system-f-omega

An intrinsically typed formalization of impredicative System F-omega, plus its normalization by hereditary substitution.

The presentation follows Altenkirch and Keller, but we need significant extra machinery to adapt it to F-omega.

We need normalization and a large part of its correctness proof for types (which is essentially a simply typed lambda calculus) in order to be able to define the term operations in an intrinsically well-typed way. This consitutes the bulk of the line count of this project; I could've probably copied Altenkirch and Keller and written some proofs to glue it to System F-omega, but I considered it a valuable effort to redo it in a way that by default mirrors System F-omega term operations.

Note: normalization by evaluation would be a much simpler and more efficient solution for type normalization! I only did hereditary substitution for types because I wanted to understand and rewrite the Altenkirch-Keller proofs.

Most notably, what we don't have is a termination proof for term substitution. It eventually dawned on me that the main reason why hereditary substitution works for STLC is because applying a term to a spine is structurally recursive in the type of the term, and this is irreparably foiled in impredicative System F. Here, if a term has type ∀ (a : k) . t, if we apply it to t' : k the resultant type would be a t [a := t'] substitution, which isn't smaller than the starting type by any measure. For example, we can have id = Λ (a : ⋆). λ (x : a) . x with id : ∀ (a : *). a → a, and then id [∀ (a : *). a → a] id has the same type (definitionally) as id.

This setup would be amenable to simple termination proofs only for predicative or stratified System F, similarly to what's in the paper of Eades and Stump, and also Mangin and Sozeau.

However, eta expansion, renaming and type substition is still total (as certified by Agda) and hopefully useful. We also have a rather large chunk of the correctness proof for STLC's normalization, and we can play around with the F-omega normalization function.