
Citation Context ... intro is the syntactic reflection of d : Φ Arg → D. 3 2 As this example shows the term “strictly positive” may no longer be wholly appropriate, since the T -argument now can appear negatively. Allen =-=[2]-=- used the alternative term “half-positive” for this reason. U always appears strictly positively however. 3 Recall that the term “universe à la Tarski” was chosen by Martin-Löf [12] because of the sim... |

35 |
Set Theory: An Introduction to Large Cardinals
- Drake
- 1974
(Show Context)
Citation Context ...c interpretations of type theory. Large cardinals in set theory. Induction-recursion gives quite a general approach to type-theoretic analogues of large cardinals in set theory. See for example Drake =-=[6]-=- for an introduction to large cardinals. Induction-recursion gives rise to analogues of for example inaccessible, hyper-inaccessible cardinals, and more generally Mahlo’s π-numbers [19], but does not ... |

25 | The strength of some Martin-Löf type theories
- Griffor, Rathjen
- 1994
(Show Context)
Citation Context ...can be seen as being almost a Mahlo-universe, since we have induction over arbitrary types. What is missing to get the full strength is the possibility of having the W-type on top of the universe. In =-=[10]-=- together with [22], [24], [25] it was shown that in case of one universe such a restriction reduces the strength from |KPI + | to |KPI| and with a similar argument for the lower bound as in [10] it i... |

24 | Proof theoretical strength of Martin-Löf Type Theory with W-type and one universe
- Setzer
- 1993
(Show Context)
Citation Context ...n as well define a model in a theory of the same strength by giving a realizability interpretation in Kripke-Platek set theory extended by a recursive Mahlo ordinal and ω admissibles above, extending =-=[21, 24, 23]-=-. Both models require some extra work, which exceeds the space available in this article. 10 Proof-theoretic strength of type theory. It should be easy to develop a term model of the theory in KPM + u... |

21 | On relating type theories and set theories
- Aczel
- 1999
(Show Context)
Citation Context ...l. Ordinary dependent type theory with generalized inductive definitions (that is, Martin-Löf’s type theory without universes) has a natural full function space interpretation in classical set theory =-=[1, 8]-=-. As shown by our construction of a set-theoretic model the step from inductive to inductive-recursive definitions in type theory is roughly analogous to moving from ordinary ZF set theory to ZF set t... |

20 |
Inductively defined types (preliminary version
- Coquand, Paulin
- 1990
(Show Context)
Citation Context ...roduced by Dybjer [7]. Categorical semantics of inductive types and of universes. The categorical semantics of inductively defined dependent types has been discussed for example by Coquand and Paulin =-=[5]-=- and Mendler [14]. The latter article also discusses categorical semantics of universes in type theory. In a future article we plan to extend Mendler’s work, by giving categorical semantics of inducti... |

19 | Well-ordering proofs for Martin-Löf’s type theory with W-type and one universe
- Setzer
- 1998
(Show Context)
Citation Context ...ahlo-universe, since we have induction over arbitrary types. What is missing to get the full strength is the possibility of having the W-type on top of the universe. In [10] together with [22], [24], =-=[25]-=- it was shown that in case of one universe such a restriction reduces the strength from |KPI + | to |KPI| and with a similar argument for the lower bound as in [10] it is very likely that using 10 The... |

16 |
Predicative type universes and primitive recursion
- Mendler
- 1991
(Show Context)
Citation Context ...r [7]. Categorical semantics of inductive types and of universes. The categorical semantics of inductively defined dependent types has been discussed for example by Coquand and Paulin [5] and Mendler =-=[14]-=-. The latter article also discusses categorical semantics of universes in type theory. In a future article we plan to extend Mendler’s work, by giving categorical semantics of inductive-recursive defi... |

16 | Inaccessibility in constructive set theory and type theory
- Rathjen, Griffor, et al.
- 1998
(Show Context)
Citation Context ...definition in type theory was Martin-Löf’s universe à la Tarski [12]. 9 Then Palmgren [17] defined external and internal universe hierarchies and also a super universe. Rathjen, Griffor, and Palmgren =-=[19]-=- defined quantifier universes and Palmgren [16] defined higher order universe hierarchies. All these constructions use induction-recursion, whereas Setzer [20] defined a Mahlo universe, which goes bey... |

