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(* as published by the Free Software Foundation; either version 2.1   *)
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(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
Require Export Qpositive_order.
Definition Qpositive_sub (w w' : Qpositive) :=
let (p, q) := Qpositive_i w in
let (p', q') := Qpositive_i w' in
Qpositive_c (p * q' - p' * q) (q * q') (p * q' + p' * q + q * q').
intros x y z.
elim (le_or_lt z y).
intros H;
generalize ((fun m n p : nat => mult_le_compat_l p n m) (S x) _ _ H).
intros H1 H2; elim (le_not_gt _ _ H1); auto.
auto.
Qed.

Theorem Qpositive_sub_correct :
forall w w' : Qpositive, Qpositive_sub (Qpositive_plus w w') w' = w.