## Inductive Families (1997)

Venue: | Formal Aspects of Computing |

Citations: | 70 - 13 self |

### BibTeX

@ARTICLE{Dybjer97inductivefamilies,

author = {Peter Dybjer},

title = {Inductive Families},

journal = {Formal Aspects of Computing},

year = {1997},

volume = {6},

pages = {440--465}

}

### Years of Citing Articles

### OpenURL

### Abstract

A general formulation of inductive and recursive definitions in Martin-Lof's type theory is presented. It extends Backhouse's `Do-It-Yourself Type Theory' to include inductive definitions of families of sets and definitions of functions by recursion on the way elements of such sets are generated. The formulation is in natural deduction and is intended to be a natural generalization to type theory of Martin-Lof's theory of iterated inductive definitions in predicate logic. Formal criteria are given for correct formation and introduction rules of a new set former capturing definition by strictly positive, iterated, generalized induction. Moreover, there is an inversion principle for deriving elimination and equality rules from the formation and introduction rules. Finally, there is an alternative schematic presentation of definition by recursion. The resulting theory is a flexible and powerful language for programming and constructive mathematics. We hint at the wealth of possible applic...

### Citations

717 | A framework for defining logics
- Harper, Honsell, et al.
- 1993
(Show Context)
Citation Context ...where there are formal rules for a correct current theory T as well as for a correct context \Gamma. (Compare the presentation of the rules for the Edinburgh LF system in Harper, Honsell, and Plotkin =-=[17]-=-, which contain rules for correct signatures as well as for contexts. Signatures play a similar rule to our current theories, but cannot contain definitional equalities. Moreover, correctness of a sig... |

548 |
A computational logic
- Boyer, Moore
- 1979
(Show Context)
Citation Context ...etimes machine-assisted) program derivation and theorem proving. One example is the normalization of if-expression, which occurs as part of Boyer and Moore's tautology-checker for propositional logic =-=[4]-=-. The set of if-expressions is an inductively defined 21 set and the subset of normalized if-expressions is naturally represented in type theory as an inductively defined predicate, see Dybjer [12]. M... |

500 |
The calculus of constructions
- Coquand, Huet
- 1988
(Show Context)
Citation Context ...y of Technology and the University of Goteborg, pages 177-190. 1 An example of a general purpose construction is impredicative quantification, which is part of system F, the calculus of constructions =-=[6]-=-, and related systems. This construction is not part of Martin-Lof's type theory, which is predicative. But in the extensional version [26] wellorderings W can be used instead. It can for example be p... |

434 | The Foundation of a Generic Theorem Prover
- Paulson
- 1989
(Show Context)
Citation Context ...rd identification), but also as an `external logic' which can be used for verifying an externally given general recursive program. Hedberg has implemented his examples using Paulson's Isabelle-system =-=[31]-=-. Nora Szasz [36] has formalized the proof that Ackermann's function is not primitive recursive. The basic definition is a binary inductive family of tuples of primitive recursive functions TPR(m;n), ... |

411 | Explicit Substitutions
- Abadi, Cardelli, et al.
- 1990
(Show Context)
Citation Context ...stitution function is explicit in the sense that it is formalized explicitly in type theory, which serves as a metalanguage here. It is not explicit in the sense of Abadi, Cardelli, Curien, and L'evy =-=[1]-=- however, since it is not a constructor for -terms, but instead defined by -recursion on n : N; g :sn : sub : (n : N)(g :sn )(m : N)(fs :sn m ) m ; where /[n; g] = (m : N)(fs :sn m ) m . sub n (var n ... |

355 |
Intuitionistic Type Theory
- Martin-Lof
- 1984
(Show Context)
Citation Context ...nto three parts: ffl general rules; ffl rules for ordinary set formers; ffl rules for universes. Typically the second part includes the set formers \Pi, \Sigma, +, I, N n , N , W and List (Martin-Lof =-=[26]-=-), but it is often remarked that this collection can be extended when there is a need for it. However, the desire for such extensions is so common that general principles need to be layed down which e... |

285 |
An initial algebra approach to the specification, correctness, and implementation of abstract data types
- Goguen, Thatcher, et al.
- 1978
(Show Context)
Citation Context ...ules Omitted. 6.5 Scheme for recursive definitions Omitted. 6.6 More examples Initial Many-Sorted Algebras. With simultaneous induction it is easy to see how to construct initial many sorted algebras =-=[15]-=-. Each sort will denote a (constant) set and each operator an introduction rule with only ordinary recursive premises. If there are equations we associate an inductively defined relation on each set. ... |

