Constructor Subtyping

Constructor Subtyping in the Calculus of Inductive Constructions

“…In particular, we show that adding constructor subtyping preserves some fundamental properties of CIC, including confluence, subject reduction, strong normalization and decidability of type-checking. These results scale up to dependent types those of previous papers [7,8], and open the road for an integration of constructor subtyping in proof-development systems such as Coq and Lego.…”

“…The immediate benefit of coherence is that the above definitions may be viewed as deterministic rule sets in the sense of [1]-see Section 2 for the difficulties with datatypes that do not yield deterministic rule sets. Therefore they support recursive definitions and may be integrated safely to typed λ-calculi [7,8].…”

“…Constructor subtyping [7,8] is 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 inhabitants than σ. It is fully compatible with the (co-)inductive approach to formalization and may be used to specify most of the examples arising in natural semantics [23].…”

“…It is fully compatible with the (co-)inductive approach to formalization and may be used to specify most of the examples arising in natural semantics [23]. For example, constructor subtyping may be used to formalize the expressions of the call-by-value λ-calculus and of the ς-calculus, or the set of Harrop formulae [7,8]. It may also be used to formalize the datatype of lists/nonempty lists:…”

Type Isomorphisms and Proof Reuse in Dependent Type Theory

“…Subtyping [16,17] is a basic mechanism to enhance proof reuse in type systems. Specific forms of subtyping connected to reusability include record subtyping [11] and constructor subtyping [8] but these forms of subtyping are too specialised to provide a general solution to the problem of reusability.…”