1 Conclusion and Future Work Conversion-based Rules Observe That Rules S-apr, S-apsl and S-apsr Con- S-apt

J. Tiuryn. A sequent calculus for subtyping polymorphic types. unpublished , 1996. TU96] J. Tiuryn and P. Urzyczyn. The subtyping problem for second-order types is undecidable. In this short survey, we have presented subtyping for two type systems: Cò for System-F(c) and P ; for. There are several common features in these studies. From the syntactic point of view, both in Cò and transitivity is admissible. From the semantical side, the interpretation of subtyping as logical entailment with coherent coercions has been rst developed in the study of Cò and then applied to. We are currently studying the coercion version of and its properties. Another direction is to use the technique developed in these works to establish subtyping extensions of Calculus of Constructions or other systems in the-cube. The aim is that these extensions by subtyping could be used in the designs of programming languages and in the proof development environments. A lter lambda model and the completeness of type assignment. 20 c from to such that: (a) cM is a proof of ; (b) c is "almost" an identity function, formally, erase(c) is equal to the identity function x:x in type-free-calculus. This interpretation captures two aspects of subtyping: 1. subtyping as logical implication ; 2. subtyping as set inclusion. So the subtyping relation can be understood as "any proof of the proposition can be used as a proof of with straightforward modiications in the type labels". If we take ; as speciications for programs, then the subtyping relation means that any programs satisfying the speciication of can be straightforwardly adapted for the speciication. The second point in the above logical interpretation of subtyping has made it clear what this straightforward modiication is. In this section, we mainly analyse the subtyping from the logical implication point of view. In P and , there are subtyping relations between type families. Type families are not types, so they can neither be interpreted as sets nor as formulas. But we use subtyping rules only in the subsumption rule for checking subtyping relations between "true" types, that is, we only want to check ? ` such that ? ` ; : ?. In , one can observe that all judgements in the derivation for such "true" subtyping judgement contains only types of kind ?. Therefore, subtyping between type families is not directly used, especially the rule S-A of-abstraction can be eliminated. Now we give …