280 |
Constructive mathematics and computer programming
- Martin-Löf
- 1982
(Show Context)
Citation Context ...of having a general notion of parameter is a new definition of the equality relation. The usual definition of equality in type theory is as the least reflexive relation on a given set, see Martin-Lof =-=[24]-=-. An alternative definition due to Christine Paulin, which is also an instance of the scheme, is to define it as a unary predicate `to be equal to a' given a set A and an element a : A. The eliminatio... |

275 |
J.M.: Programming in Martin-Löf type theory: an introduction
- Nordström, Petersson, et al.
- 1990
(Show Context)
Citation Context ...representation for the set generated inductively by \Phi. But this method does not work in the intensional version of type theory given by Martin-Lof in 1986 [25] (see Nordstrom, Petersson, and Smith =-=[29]-=-), since it makes use of `extensional isomorphisms', such as N 1 = X N0 and X = X N1 . Another possibility is to add fixed point operators to type theory. But the formulations of Mendler [27] has the ... |

172 |
An intuitionistic theory of types: predicative part, logic colloquium '73 , Rose and Shepherdson (eds
- Martin-Lof
- 1973
(Show Context)
Citation Context ...ion of the scheme. The present scheme uses Martin-Lof's theory of logical types 1 , whereas the theory of the other paper is formulated without this in the style of Martin-Lof's earlier presentations =-=[23, 24, 26]-=-. Another difference is that the present paper shows explicitly how to derive elimination and equality rules for new set formers. Moreover, the class of admissible schematic recursive definitions has ... |

170 | Inductive definitions in the system Coq: Rules and properties
- Paulin-Mohring
- 1993
(Show Context)
Citation Context ... scheme in a natural way to include universes and similar constructions, see Dybjer [14].) The present scheme is very close to a scheme developed independently by Thierry Coquand and Christine Paulin =-=[7, 30]-=-. They have arrived at essentially the same combinatorial structure, but develop it in the context of an impredicative system, so there are some differences in the type structure. The resulting system... |

88 | Pattern matching with dependent types
- Coquand
- 1992
(Show Context)
Citation Context ...hermore, this view is essential, and not only a matter of convenience, both for enlarging the scheme to capture universes and other simultaneous inductive-recursive definitions [14] and for Coquand's =-=[5]-=- approach to pattern matching with dependent types. We shall now give precise criteria for schematic recursive definitions in a similar way as we did for schematic inductive definitions above. This sc... |

77 | Inductive sets and families in Martin-Löf’s type theory and their set-theoretic semantics
- Dybjer
- 1991
(Show Context)
Citation Context ... the other 3 hand it is easy to introduce quotients arising from propositional equalities as in section 5.3 where the untyped -calculus with fi-conversion is presented. There is an accompanying paper =-=[13]-=-, which presents essentially the same scheme. The purpose of that paper is to prove consistency by constructing a set-theoretic model. The purpose of this paper is to show examples and hint at the wea... |

69 | Recursive definitions in type theory - Constable, Mendler - 1985 |

64 |
An abstract framework for environment machines
- Curien
- 1991
(Show Context)
Citation Context ...inite sets and n-tuples In this and the following subsection we use our theory for formalizing basic notions of the untypeds-calculus. The formalization uses bounded de Bruijn-indices (compare Curien =-=[8]-=-) rather 16 than the more common unbounded ones (see, for example, the -calculus theory developed by Huet [19]). Our bounded de Bruijn-indices are elements of the inductive family of finite sets index... |

62 |
Hauptsatz for the intuitionistic theory of iterated inductive definitions
- Martin-Löf
- 1971
(Show Context)
Citation Context ...ble 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, N n and N as special cases. See Martin-Lof 1971 =-=[21]-=- for a general formulation of inductive definitions in the language of ordinary first order predicate logic.' The present scheme generalizes a scheme given by Backhouse [2, 3] under the name `DoIt -Yo... |

48 | A natural extension of natural deduction
- Schroeder-Heister
- 1984
(Show Context)
Citation Context ...' The present scheme generalizes a scheme given by Backhouse [2, 3] under the name `DoIt -Yourself Type Theory'. The existence of a scheme relies on Schroeder-Heister's notion of rule of higher level =-=[33]-=- which made it possible for Martin-Lof to formulate the rules for \Pi so as to conform to the pattern for the other set formers. (Note that \Pi was not mentioned in the quotation above.) Schroeder-Hei... |

