We provide in this article two characterisation results, describing exactly which terms verify the dinaturality diagram, in Church-style system F and in Curry-style system F. The proof techniques we use here are purely syntactic, giving in particular a direct construction of the two terms generated by the dinaturality diagram. But the origin of these techniques lies in fact directly on the analysis of system F through game semantics. Thus, this article provides an example of backward engineering, where powerful syntactic results can be extracted from a semantic analysis.

Curry-style system F, that is, system F with no explicit types in terms, may be viewed as a core presentation of polymorphism from the point of view of programming languages. This paper gives a characterisation of type isomorphisms for this language using a game model, whose intuitions come from both the syntax and the game semantics universe. The model is composed of an untyped part to interpret terms, a notion of arena to interpret types and a typed part to express the fact that an untyped strategy σ plays on an arenaA. By analysing isomorphisms in the model, we prove that the equational system corresponding to type isomorphisms for Curry-style system F is the extension of the equational system for Church-style isomorphisms with a new, non-trivial equation: ∀X.A A[∀Y.Y/X] ifXappears only positively inA.