Constructor Subtyping in the Calculus of Inductive Constructions

Abstract

The Calculus of Inductive Constructions (CIC) is a powerful type system, featuring dependent types and inductive definitions, that forms the basis of proof-assistant systems such as Coq and Lego. We extend CIC with constructor subtyping, a basic form of subtyping in which an inductive type σ is viewed as a subtype of another inductive type τ if τ has more elements than σ. It is shown that the calculus is wellbehaved and provides a suitable basis for formalizing natural semantics in proof-development systems.

Structural subtyping for inductive types with functorial equality rules

“…The early development of the framework of coercive subtyping is closely related to Aczel's idea of type-checking methods in classes with overloading (Aczel 1994) and the work by Breazu-Tannen et al on giving coercion semantics to lambda calculi with subtyping (Breazu-Tannen et al 1991). Barthe and his colleagues have studied constructor subtyping and its possible applications in proof systems (Barthe and Frade 1999;Barthe and van Raamsdonk 2000). Among other research work on coercive subtyping, Y. Luo's Ph.D. work provided an extensive study of structural subtyping for inductive types, especially the notion of weak transitivity (Luo 2005;Luo and Luo 2005).…”