(* nat misc *)

Definition ltb (x y: nat): bool := negb (leb y x).Definition geb (x y: nat): bool := leb y x.Ltac subst_tac x y z := (* todo: rename *)
  match z with
  | x => y
  | ?l + ?r =>
      let l' := subst_tac x y l in
      let r' := subst_tac x y r in
        constr: (l' + r')
  | ?l * ?r =>
      let l' := subst_tac x y l in
      let r' := subst_tac x y r in
        constr: (l' * r')
  | _ => z
  end.Ltac deep_le_trans h :=
  match type of h with
  | ?n <= ?u =>
    match goal with
    | |- ?l <= _ =>
      let q := subst_tac n u l
      in apply le_trans with q
    end
  end.Instance Transitive_le: Transitive le := le_trans.

Lemma lt_exists_plus (x y: nat) (p: x < y): exists d, y = S (x + d).