(* Transport equality, only matching on eq_refl in contradictory cases, to make sure
   terms using this function reduce *)
Fixpoint nat_cast (P : nat -> Type) {n m} : n = m -> P n -> P m.

Lemma nat_cast_fuse: forall (P: nat -> Type) (n1 n2 n3: nat) (e12: n1 = n2) (e23: n2 = n3) (x: P n1), nat_cast P e23 (nat_cast P e12 x) = nat_cast P (eq_trans e12 e23) x.