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 well-behaved and provides a suitable basis for formalizing natural semantics in proof-development systems.

Coercive subtyping is a general approach to subtyping, inheritance and abbreviation in dependent type theories. A vital requirement for coercive subtyping is that of coherence which essentially says that coercions between any two types must be unique. Another important task for coercive subtyping is to prove the admissibility or elimination of transitivity and substitution. In this paper, we propose and study the notion of Weak Transitivity, consider suitable subtyping rules for certain parameterised inductive types and prove its coherence and the admissibility of substitution and weak transitivity in the coercive subtyping framework.

ing from all possible type annotations to constructors and case-expressions occurring in e. The produced terms can then be taken and typed without constructor overloading and, following Mitchell's subtyping ideas in [2], a most general typing judgment for each of these terms can now be generated. We show that this method is a correct and complete way of generating a set of most general typings to a term. The main issues raised in the extension of Mitchell's ideas to these subtyping systems are related to the presence of parametric datatypes as a basic type constructor. In extending such ideas, one needs also to take into account the presence of a partial order on datatypes, induced by the constructor subtyping mechanism, as well as the presence of case-expressions built over datatypes. References

Coherence is a vital requirement for the correct use of coercive subtyping for abbreviation and other applications. However, some coercions are incoherent, although very useful. A typical example of such is the subtyping rules for -types: the component-wise rules and the rule of the rst projection. Both of these groups of rules are often used in practice (and coherent themselves), but they are incoherent when put together directly. In this paper, we study this case for -types by introducing a new subtyping relation and the resulting system enjoys the properties of coherence and admissibility of substitution and transitivity.