Inductive Types in the Calculus of Algebraic Constructions
نویسنده
چکیده
In a previous work, we proved that an important part of the Calculus of Inductive Constructions (CIC), the basis of the Coq proof assistant, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. In this paper, we prove that almost all CIC can be seen as a CAC, and that it can be further extended with non-strictly positive types and inductive-recursive types together with non-free constructors and pattern-matching on defined symbols.
منابع مشابه
The Calculus of algebraic Constructions
This paper is concerned with the foundations of the Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions by inductive data types. CAC generalizes inductive types equipped with higher-order primitive recursion, by providing definitions of functions by patternmatching which capture recursor definitions for arbitrary non-dependent and non-polymorphic inductive t...
متن کاملThe Rooster and the Syntactic Bracket
We propose an extension of pure type systems with an algebraic presentation of inductive and co-inductive type families with proper indices. This type theory supports coercions toward from smaller sorts to bigger sorts via explicit type construction, as well as impredicative sorts. Type families in impredicative sorts are constructed with a bracketing operation. The necessary restrictions of pa...
متن کاملTowards an Implicit Calculus of Inductive Constructions. Extending the Implicit Calculus of Constructions with Union and Subset Types
We present extensions of Miquel’s Implicit Calculus of Constructions (ICC) and Barras and Bernardo’s decidable Implicit Calculus of Constructions (ICC*) with union and subset types. The purpose of these systems is to solve the problem of interaction betweeen logical and computational data. This is a work in progress and our long term goal is to add the whole inductive types to ICC and ICC* in o...
متن کاملFirst Steps Towards Cumulative Inductive Types in CIC
We discuss our on-going research on making inductive types cumulative in the predicative calculus of inductive constructions (pCIC) – the logic of the Coq proof assistant. 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 induc...
متن کاملThe Extended Calculus of Constructions (ECC) with Inductive Types
Luo’s Extended Calculus of donstructions (ECC) is a higher order functional calculus based on Coquand’s and Huet’s Calculus of Constructions, but has in addition strong sums and a predicative cumulative type hierarchy. In this paper I introduce inductive types on the predicative type levels of ECC. I also show how the o-Set model for ECC can be extended to a model for this augmented calculus. '...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2003