(** * Equalities on dependent types *)
Hint Rewrite eq_rect_nat_double.
Hint Rewrite <- (eq_rect_eq_dec eq_nat_dec).
Ltac generalize_proof :=
match goal with
| [ |- context[eq_rect _ _ _ _ ?H ] ] => generalize H
end.
Ltac eq_rect_simpl :=
unfold eq_rec_r, eq_rec;
repeat rewrite eq_rect_nat_double;
repeat rewrite <- (eq_rect_eq_dec eq_nat_dec).
Ltac destruct_existT :=
repeat match goal with
| [H: existT _ _ _ = existT _ _ _ |- _] =>
(apply Eqdep.EqdepTheory.inj_pair2 in H; subst)
end.

Lemma existT_eq_rect:
forall (X: Type) (P: X -> Type) (x1 x2: X) (H1: P x1) (Hx: x1 = x2),
existT P x2 (eq_rect x1 P H1 x2 Hx) =
existT P x1 H1.