31 | Terminating general recursion
- Nordström
- 1988
(Show Context)
Citation Context ...nd A 2 is a binary relation on that set, then Acc A 1 ;A 2 (a) is true iff a is in the well-founded part of A 2 . An application of this notion in the context of type theory can be found in Nordstrom =-=[28]-=-. He suggested to add general recursion along a well-founded relation to a version of type theory in which propositions and sets are not identified and which is also extended with subset formation. Co... |

22 |
A set constructor for inductive sets in Martin-Löf’s type theory
- Petersson, Synek
- 1989
(Show Context)
Citation Context ... Backhouse, Chisholm, Malcolm, and Saaman [3].) ffl General parameters, which can be elements as well as (families of) sets are allowed. Thus the generalized well-founded trees of Petersson and Synek =-=[32]-=- are covered by the scheme. Another nice application of having a general notion of parameter is a new definition of the equality relation. The usual definition of equality in type theory is as the lea... |

21 |
Inductively defined types, preliminary version
- Coquand, Paulin-Mohring
(Show Context)
Citation Context ...drawback that it needs a notion of subtype and therefore require fundamental changes of the theory. A new formulation which does not assume a notion of subtype has been proposed by Coquand and Paulin =-=[7]-=-. The second possibility is the topic of this paper: to specify a scheme which determines correct extensions of a theory. Thus we do not deal with a fixed theory but an open theory. But note that what... |

19 |
The Coq proof assistant version 5.6, users guide, rapport de recherche 134
- Dowek
- 1991
(Show Context)
Citation Context ...ture, but develop it in the context of an impredicative system, so there are some differences in the type structure. The resulting system `the Calculus of Inductive Constructions' is the basis of Coq =-=[10]-=- - a system for interactive proof in an extension of the Calculus of Constructions [6]. I am very grateful to Thierry Coquand and Christine Paulin for many interesting discussions on the topic of indu... |

15 |
On the meaning and construction of the rules in Martin-Löf’s theory of types
- Backhouse
- 1987
(Show Context)
Citation Context ...cases. See Martin-Lof 1971 [21] for a general formulation of inductive definitions in the language of ordinary first order predicate logic.' The present scheme generalizes a scheme given by Backhouse =-=[2, 3]-=- under the name `DoIt -Yourself Type Theory'. The existence of a scheme relies on Schroeder-Heister's notion of rule of higher level [33] which made it possible for Martin-Lof to formulate the rules f... |

13 | How to use Lego (a preliminary user's manual - Luo, Pollack, et al. - 1989 |

12 |
Foundation of Logic Programming Based on Inductive Definition
- Hagiya, Sakurai
- 1984
(Show Context)
Citation Context ...der in which it can be introduced and linked predicate symbols have to be introduced in a block. An application of this theory is as foundation for logic programming as proposed by Hagiya and Sakurai =-=[16]-=-. 7 Further references Several people have used the present formulation of inductively defined sets, families of sets, and predicates in type theory for formal (and sometimes machine-assisted) program... |

11 |
De Bruijn. “Telescopic mappings in typed lambda calculus
- G
- 1991
(Show Context)
Citation Context ...nces of the scheme. An s-type is either a set or a types of functions from an s-type to an s-types. (Another possible name for s-type is purely functional type.) We use de Bruijn's telescope notation =-=[9]-=- and write (a :: oe) as an abbreviation of the sequence (a 1 : oe 1 ) \Delta \Delta \Delta (a n : oe n ) and refer to oe as a `sequence of types'. 3 A scheme for inductive definitions In this section ... |

10 |
Comparing integrated and external logics of functional programs
- Dybjer
- 1990
(Show Context)
Citation Context ...ion along a well-founded relation to a version of type theory in which propositions and sets are not identified and which is also extended with subset formation. Compare also the discussion in Dybjer =-=[12]-=-. Formation rule. Acc : (A 1 : set) (A 2 : (A 1 )(A 1 )set) (a : A 1 ) set : Introduction rule. acc : (A 1 : set) (A 2 : (A 1 )(A 1 )set) (b : A 1 ) (u : (x 1 : A 1 )(x 2 : A 2 (x 1 ; b))Acc A1;A2 (x ... |

