An Alpha-Corecursion Principle for the Infinitary Lambda Calculus

Abstract

Abstract. Gabbay and Pitts proved that lambda-terms up to alphaequivalence constitute an initial algebra for a certain endofunctor on the category of nominal sets. We show that the terms of the infinitary lambda-calculus form the final coalgebra for the same functor. This allows us to give a corecursion principle for alpha-equivalence classes of finite and infinite terms. As an application, we give corecursive definitions of substitution and of infinite normal forms (Böhm, Lévy-Longo and Berarducci trees).

Nominal Coalgebraic Data Types with Applications to Lambda Calculus

“…Moreover the map [−] α : T ∞ Σ → (T Σ /= α ) ∞ is not surjective in general, as shown by Example 5.20. In the case of the infinitary λ-calculus we solved this issue by restricting our attention to terms with finitely many free variables [KPSdV12]. We do the same in the case of a general binding signature.…”

“…In [KPSdV12] we showed that this nominal set is isomorphic to (Λ ∞ ffv /= α , ·) and to the carrier of the final coalgebra of the Nom-functor L α . Hence, for each α-equivalence class of infinitary terms with finitely many variables we can find a representative.…”

“…is surjective can be proved by going back to the syntax as in [KPSdV12], just that this time, due to generalising from λ-calculus to binding signatures, the notation becomes even heavier and quite unpleasant. Therefore, we will give a semantic proof in the next section, so that surjectivity becomes a consequence of Theorem 5.72.…”

“…Then the bound variables of xλy.y can be computed as supp(xλy.y) \ supp([xλy.y] α ) = {x, y} \ {x} = {y}. Of course, this calculation depends on being able to assume that the bound variables and free variables of xλy.y do not overlap, or, in the terminology of [KPSdV12], that xλy.y is α-safe. The next definition gives a semantic formulation of an element being safe with respect to a map, which now does not need to be a quotient by α-equivalence.…”

“…We prove that the set of equivalence classes of infinitary terms with finitely many free variables is the final coalgebra of a Nom-functor. Our running example is the infinitary λ-calculus, and indeed the results of [KPSdV12] are particular instances of the main theorems in this section.…”