## Understanding Inductive Types in Constructions (1993)

Citations: | 2 - 1 self |

### BibTeX

@TECHREPORT{Fu93understandinginductive,

author = {Yuxi Fu},

title = {Understanding Inductive Types in Constructions},

institution = {},

year = {1993}

}

### OpenURL

### Abstract

In this paper we extend the Calculus of Constructions with generalized inductive types. The extension is justified by showing that the usual set theoretical model can be effectivized. It is also pointed out that the model given in a published paper for a collection of inductive types in a different style is wrong.

### Citations

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A formulation of the simple theory of types
- Church
- 1940
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Citation Context ...er, a counter example is given to show that if we identify sets with propositions, the Calculus of Constructions is not conservative over HOL, the constructive version of Church's higher order logic (=-=[Chu40]). In the -=-second one, it is shown that, when properly separated, it 1 To my way of thinking, there is no such thing as a "ramified logic". Logics are intrinsically impredicative, while sets must be st... |

471 | The calculus of constructions - Coquand, Huet - 1988 |

441 |
The formulae-as-types notion of construction
- Howard
- 1980
(Show Context)
Citation Context ...of Computer Science, The University, Oxford Road, Manchester M13 9PL, U.K. 1 Introduction Constructive type theory has been developed around three principles. The oft-quoted Curry-Howard's principle (=-=[How80]-=-) may be stated as: Constructive propositions are types. This is what underlies Girard's work on system F and some recent interest in polymorphism and functional programming languages. Here the notion... |

263 | Foundations of Constructive Mathematics - Beeson - 1985 |

177 |
An introduction to inductive definitions
- Aczel
- 1977
(Show Context)
Citation Context ...f another class of inductive types. 2 The Traditional Approach In [Dyb91], a set-theoretic model is given to the inductive types defined in MartinL of's set theory, using Aczel's notion of rule sets (=-=[Acz77]). A -=-similar model using - continuous functors is described in [CPM90]. Neither of them can treat inductive types in Constructions, which is impredicative. In [Dyb91] the author says, we quote, "This ... |

119 |
The type theoretic interpretation of constructive set theory
- Aczel
- 2001
(Show Context)
Citation Context ...o have a good understanding of the relationships. In doing so, one should take into account of the fact that ECC contains a logic. The usual interpretation in Martin-Lof's set theory, see for example =-=[Acz86]-=-, might have a more faithful counterpart in ECC---that is logical formulas are interpreted as (impredicative) propositions while the notion of sets is kept at the Type-level as it were. See [Fu93c] fo... |

109 |
Irlductively defined types
- Coquand, Paulin
- 1990
(Show Context)
Citation Context ...91], a set-theoretic model is given to the inductive types defined in MartinL of's set theory, using Aczel's notion of rule sets ([Acz77]). A similar model using - continuous functors is described in =-=[CPM90]. Nei-=-ther of them can treat inductive types in Constructions, which is impredicative. In [Dyb91] the author says, we quote, "This 2 interpretation does of course not 2 extend to the full system of the... |

84 | An Extended Calculus of Constructions
- Luo
- 1990
(Show Context)
Citation Context ...anism for defining and manipulating data types. But applications to program specifications and verifications are limited due to the absence of a logic. Since [Coq86], and especially [Luo91] (see also =-=[Luo90a]-=-), we now know that it is perfectly sound to unify those two kinds of languages. I will call Russell's principle the following: The range of a constructive significance is a type. The idea has its ori... |

76 | Inductive sets and families in Martin-Löfs Type Theory and their set-theoretic semantics: An inversion principle for Martin-Löfs type theory
- Dybjer
- 1991
(Show Context)
Citation Context ...d with another class of inductive types can be interpreted in !-Set. This general categorical results then motivates the formulation of another class of inductive types. 2 The Traditional Approach In =-=[Dyb91]-=-, a set-theoretic model is given to the inductive types defined in MartinL of's set theory, using Aczel's notion of rule sets ([Acz77]). A similar model using - continuous functors is described in [CP... |

