(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
Require Export Qpositive.
Fixpoint Qpositive_le_bool (w w' : Qpositive) {struct w'} : bool :=
match w with
| One => match w' with
| dL y => false
| _ => true
end
| dL y => match w' with
| dL y' => Qpositive_le_bool y y'
| _ => true
end
| nR y => match w' with
| nR y' => Qpositive_le_bool y y'
| _ => false
end
end.
Definition Qpositive_le (w w' : Qpositive) := Qpositive_le_bool w w' = true.
Definition Qpositive_le' (w w' : Qpositive) :=
match Qpositive_i w, Qpositive_i w' with
| (p, q), (p', q') => p * q' <= p' * q
end.
intros w w'; generalize w; elim w'; clear w w'.
intros w'1 Hrec w; case w.
(* 1 *)
unfold Qpositive_le in |- *; simpl in |- *.
intros w1 H; generalize (Hrec w1 H).
unfold Qpositive_le' in |- *.
simpl in |- *.
case (Qpositive_i w1); case (Qpositive_i w'1).
intros p' q' p q Hle; repeat rewrite mult_plus_distr_r.
rewrite (mult_comm q').
apply plus_le_compat_r.
auto.
(* 1 *)
(* 2 *)
intros w1; unfold Qpositive_le in |- *.
simpl in |- *.
unfold Qpositive_le' in |- *; simpl in |- *; intros H; case (Qpositive_i w1);
case (Qpositive_i w'1); intros p' q' p q.
rewrite mult_plus_distr_r.
repeat rewrite <- (mult_comm (p + q)).
repeat rewrite mult_plus_distr_r.
rewrite (plus_comm (p * p' + q * p')).
repeat rewrite <- plus_assoc.
auto with arith.
(* 2 *)
(* Focus 3 *)
unfold Qpositive_le, Qpositive_le' in |- *.
simpl in |- *.
case (Qpositive_i w'1).
intros; omega.
(* 3 *)
intros w'1 Hrec w; case w; clear w; unfold Qpositive_le in |- *;
simpl in |- *.
(* 1 *)
intros q H; discriminate H.
(* 2 *)
intros w1 H; generalize (Hrec w1 H); unfold Qpositive_le' in |- *.
simpl in |- *; case (Qpositive_i w1); case (Qpositive_i w'1);
intros p' q' p q.
repeat rewrite (mult_comm p); repeat rewrite (mult_comm p');
repeat rewrite mult_plus_distr_r.
intros H1.
rewrite (mult_comm p').
apply plus_le_compat_l.
auto.
(* 3 *)
intros H; discriminate H.
unfold Qpositive_le, Qpositive_le' in |- *.
intros w; case w.
3: simpl in |- *; auto.
intros w1; simpl in |- *.
intros Heq; discriminate Heq.
intros w'; simpl in |- *.
case (Qpositive_i w').
intros; omega.
Qed.

Theorem Qpositive_le'_to_Qpositive_le :
forall w w' : Qpositive, Qpositive_le' w w' -> Qpositive_le w w'.