Ornaments for Proof Reuse in Coq

Ornaments express relations between inductive types with the same inductive structure. We implement fully automatic proof reuse for a particular class of ornaments in a Coq plugin, and show how such a tool can give programmers the rewards of using indexed inductive types while automating away many of the costs. The plugin works directly on Coq code; it is the first ornamentation tool for a non-embedded dependently typed language. It is also the first tool to automatically identify ornaments: To lift a function or proof, the user must provide only the source type, the destination type, and the source function or proof. In taking advantage of the mathematical properties of ornaments, our approach produces faster functions and smaller terms than a more general approach to proof reuse in Coq

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

First Steps Towards Cumulative Inductive Types in CIC

Having the type of all types in a type system results in paradoxes like Russel’s paradox. Therefore type theories like predicative calculus of inductive constructions (pCIC) - the logic of the Coq proof assistant - have a hierarchy of types Type0, Type1, Type2, . . . , where Type0 : Type1, Type1 : Type2, . . . . In a cumulative type system, e.g., pCIC, for a term t such that t: Type i we also have that t: Typei + 1. The system pCIC has recently been extended to support universe polymorphism, i.e., definitions can be parametrized by universe levels. This extension does not support cumulativity for inductive types. For example, we do not have that a pair of types at levels i and j is also considered a pair of types at levels i + 1 and j + 1. In this paper, we discuss our on-going research on making inductive types cumulative in the pCIC. Having inductive types be cumulative alleviates some problems that occur while working with large inductive types, e.g., the category of small categories, in pCIC. We present the pCuIC system which adds cumulativity for inductive types to pCIC and briefly discuss some of its properties and possible extensions. We, in addition, give a justification for the introduced cumulativity relation for inductive types.status: publishe