10 |
Residual theory in -calculus: a complete Gallina development
- Huet
- 1994
(Show Context)
Citation Context ...of the untypeds-calculus. The formalization uses bounded de Bruijn-indices (compare Curien [8]) rather 16 than the more common unbounded ones (see, for example, the -calculus theory developed by Huet =-=[19]-=-). Our bounded de Bruijn-indices are elements of the inductive family of finite sets indexed by N and defined in a similar way to List 0 above. We have the following rules: N 0 -formation: N 0 : (N)se... |

9 |
Universes and a general notion of simultaneous inductive–recursive de nition in type theory
- Dybjer
- 1992
(Show Context)
Citation Context ...the U-introduction rule a : U (x : T (a)) b(x) : U (a; b) : U ; 2 for example. (It is however possible to extend the scheme in a natural way to include universes and similar constructions, see Dybjer =-=[14]-=-.) The present scheme is very close to a scheme developed independently by Thierry Coquand and Christine Paulin [7, 30]. They have arrived at essentially the same combinatorial structure, but develop ... |

7 |
Normalizing the associative law: An experiment with Martin-Lof's type theory
- Hedberg
- 1991
(Show Context)
Citation Context ... in type theory as an inductively defined predicate, see Dybjer [12]. Michael Hedberg has looked at the similar but simpler example of normalization of binary trees: `normalizing the associative law' =-=[18]-=-. He uses inductively defined sets and predicates in a variety of ways to show that Martin-Lof type theory can be used not only as an `integrated logic' (based on the Curry-Howard identification), but... |

7 |
The domain interpretation of type theory, lecture notes
- Martin-Löf
- 1983
(Show Context)
Citation Context ... X B(x) we are justified in using it as a representation for the set generated inductively by \Phi. But this method does not work in the intensional version of type theory given by Martin-Lof in 1986 =-=[25]-=- (see Nordstrom, Petersson, and Smith [29]), since it makes use of `extensional isomorphisms', such as N 1 = X N0 and X = X N1 . Another possibility is to add fixed point operators to type theory. But... |

7 | A machine checked proof that Ackermann’s function is not primitive recursive
- Szasz
- 1993
(Show Context)
Citation Context ...), but also as an `external logic' which can be used for verifying an externally given general recursive program. Hedberg has implemented his examples using Paulson's Isabelle-system [31]. Nora Szasz =-=[36]-=- has formalized the proof that Ackermann's function is not primitive recursive. The basic definition is a binary inductive family of tuples of primitive recursive functions TPR(m;n), where n is the nu... |

6 |
Inductively defined sets in Martin-Lof's type theory
- Dybjer
- 1987
(Show Context)
Citation Context ...related systems. This construction is not part of Martin-Lof's type theory, which is predicative. But in the extensional version [26] wellorderings W can be used instead. It can for example be proved =-=[11]-=- that for any strictly positive set operator \Phi built up by constants, variables, +, \Theta, and !, there is a set A and a family of sets B over A, such that \Phi(X) = \Sigma x:A X B(x) : Since W x:... |

6 | An Intuitionistic Theory of Types. Unpublished manuscript - Martin-Lof - 1971 |

5 |
Do-it-yourself type theory (part 1
- Backhouse, Chisholm, et al.
- 1989
(Show Context)
Citation Context ...cases. See Martin-Lof 1971 [21] for a general formulation of inductive definitions in the language of ordinary first order predicate logic.' The present scheme generalizes a scheme given by Backhouse =-=[2, 3]-=- under the name `DoIt -Yourself Type Theory'. The existence of a scheme relies on Schroeder-Heister's notion of rule of higher level [33] which made it possible for Martin-Lof to formulate the rules f... |

3 |
Propositional functions and families of types. Notre Dame
- Smith
- 1989
(Show Context)
Citation Context ...y into the scheme, but we cannot simply replace set above by type in the typing of C, because we do not have type:type. One solution is to index the elimination rule by a family of types C, see Smith =-=[35]-=-, but if we want to keep the notion of a finitary theory (and not extend the theory of logical types) then we must treat the theory as open (potentially infinite) with respect to recursive definitions... |

2 |
Judgements of higher levels and standardized rules for logical constants in Martin-Lof's theory of logic. Unpublished paper
- Schroeder-Heister
- 1985
(Show Context)
Citation Context ...h made it possible for Martin-Lof to formulate the rules for \Pi so as to conform to the pattern for the other set formers. (Note that \Pi was not mentioned in the quotation above.) Schroeder-Heister =-=[34]-=- also formulated a scheme for Martin-Lof's logical theory which is close to Backhouse's scheme for type theory. The present scheme subsumes both Martin-Lof's scheme for the intuitionistic theory of it... |