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 { computational uniqueness of coercions between any two types. In this paper, we develop techniques useful in proving coherence and its related result on admissibility of transitivity and substitution. In particular, we consider suitable subtyping rules for -types and -types and prove its coherence and the admissibility of substitution and transitivity rules at the type level in the coercive subtyping framework. 1

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.