72 |
Basic Set Theory
- Levy
- 1979
(Show Context)
Citation Context ...(x). Obviously, for ordinals fi 0 2 fi 00 , r fi 0sr fi 00 . By simple cardinality argument, we know that there exists an ordinal fi such that for all `sfi, r ` = r fi . By the least ordinal theorem (=-=[Lev79]-=-), we can choose the least such fi. Define !(I(R ff )) to be ! fi . We can now interpret (\Delta 1 ; : : : ; \Delta n ):[\Phi 11 ; : : : ; \Phi 1n 1 ; : : : ; \Phi n1 ; : : : ; \Phi nnn ] by ff : j\Ga... |

67 | An Analysis of Girard’s Paradox
- Coquand
- 1986
(Show Context)
Citation Context ...ts. Languages in this group have good mechanism for defining and manipulating data types. But applications to program specifications and verifications are limited due to the absence of a logic. Since =-=[Coq86]-=-, and especially [Luo91] (see also [Luo90a]), we now know that it is perfectly sound to unify those two kinds of languages. I will call Russell's principle the following: The range of a constructive s... |

60 | Metamathematical investigations of a calculus of constructions. Rapport de recherche de l’INRIA - Coquand - 1989 |

48 | Polymorphism is set theoretic, constructively - Pitts - 1987 |

43 | Inductively Defined Types in the Calculus of Constructions
- Pfenning, Paulin-Mohring
- 1990
(Show Context)
Citation Context ...ustrated in x 2.3. Other formulations of generalized inductive types. Once the essence of the generalized inductive types is understood, different formulations should present no problems. [CPM90] and =-=[PPM91]-=- contain two other formulations. The advantage of the formulation in [CPM90] is that it is easier to see how to generalize(!) the generalized inductive types. We call general inductive types those ind... |

24 | A higher-order calculus and theory abstraction
- Luo
- 1991
(Show Context)
Citation Context ...oup have good mechanism for defining and manipulating data types. But applications to program specifications and verifications are limited due to the absence of a logic. Since [Coq86], and especially =-=[Luo91]-=- (see also [Luo90a]), we now know that it is perfectly sound to unify those two kinds of languages. I will call Russell's principle the following: The range of a constructive significance is a type. T... |

22 |
A set constructor for inductive sets in Martin-Löf’s type theory
- Petersson, Synek
- 1989
(Show Context)
Citation Context ... interpretation. 2.4 Inductive Families In [Dyb91], a class of types---inductive families---is introduced, which bear some resemblance to the combination of the two variants of tree types proposed in =-=[PS89]-=-. On occasions, it is useful to be able to deal with a family of inductive types indexed over another type. We take the attitude that an indexing type is different from a type in a context. The former... |

19 | A note on categorical datatypes - Wraith - 1989 |

16 |
Predicative type universes and primitive recursion
- Mendler
- 1991
(Show Context)
Citation Context ...any model theoretical problems in the traditional sense. The semantic interest here is about how to relate two levels of models. See [Pit87, Fu93a, Fu92a] for details. Categorical Inductive Types. In =-=[Men91]-=-, a categorical formulation of recursion is given. It is interesting to see what this general definition means in concrete categories in which dependent typed calculus can be modeled. See [Fu92b] for ... |

14 | Recursive models for constructive set theories - Beeson - 1982 |

14 |
A unifying theory of dependent types: the schematic approach
- Luo
- 1992
(Show Context)
Citation Context ...ages of using a logical framework is the possibility of separating the computational content of a language to be defined from the extensional properties of the meta calculus. The language proposed in =-=[Luo92]-=- is designed to maximize this gain. To be able to define the generalized inductive types, the Martin-Lof's logical framework is extended with kind schemata. These schemata are `small' in the sense tha... |

13 |
Least fixed point of a functor
- Adámek, Koubek
- 1979
(Show Context)
Citation Context ...nd an endofunctor F on the category within which the model is taken, how do we construct the initial F -algebra? The answer is given by the following theorem 3.5. This theorem generalizes a result in =-=[AK79]-=-. The proof of theorem 3.5 also generalizes that in [AK79]. Suppose T is an endofunctor on C. A T -algebra is a pair (A; TA f \Gamma! A) where A is an object in C and f a morphism in C. A homomorphism... |

8 | Inductive data types: Well-ordering types revisited
- Goguen, Luo
(Show Context)
Citation Context ...iverses in Martin-Lof's set theory are closed while Type i (i 2 !) are not. This is really a misconception. The closed universes should sit inside each Type i . The Unifying Theory of Dependent Types =-=[GL92]-=- is a calculus proposed along this line, although the author's emphasis is on the decidability of the language. The main purpose of this paper is to reinforce our confidence in the above slogan by tak... |

