## The not so simple proof-irrelevant model of CC (2002)

Venue: | OF LECTURE NOTES IN COMPUTER SCIENCE |

Citations: | 10 - 1 self |

### BibTeX

@INPROCEEDINGS{Miquel02thenot,

author = {Alexandre Miquel and Benjamin Werner},

title = { The not so simple proof-irrelevant model of CC},

booktitle = {OF LECTURE NOTES IN COMPUTER SCIENCE},

year = {2002},

pages = {240--258},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

It is well-known that the Calculus of Constructions (CC) bears a simple set-theoretical model in which proof-terms are mapped onto a single object—a property which is known as proof-irrelevance. In this paper, we show that when going into the (generally omitted) technical details, this naive model raises several unexpected difficulties related to the interpretation of the impredicative level, especially for the soundness property which is surprisingly difficult to be given a correct proof in this simple framework. We propose a way to tackle these difficulties, thus giving a (more) detailed elementary consistency proof of CC without going back to a translation to Fω. We also discuss some possible alternatives and possible extensions of our construction.

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Citation Context ...able x in t. The one step β-reduction is defined as usual and will be denoted by →β. Its reflexive and transitive closure is denoted by ↠β, and the β-conversion equivalence by =β. As in any other PTS =-=[3]-=-, the β-reduction enjoys the Church-Rosser property : If t1 =β t2, then there exists t ′ such that t1 ↠β t ′ and t2 ↠β t ′ . Typing contexts are given by Contexts Γ ::= [] | Γ ; [x : T ] We will not r... |

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Citation Context ...atization. Some of these type theories are impredicative, meaning a proposition may be formed by quantification over all propositions. Technically, this means they are extensions of Girard’s system F =-=[12]-=-. As pointed out by Reynolds [20], the only way to give a set-theoretical account for system F (and thus its extensions) is to identify all elements of a given type in the denotation. Thus types are e... |

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Citation Context ...one sort. The sort of any term is given unambiguously once the sorts of its free variables are known. We therefore build the model for a sorted version of the type system. This idea is due to Geuvers =-=[11]-=- (and is also used in [23] for more syntactical purposes). Definition 1. The definition of terms is unchanged, but we take a set of variables indexed by the sorts. In other words, we can describe the ... |

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Citation Context ..., causing a dramatic increase of the proof-terms. This gets drasticly simplified in the proof-irrelevant 6 For this reason, this model is actually a model of the domain-free Calculus of Constructions =-=[6]-=- with its untyped β-conversion rule.sversion of the type theory. Notice that practical type-checking for such theories is indeed much simpler if proof-terms are tagged in the style of the sorted syste... |

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Citation Context ...β-reduction or expansion. When modeling such theories, it is possible to prove soundness for the model and with respect to judgemental equality in a single mutual induction. This is done by Streicher =-=[21]-=- using a presentation involving a form of judgemental equality. As pointed out to us, one may notice that he also uses two different interpretations of the universal quantification depending of the pr... |

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Citation Context ...term is given unambiguously once the sorts of its free variables are known. We therefore build the model for a sorted version of the type system. This idea is due to Geuvers [11] (and is also used in =-=[23]-=- for more syntactical purposes). Definition 1. The definition of terms is unchanged, but we take a set of variables indexed by the sorts. In other words, we can describe the algebra of sorted terms by... |

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Citation Context ...d inside proof-terms. The afferent type system is obviously validated by the constructed model. This relaxed conversion rule is particularly interesting when objects live in the predicative level. In =-=[2]-=-, Altenkirch presents such a system together with a categorical model; he shows that proof-irrelevance combined with η-conversion in the conversion rule entails the extensionality principle for functi... |

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Le Calcul des Constructions Implicite: Syntaxe et Sémantique. Theèses de Doctorat, Université
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Citation Context ...is not problematic. The equivalence with conventional PTSs however is proved after strong normalization and heavily relies on the latter property.s5.4 Proof-irrelevance in a domain-theoretic model In =-=[18, 19]-=-, Miquel presents a proof-irrelevant model of (an extension of) the Calculus of Constructions by interpreting type-theoretic functions as (traces of) stable functions in a suitable coherence space [13... |

