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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.
Definition Qpositive_plus (w w' : Qpositive) : 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 w; elim (interp_non_zero w); intros p Hex; elim Hex; intros q Heq.
unfold Qpositive_plus in |- *.
replace (Qpositive_i One) with (1, 1).
rewrite Heq.
rewrite mult_1_r; rewrite mult_1_l.
rewrite (plus_comm (S q)).
replace (S p + S q + S q) with (S (p + S q + S q)).
rewrite (Qpositive_c_unfold1 p q).
rewrite (construct_correct w); auto with *.
auto.
auto.
auto.
Qed.

Theorem Qpositive_plus_sym :
forall w w' : Qpositive, Qpositive_plus w w' = Qpositive_plus w' w.