We give a machine-checked proof of the equivalence of the usual, two-sorted presentation of System F and its single-sorted pure type system variant λ2. This is established by reducing the typing problem of F to λ2 and vice versa. The difficulty lies in aligning different binding-structures and different contexts (dependent vs. non-dependent). The use of de Bruijn syntax, parallel substitutions, and context morphism lemmas leads to an elegant proof. We use the Coq proof assistant and the substitution library Autosubst.
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