Ltac normZ :=
  repeat
  match goal with
  | H : _ < _ |- _ =>  apply Zlt_le_add_1 in H
  | H : ?Y <= _ |- _ =>
    lazymatch Y with
    | 0 => fail
    | _ => apply Zle_minus_le_0 in H
    end
  | H : _ >= _ |- _ => apply Z.ge_le in H
  | H : _ > _  |- _ => apply Z.gt_lt in H
  | H : _ = _  |- _ => apply eq_incl in H ; destruct H
  | |- @eq Z _ _  => apply elim_concl_eq ; let H := fresh "HZ" in intros [H|H]
  | |- _ <= _ => apply elim_concl_le ; intros
  | |- _ < _ => apply elim_concl_lt ; intros
  | |- _ >= _ => apply Z.le_ge
  end.Inductive proof :=
| Hyp (e : Z) (prf : 0 <= e)
| Add (p1 p2: proof)
| Mul (p1 p2: proof)
| Cst (c : Z)
.Inductive proof :=
| Hyp (e : Z) (prf : 0 <= e)
| Add (p1 p2: proof)
| Mul (p1 p2: proof)
| Cst (c : Z)
.Inductive proof :=
| Hyp (e : Z) (prf : 0 <= e)
| Add (p1 p2: proof)
| Mul (p1 p2: proof)
| Cst (c : Z)
.Inductive proof :=
| Hyp (e : Z) (prf : 0 <= e)
| Add (p1 p2: proof)
| Mul (p1 p2: proof)
| Cst (c : Z)
.Inductive proof :=
| Hyp (e : Z) (prf : 0 <= e)
| Add (p1 p2: proof)
| Mul (p1 p2: proof)
| Cst (c : Z)
.

(* Specifications: *)
Lemma add_le : forall e1 e2, 0 <= e1 -> 0 <= e2 -> 0 <= e1+e2.
Lemma mul_le : forall e1 e2, 0 <= e1 -> 0 <= e2 -> 0 <= e1*e2.