15 |
On the meaning and construction of the rules in Martin-Löf’s theory of types
- Backhouse
- 1987
(Show Context)
Citation Context ...s in Martin-Löf’s type theory has mainly used external schemata in the style of Martin-Löf’s intuitionistic theory of iterated inductive definitions in predicate logic [11]. See for example Backhouse =-=[3]-=-, Dybjer [9], and Paulin [18]. A schema for inductive-recursive definitions was introduced by Dybjer [7]. Categorical semantics of inductive types and of universes. The categorical semantics of induct... |

15 | Extending Martin-Löf Type Theory by one Mahlo-universe
- Setzer
(Show Context)
Citation Context ...universe. Rathjen, Griffor, and Palmgren [19] defined quantifier universes and Palmgren [16] defined higher order universe hierarchies. All these constructions use induction-recursion, whereas Setzer =-=[20]-=- defined a Mahlo universe, which goes beyond it. 9 There are earlier examples of informal inductive-recursive definitions, for example, MartinLöf’s simultaneous definition of the notions of computable... |

8 |
On Fixed Point Operators, Inductive Definitions and Universes in Martin-Löf’s Type Theory
- Palmgren
- 1991
(Show Context)
Citation Context ... U κ+1 ⊆ U M . ⊓⊔ 7 Related and Future Work Universes in type theory. The first example of an inductive-recursive definition in type theory was Martin-Löf’s universe à la Tarski [12]. 9 Then Palmgren =-=[17]-=- defined external and internal universe hierarchies and also a super universe. Rathjen, Griffor, and Palmgren [19] defined quantifier universes and Palmgren [16] defined higher order universe hierarch... |

8 | An upper bound for the proof theoretical strength of Martin-Löf type theory with W-type and one universe
- Setzer
- 1996
(Show Context)
Citation Context ...n as well define a model in a theory of the same strength by giving a realizability interpretation in Kripke-Platek set theory extended by a recursive Mahlo ordinal and ω admissibles above, extending =-=[21, 24, 23]-=-. Both models require some extra work, which exceeds the space available in this article. 10 Proof-theoretic strength of type theory. It should be easy to develop a term model of the theory in KPM + u... |

8 |
On the syntax of Martin-Löf’s type theories
- Troelstra
- 1987
(Show Context)
Citation Context ...ve-recursive definitions. Set-theoretic semantics of type theory. It is well-known that Martin-Löf’s type theory has a “naive” full function-space model, see for example the introduction in Troelstra =-=[26]-=-. Dybjer [8] gives a full function space model of Martin-Löf’s type theory with an external schema for inductive definitions. Aczel’s recent article [1] contains further information about set-theoreti... |

7 | Pointfree Approach to Constructive Analysis in Type Theory
- Cederquist
- 1997
(Show Context)
Citation Context ...ical semantics of inductively defined sets. The resulting theory has been implemented in the Half system, a proof assistant for Martin-Löf’s type theory developed by Coquand and Synek, see Cederquist =-=[4]-=-. Plan of the paper. In Section 2 we present Martin-Löf’s Logical Framework. In Section 3 we recall how to use initial algebras for giving categorical semantics of inductive types in the simply typed ... |

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 ...n as well define a model in a theory of the same strength by giving a realizability interpretation in Kripke-Platek set theory extended by a recursive Mahlo ordinal and ω admissibles above, extending =-=[21, 24, 23]-=-. Both models require some extra work, which exceeds the space available in this article. 10 Proof-theoretic strength of type theory. It should be easy to develop a term model of the theory in KPM + u... |

2 |
On universes in type theory. To appear
- Palmgren
(Show Context)
Citation Context ...erse à la Tarski [12]. 9 Then Palmgren [17] defined external and internal universe hierarchies and also a super universe. Rathjen, Griffor, and Palmgren [19] defined quantifier universes and Palmgren =-=[16]-=- defined higher order universe hierarchies. All these constructions use induction-recursion, whereas Setzer [20] defined a Mahlo universe, which goes beyond it. 9 There are earlier examples of informa... |