Isomorphisms of types are computational witnesses of logical equivalence with additional properties. The types/formulas A and B are isomorphic if there are functions (in a certain formalism) f : A → B and g : B → A such that g ○ f and f ○ g are equal in a certain sense to the identity on A and B, respectively. Typical such formalisms are extensions of simply typed λ-calculus, with βη-convertibility as equality relation. Another view of a pair of functions f : A → B and g : B → A (besides establishing the logical equivalence of A and B) is that f is invertible with left-inverse g, and it is then natural to relax the above symmetric condition to just g ○ f being equal to the identity on A. In this situation, A is called a retract of B, which is thus a natural generalisation of the notion of an isomorphism, while both these notions are refinements of the concept of logical equivalence in operational terms, that is, in terms of computable functions.

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 arena A.

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.AA[∀Y.Y/X] if X appears only positively in A.

A common way to show the termination of the union of two abstract reduction systems, provided both systems terminate, is to prove that they enjoy a specific property (some sort of ‘commutation’ for instance). This specific property is actually used to show that, for the union not to terminate, one of the systems must itself be non-terminating, which leads to a contradiction. Unfortunately, the property may be impossible to prove because some of the objects that are reduced do not enjoy an adequate form.

Hence the purpose of this paper is threefold:

• – First, it introduces an operator enabling us to insert a reduction step on such an object, and therefore to change its shape, while still preserving the ability to use the property. Of course, some new properties will need to be verified.

• – Second, as an instance of our technique, the operator is applied to relax a well-known lemma stating the termination of the union of two termination abstract reduction systems.

• – Finally, this lemma is applied in a peculiar and then in a more general way to show the termination of some lambda calculi with inductive types augmented with specific reductions dealing with:

  1. (i) copies of inductive types;

  2. (ii) the representation of symmetric groups.

We incorporate the idea of coercive subtyping, a theory of abbreviation for dependent type theories, into the polymorphic type system in functional programming languages. The traditional type system with let-polymorphism is extended with argument coercions and function coercions, and a corresponding type inference algorithm is presented and proved to be sound and complete.

A simple type σ is retractable to a simple type τ if there are two terms Cσ→τ and Dτ→σ such that D ○ C λx.x. The retractability of types is affine if the terms C and D are affine, that is, when every bound variable occurs in them at most once in the scope of its declaration. This paper presents a system that derives affine retractability for simple types. It also studies the complexity of constructing these affine retractions. The problem of affine retractability is NP-complete even for the class of types over a single type atom and having limited functional order. In addition, a polynomial algorithm for types of orders less than three is presented.