(* 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 Q_field.
intros n m; case m; case n; auto; intros n' m'; simpl in |- *;
rewrite Qpositive_mult_sym; auto.
Qed.
intros n m p; case p; case m; case n; auto; intros n' m' p'; simpl in |- *;
rewrite Qpositive_mult_assoc; auto.
Qed.
intros n; case n; auto.
Qed.
intros n; case n; simpl in |- *; auto; intros; rewrite Qpositive_mult_One;
auto.
Qed.
intros n; case n; simpl in |- *; auto.
intros n'; Case' (Qpositive_le_dec n' n').
case (Qpositive_eq_dec n' n'); auto.
intros H; elim H; auto.
intros H; elim H; auto.
intros n'; Case' (Qpositive_le_dec n' n').
case (Qpositive_eq_dec n' n'); auto.
intros H; elim H; auto.
intros H; elim H; auto.
Qed.
Definition Q_eq (n m : Q) :=
match n, m with
| Zero, Zero => true
| Qpos n', Qpos m' =>
match Qpositive_eq_dec n' m' with
| left h => true
| right h => false
end
| Qneg n', Qneg m' =>
match Qpositive_eq_dec n' m' with
| left h => true
| right h => false
end
| _, _ => false
end.
intros x y Htrue; generalize Htrue; clear Htrue.
case y; case x; try (intros; simpl in Htrue; discriminate; fail).
(* Focus 1 *)
auto.
(* Focus 2 *)
intros x' y'; simpl in |- *; case (Qpositive_eq_dec x' y').
intros H; rewrite H; auto.
intros; discriminate.
(* Focus 3 *)
intros x' y'; simpl in |- *; case (Qpositive_eq_dec x' y').
intros H; rewrite H; auto.
intros; discriminate.
Qed.
discriminate.
Qed.
Definition Qinv (x : Q) :=
match x with
| Qpos x => Qpos (Qpositive_inv x)
| Qneg x => Qneg (Qpositive_inv x)
| Zero => Zero
end.

Theorem Qinv_def : forall x : Q, x <> Zero -> Qmult x (Qinv x) = Qpos One.