Ltac Flip :=
  apply Z.gt_lt || apply Z.lt_gt || apply Z.le_ge || apply Z.ge_le; assumption.Ltac Falsum :=
  try intro; apply False_ind;
   repeat
    match goal with
    | id1:(~ ?X1) |- ?X2 =>
        (apply id1; assumption || reflexivity) || clear id1
    end.Ltac Step_l a :=
  match goal with
  |  |- (?X1 < ?X2)%Z => replace X1 with a; [ idtac | try ring ]
  end.Ltac Step_r a :=
  match goal with
  |  |- (?X1 < ?X2)%Z => replace X2 with a; [ idtac | try ring ]
  end.Ltac CaseEq formula :=
  generalize (refl_equal formula); pattern formula at -1 in |- *;
   case formula.

(* Specifications: *)
Lemma pair_1 : forall (A B : Set) (H : A * B), H = pair (fst H) (snd H).
Lemma pair_2 : forall (A B : Set) (H1 H2 : A * B), fst H1 = fst H2 -> snd H1 = snd H2 -> H1 = H2.