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Citation Context ... the existence of enough inaccessible cardinals. In that respect, this work can be seen as a cleaning-up of the encoding of Coq’s type theory (without the impredicative sort Set) which is sketched in =-=[22]-=-. On the other hand, one should mention that adding the subsumption rule Γ ⊢ T : ∗ Γ ⊢ T : ✷ to the system cannot be handled easily in our setting and could indeed become problematic when combined wit... |

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Citation Context ... an infinite set (for instance ω), but in this case, the set U can only be built under the assumption of the existence of an inaccessible cardinal—that cannot be derived in ZermeloFraenkel set theory =-=[14]-=-. x∈T �s2.3 Defining the interpretation function The interpretation function is organized as follows: – To each context Γ we associate a set �Γ � of Γ -valuations, that is lists of objects of the form... |

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Citation Context ...ms related to the strong normalization property, which is notoriously more complex. 5.3 Typed reduction This last remark is also illustrated by the interpretation carried out by Melliès and Werner in =-=[17]-=-. In this work, the semantic interpretation is carried out for a very restricted version of β-reduction, for which soundness is not problematic. The equivalence with conventional PTSs however is prove... |

7 |
Polymorphism is not Set-Theoretic. in: Semantics of Data Types
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Citation Context ...ories are impredicative, meaning a proposition may be formed by quantification over all propositions. Technically, this means they are extensions of Girard’s system F [12]. As pointed out by Reynolds =-=[20]-=-, the only way to give a set-theoretical account for system F (and thus its extensions) is to identify all elements of a given type in the denotation. Thus types are either interpreted by the empty se... |

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Citation Context ...y : If t1 =β t2, then there exists t ′ such that t1 ↠β t ′ and t2 ↠β t ′ . Typing contexts are given by Contexts Γ ::= [] | Γ ; [x : T ] We will not recall the typing rules of CC that can be found in =-=[8, 10, 3]-=-, and we will just mention the subject-reduction property: If Γ ⊢ t : T and t ↠β t ′ , then Γ ⊢ t ′ : T . 2.2 Defining the model The proof-irrelevant model of CC is based on the simple idea that each ... |

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Citation Context ...g mechanism we introduced in paragraph 2.2, but to the fact that the identification of all proofterms requires to forget the domain of the corresponding functions. In particular, P. Aczel presents in =-=[1]-=- another mechanism (for the same purpose) which is based on the following definitions • = ∅ lam(f) = � (x, z); x ∈ Dom(f) ∧ z ∈ f(x) � app(u, x) = � z; (x, z) ∈ u � and it is easy to check that our co... |

2 |
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Citation Context ...is not problematic. The equivalence with conventional PTSs however is proved after strong normalization and heavily relies on the latter property.s5.4 Proof-irrelevance in a domain-theoretic model In =-=[18, 19]-=-, Miquel presents a proof-irrelevant model of (an extension of) the Calculus of Constructions by interpreting type-theoretic functions as (traces of) stable functions in a suitable coherence space [13... |

1 |
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Citation Context ... well as some alternative ways to proceed. 5.1 Extensions and Axioms A particular useful application of set-theoretical models is the validation of axioms. In particular, Barbanera and Berardi’s work =-=[4]-=- states that any model validating the excluded middle is necessarily proof-irrelevant. In addition to being proof-irrelevant, set-theoretical models are simple enough to easily validate a large class ... |

1 |
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Citation Context ...at follows, we try to be as precise as possible, but for the usual choice between de Bruijn indices and named variables. Both choices require specific technical developments (see respectively [5] and =-=[16]-=-). We believe the present work can be fitted in both formalization styles. 2 A problematic proof-irrelevant model 2.1 The Calculus of Constructions In the following, we shall assume that the calculus ... |