Cumulative Inductive Types In Coq

Authors Amin Timany, Matthieu Sozeau

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Amin Timany
  • imec-Distrinet, KU Leuven, Leuven, Belgium
Matthieu Sozeau
  • Inria Paris & IRIF, Paris, France

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Amin Timany and Matthieu Sozeau. Cumulative Inductive Types In Coq. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Type theory
  • Theory of computation → Lambda calculus
  • Coq
  • Proof Assistants
  • Inductive Types
  • Universes
  • Cumulativity


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