114 research outputs found
Equations for Hereditary Substitution in Leivant's Predicative System F: A Case Study
This paper presents a case study of formalizing a normalization proof for
Leivant's Predicative System F using the Equations package. Leivant's
Predicative System F is a stratified version of System F, where type
quantification is annotated with kinds representing universe levels. A weaker
variant of this system was studied by Stump & Eades, employing the hereditary
substitution method to show normalization. We improve on this result by showing
normalization for Leivant's original system using hereditary substitutions and
a novel multiset ordering on types. Our development is done in the Coq proof
assistant using the Equations package, which provides an interface to define
dependently-typed programs with well-founded recursion and full dependent
pattern- matching. Equations allows us to define explicitly the hereditary
substitution function, clarifying its algorithmic behavior in presence of term
and type substitutions. From this definition, consistency can easily be
derived. The algorithmic nature of our development is crucial to reflect
languages with type quantification, enlarging the class of languages on which
reflection methods can be used in the proof assistant.Comment: In Proceedings LFMTP 2015, arXiv:1507.07597. www:
http://equations-fpred.gforge.inria.fr
Univalence for free
We present an internalization of the 2-groupoid interpretation of the calculus of construction that allows to realize the univalence axiom, proof irrelevance and reasoning modulo. As an example, we show that in our setting, the type of Church integers is equal to the inductive type of natural numbers
Cumulative Inductive Types In Coq
In order to avoid well-known paradoxes associated with self-referential definitions, higher-order dependent type theories stratify the theory using a countably infinite hierarchy of universes (also known as sorts), Type_0 : Type_1 : *s. Such type systems are called cumulative if for any type A we have that A : Type_i implies A : Type_{i+1}. The Predicative Calculus of Inductive Constructions (pCIC) which forms the basis of the Coq proof assistant, is one such system. In this paper we present the Predicative Calculus of Cumulative Inductive Constructions (pCuIC) which extends the cumulativity relation to inductive types. We discuss cumulative inductive types as present in Coq 8.7 and their application to formalization and definitional translations
Gradual Certified Programming in Coq
Expressive static typing disciplines are a powerful way to achieve
high-quality software. However, the adoption cost of such techniques should not
be under-estimated. Just like gradual typing allows for a smooth transition
from dynamically-typed to statically-typed programs, it seems desirable to
support a gradual path to certified programming. We explore gradual certified
programming in Coq, providing the possibility to postpone the proofs of
selected properties, and to check "at runtime" whether the properties actually
hold. Casts can be integrated with the implicit coercion mechanism of Coq to
support implicit cast insertion a la gradual typing. Additionally, when
extracting Coq functions to mainstream languages, our encoding of casts
supports lifting assumed properties into runtime checks. Much to our surprise,
it is not necessary to extend Coq in any way to support gradual certified
programming. A simple mix of type classes and axioms makes it possible to bring
gradual certified programming to Coq in a straightforward manner.Comment: DLS'15 final version, Proceedings of the ACM Dynamic Languages
Symposium (DLS 2015
The Definitional Side of the Forcing
International audienceThis paper studies forcing translations of proofs in dependent type theory, through the Curry-Howard correspondence. Based on a call-by-push-value decomposition, we synthesize two simply-typed translations: i) one call-by-value, corresponding to the translation derived from the presheaf construction as studied in a previous paper ; ii) one call-by-name, whose intuitions already appear in Kriv-ine and Miquel's work. Focusing on the call-by-name translation, we adapt it to the dependent case and prove that it is compatible with the definitional equality of our system, thus avoiding coherence problems. This allows us to use any category as forcing conditions , which is out of reach with the call-by-value translation. Our construction also exploits the notion of storage operators in order to interpret dependent elimination for inductive types. This is a novel example of a dependent theory with side-effects, clarifying how dependent elimination for inductive types must be restricted in a non-pure setting. Being implemented as a Coq plugin, this work gives the possibility to formalize easily consistency results, for instance the consistency of the negation of Voevodsky's univalence axiom
Cohérence du Calcul Prédicatif des Constructions Inductives Cumulatives
Version 2 fixes some typos from version 1.Version 3 fixes a typo in a typing rule from version 2.In order to avoid well-know paradoxes associated with self-referential definitions, higher-order dependent type theories stratify the theory using a countably infinite hierarchy of universes (also known as sorts), Type 0 : Type 1 : · · ·. Such type systems are called cumulative if for any type A we have that A : Type i implies A : Type i+1. The predicative calculus of inductive constructions (pCIC) which forms the basis of the Coq proof assistant, is one such system. In this paper we present and establish the soundness of the predicative calculus of cumulative inductive constructions (pCuIC) which extends the cumulativity relation to inductive types.Les théories des types d’ordre supérieur sont stratifiées afin d’éviter les paradoxes bien connus associés aux définitions circulaires. Elles utilisent une hiérarchie dénombrable d’univers (aussi appelé sortes), Type0 : Type1 : · · · . Ces systèmes de types sont appelés cumulatifs si pour tout type A on a A : Typei implique A : Typei+1. Le calcul prédicatif des constructions inductives (pCIC), qui forme la base de l’assistant de preuve Coq, est un tel système. Dans cet article, nous présentons une extension du calcul, dont nous prouvons la cohérence relative vis à vis de la théorie des ensembles. Ce nouveau calcul étend la relation de cumulativité aux types inductifs
Partiality and Recursion in Interactive Theorem Provers - An Overview
To appearInternational audienceThe use of interactive theorem provers to establish the correctness of critical parts of a software development or for formalising mathematics is becoming more common and feasible in practice. However, most mature theorem provers lack a direct treatment of partial and general recursive functions; overcoming this weakness has been the objective of intensive research during the last decades. In this article, we review many techniques that have been proposed in the literature to simplify the formalisation of partial and general recursive functions in interactive theorem provers. Moreover, we classify the techniques according to their theoretical basis and their practical use. This uniform presentation of the different techniques facilitates the comparison and highlights their commonalities and differences, as well as their relative advantages and limitations. We focus on theorem provers based on constructive type theory (in particular, Agda and Coq) and higher-order logic (in particular Isabelle/HOL). Other systems and logics are covered to a certain extent, but not exhaustively. In addition to the description of the techniques, we also demonstrate tools which facilitate working with the problematic functions in particular theorem provers
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