---
title: Inductive types in the Calculus of Algebraic Constructions
url: https://www.emergentmind.com/papers/0610073
type: paper
arxiv_id: '0610073'
arxiv_url: https://arxiv.org/abs/0610073
published: '2006-10-12'
categories:
- cs.LO
---

# Inductive types in the Calculus of Algebraic Constructions

## Abstract

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.