Definition projP1 (H : sig P) := let (x, h) := H in x.Definition projP2 (H : sig P) :=
   let (x, h) as H return (P (projP1 H)) := H in h.

Lemma Zlt_le_eq_S : forall x y : Z, (x < y)%Z -> (y <= x + 1)%Z -> y = (x + 1)%Z.