A Presheaf Model of Parametric Type Theory

We propose a new type theory with internalized parametricity. Compared to previous similar proposals, this version comes with a denotational semantics which is a refinement of the standard presheaf semantics of dependent type theory. Further, this presheaf semantics is a refinement of the one used to interpret nominal sets with restriction. The present calculus is a candidate for the core of a proof assistant with internalized parametricity. Reynolds’s abstraction theorem can be stated in a purely syntactical way: for instance, if a function f has type (A : ?) → A → A — the type of the polymorphic identity — then the proposition (A : ?) → (P : A → ?) → (x : A) → P x → P (f Ax) holds. However this result is not provable internally, i.e., (f : (A : ?) → A → A) → (A : ?) → (P : A → ?) → (x : A) → P x → P (f Ax) is not provable. Several attempts have been made for designing an extension of dependent type theory in which such an internal form of parametricity holds. We propose another such system here. Our technical contributions are as follows: • We present a type theory which internalizes parametricity and can be seen as a simplification and generalization of the systems of [1, 2] • We provide a denotational semantics, in the form of a presheaf model, for this type theory. This model is a refinement of the presheaf semantics used to interpret nominal sets with restrictions [3, 4]. Syntax We assume a special symbol ‘0’, and a countable infinite set I of other symbols, called colors. The metasyntactic variables i, j, . . . range over colors, while φ range over I ∪ {0}. The main innovation of the type theory presented here is that terms may depend on (a finite number of) colors. We add the following constructions to the usual syntax of lambda calculi: a, p, t, A, P, T := . . . | (a,i p) | (x : A)×i P | A 3i a | a·i Remark. Here is some intuition for these new constructions: • Any type is associated with a predicate for every color. The type A 3i a expresses that a satisfies the parametricity predicate associated with the type A on color i. For each term a and color i, the term a (i 0) is the erasure of i in a. It is defined by induction on a and can be understood as a realizer of a. • The term a·i yields a proof of A 3i a (i 0). • The forms (a,i p) and (x : A) ×i P allow to locally associate parametricity proofs with a given realizer.