## A Simple Model for Quotient Types (1995)

Venue: | Proceedings of TLCA'95, volume 902 of Lecture Notes in Computer Science |

Citations: | 17 - 0 self |

### BibTeX

@INPROCEEDINGS{Hofmann95asimple,

author = {Martin Hofmann},

title = {A Simple Model for Quotient Types},

booktitle = {Proceedings of TLCA'95, volume 902 of Lecture Notes in Computer Science},

year = {1995},

pages = {216--234},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We give an interpretation of quotient types within in a dependent type theory with an impredicative universe of propositions (Calculus of Constructions). In the model, type dependency arises only at the propositional level, therefore universes and large eliminations cannot be interpreted. In exchange, the model is much simpler and more intuitive than the one proposed by the author in [10]. Moreover, we interpret a choice operator for quotient types that, under certain restrictions, allows one to recover a representative from an equivalence class. Since the model is constructed syntactically, the interpretation function from the syntax with quotient types to the model gives rise to a procedure which eliminates quotient types by replacing propositional equality by equality relations defined by induction on the type structure ("book equalities"). 1 Introduction Intensional type theories like the Calculus of Constructions have been proposed as a framework in which to formalise mathemati...

### Citations

489 |
The calculus of constructions
- COQUAND, HUET
- 1988
(Show Context)
Citation Context ...: oe to mean that M and N are definitionally equal terms of type oe in context \Gamma . So we have a typed definitional equality which is given inductively by a system of rules. It is known (see e.g. =-=[6, 1]-=-) that there is a decision procedure for it using untyped conversion, which is often regarded as the very definition of definitional equality. Another thing which is slightly unconventional is that if... |

429 | Constructive Analysis
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ...valence relation on �� such that these partial equivalence relations are compatible with the relation on oe. By analogy to Bishop's definition of sets as assemblies together with an equality relat=-=ion [4]-=- we call the pairs of types and partial equivalence relations setoids. 4.1 Contexts of Setoids A context of setoids is a pair \Gamma = (\Gamma set ; \Gamma rel ) where \Gamma set is a (syntactic) cont... |

274 |
Foundations of Constructive Mathematics
- Beeson
- 1985
(Show Context)
Citation Context ...e choice operator we introduce seems original and hinges on the intensional nature of definitional equality. The model we give bears some resemblance with realizability interpretations of type theory =-=[3]-=- in which types are interpreted as partial equivalence relations on the natural numbers. Again, the main difference is that these models are extensional and therefore semantic equality is undecidable ... |

271 |
Programming in Martin–Löf ’s Type Theory. An introduction
- Nordström, Petersson, et al.
- 1990
(Show Context)
Citation Context ...ces of models, though. The method of interpreting type theory in itself so as to obtain additional features has also been used by Martin-Lof in order to account for proof irrelevance and subset types =-=[15]-=-. The use of categorical model theory (Section 3) to describe such translations appears to be new. Also the choice operator we introduce seems original and hinges on the intensional nature of definiti... |

142 |
Introduction to higher-order categorical logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ... be reduced to the task of verifying that one has an instance of the abstract model. This may be compared to the correspondance between typed -calculus and cartesian closed categories as described in =-=[13]-=-. The role of cartesian close categories is played here by the (by now standard) notion of categories with attributes introduced by Cartmell [5]. We only sketch this notion of model and the interpreta... |

105 |
Introduction to HOL
- Gordon, Melham
- 1993
(Show Context)
Citation Context ...ween proofs and programs which may be interesting to study in its own right. Related work. Quotient types are available in extensional frameworks without explicit proof objects like Nuprl [7] and HOL =-=[9]-=- and also in the internal language of toposes [16]. The rules given there are quite similar to the ones proposed here. For an intensional calculus with explicit proof objects no extension with quotien... |

104 |
et al . Implementing Mathematics with the Nuprl
- Constable
- 1986
(Show Context)
Citation Context ...paration between proofs and programs which may be interesting to study in its own right. Related work. Quotient types are available in extensional frameworks without explicit proof objects like Nuprl =-=[7]-=- and HOL [9] and also in the internal language of toposes [16]. The rules given there are quite similar to the ones proposed here. For an intensional calculus with explicit proof objects no extension ... |

91 | Categorical logic
- Pitts
- 1996
(Show Context)
Citation Context ...e (by now standard) notion of categories with attributes introduced by Cartmell [5]. We only sketch this notion of model and the interpretation of syntax therein. The interested reader is referred to =-=[17, 19, 12]-=-. Definition 1 A category with attributes (cwa) is given by the following data and conditions: -- A category C with terminal object 1. The objects of C are called contexts (A; B; \Gamma; \Delta; : : :... |

