Is it possible to derive induction for the church-encoded Nat?

Question

I was just wondering if it is possible to derive induction for the church-encoded Nat type on Idris, Agda, Coq and similar. Notice this is a different issue from doing it on

Answer 1

It is known not to be provable because there are models of the calculus of constructions where the impredicative encoding of the natural numbers is not initial (i.e. doesn't satisfy induction). It does follow from relational parametricity as Phil Wadler has shown long time ago. Hence combining Wadler with internal relational parametricity ala Moulin and Bernardy may do the trick.