## A Model for Impredicative Type Systems, Universes, Intersection Types and Subtyping

Citations: | 6 - 0 self |

### BibTeX

@MISC{Miquel_amodel,

author = {Alexandre Miquel},

title = {A Model for Impredicative Type Systems, Universes, Intersection Types and Subtyping},

year = {}

}

### OpenURL

### Abstract

We introduce a new model based on coherence spaces for interpreting large impredicative type systems such as the Extended Calculus of Constructions (ECC). Moreover, we show that this model is well-suited for interpreting intersection types and subtyping too, and we illustrate this by interpreting a variant of ECC with an additional intersection type binder. Furthermore, we propose a general method for interpreting the impredicative level in a non-syntactical way, by allowing the model to be parametrized by an arbitrarily large coherence space in order to interpret inhabitants of impredicative types. As an application, we show that uncountable types such as the type of real numbers or Zermelo-Frnkel sets can safely be axiomatized on the impredicative level of, say, ECC, without harm for consistency. 1

### Citations

352 |
Proofs and Types
- Girard, Lafont, et al.
- 1989
(Show Context)
Citation Context ...presence of the (STR) rule (see the discussion about strong normalization in section 6). 3 Types in coherence spaces In this section, we shall assume that the reader is familiar with coherence spaces =-=[6]-=-, and more generally with domain theory. So we will only recall some basic definitions and notations. 3.1 Coherence spaces A coherence space is a set of sets A satisfying the following criterions: 1. ... |

108 |
Computation and Reasoning, A Type Theory for Computer Science
- Luo
- 1994
(Show Context)
Citation Context ...Calculus of Constructions, which contains ECC. 2 The implicit calculus 2.1 Presentation Basically, the Implicit Calculus of Constructions (ICC) or, shortly, the implicit calculus, is a variant of ECC =-=[9-=-] in which we distinguish two kinds of products: the explicit product and the implicit product, denoted by x : T : U and 8x : T : U respectively. The explicit product is the usual dependent product of... |

43 |
Modular proof of strong normalization for the calculus of constructions
- Geuvers, Nederhof
- 1991
(Show Context)
Citation Context .... In the following, we will only recall some basic definitions and results whose proofs are detailed in [10]. 2.2 Basic notations In this section, we shall assume that the reader is familiar with PTS =-=[5]-=-. The set of sorts used in the implicit calculus is defined by S = fProp; Setg [ fType i ; i > 0g; where Prop and Set denote the impredicative sorts, and (Type i ) i>0 the usual universe hierarchy of ... |

31 | Constructions, Inductive Types and Strong Normalization
- Altenkirch
- 1993
(Show Context)
Citation Context ...aper could be used also for building strong normalization models. Technically, we can add normalization information in the atoms (X) representing types as values, by giving to them a -set structure [1=-=]-=-. But for achieving this goal, we first need to modify the model in such a way that all types becomes inhabited, since we want to interpret terms in all contexts. Actually, such a modification seems t... |

7 |
S.R.D.: Type inference: Some results, some problems
- Giannini, Honsell, et al.
- 1993
(Show Context)
Citation Context ...ystems (PTS), whereas the implicit product is much more an intersection type binder, like the impredicative product of the Curry-style system F and its extensions, such as the Type Assignment Systems =-=[3]-=-. But in contrast to these calculi, the implicit product can be used at any level---which may be impredicative or predicative. Since the two products are distinguished by the syntax and not only by th... |

7 |
in Theorie des ensembles
- Krivine
- 1998
(Show Context)
Citation Context ... 3. for each family (b i ) i2c indexed over a cardinal csa such that b isa for all i 2 c, we have sup i2c b isa. Although the existence of inaccessible cardinals cannot be proven in the ZF-set theory =-=[8]-=- 6 , such `big' cardinals have been used for a while, especially for building settheoretical models of type theories [9]. Lemma 3.5 --- Let A and B be two coherence spaces equipped with a-stable funct... |

7 |
Une thorie des constructions inductives
- Werner
- 1994
(Show Context)
Citation Context ...predicativity in a non trivial way. Remark that such a requirement is necessary if we want to reuse our model for type systems with strong elimination, such as the Calculus of Inductive Constructions =-=[11, 12-=-]. Usually, impredicative types are interpreted by PER's or saturated sets, and proof terms are interpreted by syntactic constructs such as recursive functions or -terms. Although such an interpretati... |

5 |
Arguments implicites dans le calcul des constructions: étude d’un formalisme à la curry
- Miquel
- 1998
(Show Context)
Citation Context ...nce spaces for interpreting and proving the consistency of large impredicative type theories. Originally, this model was designed for proving the consistency of the Implicit Calculus of Constructions =-=[10]-=-, a Curry-style variant of ECC with an additional intersection type binder. Nevertheless, it is powerful enough for proving the consistency of a large class of impredicative type systems, since it mak... |

1 |
The Church-Rosser property for reduction in typed lambda calculi
- Geuvers
(Show Context)
Citation Context ...on with the rules written in figure 1. These rules include: Rules for well-formed contexts (WF-E) and (WF-S). Logical rules (VAR), (SORT), (EXPPROD), (IMPPROD), (LAM) and (APP). 2 It is well-known[4=-=-=-] that thes-reduction is not confluent on raw terms in a Church-style PTS. 2 Rules for well-formed contexts ? ` (WF-E) ` T : s x = 2 Dom() ;x : T ` (WF-S) Rules for well-typed terms ` (x : T ) 2 ` x :... |

1 |
Private communication
- Solus
- 1999
(Show Context)
Citation Context ...e problem disappears by writing the axioms of ZF in the style of first-order theories. 6 Future work Connections with Ludics At the time we discovered this model, we didn't know anything about Ludics =-=[7]-=-. Nevertheless, it turns out that many notions defined in section 3 9 The same problem arises if we want to introduce a symbol expressing the l.u.b of a bounded subset of R given by a predicate over t... |

1 |
Dfinitions inductives en Thorie des Types d'Ordre Suprieur. Habilitation diriger des recherches
- Paulin-Mohring
- 1996
(Show Context)
Citation Context ...predicativity in a non trivial way. Remark that such a requirement is necessary if we want to reuse our model for type systems with strong elimination, such as the Calculus of Inductive Constructions =-=[11, 12-=-]. Usually, impredicative types are interpreted by PER's or saturated sets, and proof terms are interpreted by syntactic constructs such as recursive functions or -terms. Although such an interpretati... |