59 |
Generalised algebraic theories and contextual categories
- Cartmell
- 1986
(Show Context)
Citation Context ...ation emphasises that the variables from \Delta are free in �� . For example we may define eqzero[n: Nat] := (n L = 0). Explicit variable names in a substitution may now be omitted, for example eq=-=zero[5]-=- denotes the proposition 5 L = 0. If f : \Gamma ) \Delta and \Delta ` oe type then we write oe[f ] for the parallel substitution of f into oe. We have \Gamma ` oe[f ] type and similarly for terms. 3 C... |

23 |
An introduction to fibrations, topos theory, the effective topos, and modest sets
- Phoa
- 1992
(Show Context)
Citation Context ... to study in its own right. Related work. Quotient types are available in extensional frameworks without explicit proof objects like Nuprl [7] and HOL [9] and also in the internal language of toposes =-=[16]-=-. The rules given there are quite similar to the ones proposed here. For an intensional calculus with explicit proof objects no extension with quotient types exists in the literature to our knowledge.... |

18 | The independence of Peano’s fourth axiom from Martin-Löf ’s type theory without Universes
- Smith
- 1988
(Show Context)
Citation Context ... to be a function ? �� : oe set ! �� for every type �� which contradicts the second equation if �� = 0. A major application of such families of types is that they allow to derive Peano=-='s fourth axiom [15, 18]-=-. On the level of propositions we are still able to interpret this axiom, that is we have 0 L = Suc(0) ) ff, provided this holds in the target type theory. The difference is that Prf(ff) is weaker tha... |

14 |
On the church-rosser property for expressive type systems and its consequences for their metatheoretic study
- Geuvers, Werner
- 1994
(Show Context)
Citation Context ...Martin-Lof's identity type in the sense that a Leibniz principle for dependent types can be interpreted from which the definability of MartinL of's elimination rule [15] follows using Pr-Ir (see e.g. =-=[8]). Propositi-=-on 10 For each oe 2 Fam(\Gamma ) and �� 2 Fam(\Gamma \Delta oe) and M;N 2 Sect(oe) and P 2 L Eq(M;N ) and U 2 Sect(��fMg) there exists a well-determined section Subst oe;�� (P; U ) 2 Sect(... |

13 |
Elimination of extensionality in Martin-Löf type theory, in: Henk Barendregt, Tobias Nipkow (Eds
- Hofmann
- 1993
(Show Context)
Citation Context ...ses only at the propositional level, therefore universes and large eliminations cannot be interpreted. In exchange, the model is much simpler and more intuitive than the one proposed by the author in =-=[10]-=-. Moreover, we interpret a choice operator for quotient types that, under certain restrictions, allows one to recover a representative from an equivalence class. Since the model is constructed syntact... |

8 |
Semantics of Type Theory Birkhäuser
- Streicher
- 1991
(Show Context)
Citation Context ...e (by now standard) notion of categories with attributes introduced by Cartmell [5]. We only sketch this notion of model and the interpretation of syntax therein. The interested reader is referred to =-=[17, 19, 12]-=-. Definition 1 A category with attributes (cwa) is given by the following data and conditions: -- A category C with terminal object 1. The objects of C are called contexts (A; B; \Gamma; \Delta; : : :... |

4 |
Comprehension categories and the semantics of type theory
- Jacobs
- 1993
(Show Context)
Citation Context ...e (by now standard) notion of categories with attributes introduced by Cartmell [5]. We only sketch this notion of model and the interpretation of syntax therein. The interested reader is referred to =-=[17, 19, 12]-=-. Definition 1 A category with attributes (cwa) is given by the following data and conditions: -- A category C with terminal object 1. The objects of C are called contexts (A; B; \Gamma; \Delta; : : :... |

2 |
Quotients in simple Type Theory, submitted
- Jacobs
(Show Context)
Citation Context ...f objects no extension with quotient types exists in the literature to our knowledge. Categorical formulations of extensional quotient types in simple type theory have recently been studied by Jacobs =-=[11]-=- without giving particular instances of models, though. The method of interpreting type theory in itself so as to obtain additional features has also been used by Martin-Lof in order to account for pr... |

1 |
Constructions, normalization, and inductive types
- Altenkirch
- 1994
(Show Context)
Citation Context ...: oe to mean that M and N are definitionally equal terms of type oe in context \Gamma . So we have a typed definitional equality which is given inductively by a system of rules. It is known (see e.g. =-=[6, 1]-=-) that there is a decision procedure for it using untyped conversion, which is often regarded as the very definition of definitional equality. Another thing which is slightly unconventional is that if... |

1 |
Functional programming and lambda calculus. Handbook
- Barendregt
(Show Context)
Citation Context ...h is often regarded as the very definition of definitional equality. Another thing which is slightly unconventional is that if M : Prop we have Prf(M ) type rather than M type as in the PTS-tradition =-=[2]-=-. So an expression is either a term or a type but never both. The source type theory is meant to be the internal language of the model which we are going to describe. It supports all the type and term... |