8 |
The extended calculus of constructions (ECC) with inductive types
- Ore
- 1992
(Show Context)
Citation Context ... [Fu92b] for some examples. 3 The Initial Algebraic Approach There is another way of formalizing the ideas embodied in the generalized inductive types by using initial T -algebras. This is studied in =-=[Ore92]-=-. For this formulation to be possible, the calculus must be extensional. The reason is that we use functors to define rules about the inductive types and the functoriality of a type constructor is equ... |

7 | An inversion principle for Martin-Lof's type theory. Talk presented at the Workshop on Programming Logic - Dybjer - 1989 |

5 |
A problem of adequacy: conservativity of calculus of constructions over higher-order logic
- Luo
- 1990
(Show Context)
Citation Context ...n the CC-like calculi. Part of the expressive power of ECC comes from the fact that Type i (i 2 !) are open universes, which is a consequence of Prop : Type 0 . The studies carried out in [Ber89] and =-=[Luo90b]-=- clarify, both negatively and positively, the role of propositions in Constructions. In the first paper, a counter example is given to show that if we identify sets with propositions, the Calculus of ... |

4 |
Some Semantic Issues In Type Theory
- Fu
- 1992
(Show Context)
Citation Context ...stions are closely related to a much harder one: what are the endofunctors on !-Set that have initial T -algebras? 6 Acknowledgement The material in this paper is taken from the author's Ph.D thesis (=-=[Fu92d]-=-). My supervisor David Rydeheard has been very helpful all the way over the last two years. The examiners Peter Aczel and Martin Hyland have raised some interesting points and made quite a few comment... |

3 |
Non-conservativity of Coquand's Calculus with respect to Higher-order Intuitionistic Logic, Talk given in the 3rd Jumelage meeting on Typed Lambda Calculi
- Berardi
- 1989
(Show Context)
Citation Context ...esentative in the CC-like calculi. Part of the expressive power of ECC comes from the fact that Type i (i 2 !) are open universes, which is a consequence of Prop : Type 0 . The studies carried out in =-=[Ber89]-=- and [Luo90b] clarify, both negatively and positively, the role of propositions in Constructions. In the first paper, a counter example is given to show that if we identify sets with propositions, the... |

2 | Recursive Models of General Inductive Types
- Fu
- 1992
(Show Context)
Citation Context ...the elements and the tuples are natural numbers. A transfinite induction should be used to define a partial equivalence relation on I(R ff ). This process produces a per P(I(R ff )). For details, see =-=[Fu92c]-=-. 2.7 Models of Other Variants Inductive types defined using a logical framework. One of the advantages of using a logical framework is the possibility of separating the computational content of a lan... |

2 |
Topics in type theory
- Fu
- 1992
(Show Context)
Citation Context ...he Calculus of separated Constructions. The second language has elimination rules for closed universes, but the usefulness of it is questionable. For a detailed description of the third language, see =-=[Fu93c]-=-. ECC is the best representative in the CC-like calculi. Part of the expressive power of ECC comes from the fact that Type i (i 2 !) are open universes, which is a consequence of Prop : Type 0 . The s... |

1 | A Note on Topos Models - Fu - 1992 |

1 |
Recursion and Universes in !-Set
- Fu
- 1992
(Show Context)
Citation Context .... In [Men91], a categorical formulation of recursion is given. It is interesting to see what this general definition means in concrete categories in which dependent typed calculus can be modeled. See =-=[Fu92b]-=- for some examples. 3 The Initial Algebraic Approach There is another way of formalizing the ideas embodied in the generalized inductive types by using initial T -algebras. This is studied in [Ore92].... |

1 | Categorical Properties of Logical Frameworks
- Fu
- 1993
(Show Context)
Citation Context ... 2.3 live in \Omega\Gamma SET. The `smallness' of the kind schemata implies that the least closed sets of these rule sets are in !-Set. For more about categorical semantics of logical frameworks, see =-=[Fu93a]-=-. Inductive types via W -types. It is well-known that inductive types are definable in an extensional dependent calculus with W -types ([Dyb89, GL92]). The interpretation given in x 2.3 is extensional... |