intros w w' w''.
elim (interp_non_zero w); intros p Hex; elim Hex; intros q Heq; clear Hex.
elim (interp_non_zero w'); intros p' Hex; elim Hex; intros q' Heq'; clear Hex.
elim (interp_non_zero w''); intros p'' Hex; elim Hex; intros q'' Heq'';
clear Hex.
unfold Qpositive_mult, Qpositive_plus in |- *.
rewrite Heq; rewrite Heq'; rewrite Heq''.
elim
(construct_correct2' (S p * S q' + S p' * S q + S q * S q')
(S p * S q' + S p' * S q) (S q * S q')).
intros d;
elim
(interp_non_zero
(Qpositive_c (S p * S q' + S p' * S q) (S q * S q')
(S p * S q' + S p' * S q + S q * S q'))); intros p3 Hex;
elim Hex; intros q3 Heq3; rewrite Heq3; intros (Heq1, Heq2);
clear Hex.
elim
(construct_correct2' (S p * S p'' + S q * S q'') (S p * S p'') (S q * S q'')).
intros d';
elim
(interp_non_zero
(Qpositive_c (S p * S p'') (S q * S q'') (S p * S p'' + S q * S q'')));
intros p4 Hex; elim Hex; intros q4 Heq6; rewrite Heq6;
intros (Heq4, Heq5); clear Hex.
elim
(construct_correct2' (S p' * S p'' + S q' * S q'')
(S p' * S p'') (S q' * S q'')).
intros d'';
elim
(interp_non_zero
(Qpositive_c (S p' * S p'') (S q' * S q'') (S p' * S p'' + S q' * S q'')));
intros p5 Hex; elim Hex; intros q5 Heq9; rewrite Heq9;
intros (Heq7, Heq8); clear Hex.
apply construct_correct4'.
simpl in |- *; auto with arith.
simpl in |- *; auto with arith.
simpl in |- *; auto with arith.
simpl in |- *; auto with arith.
auto.
auto.
unfold fst, snd in Heq1, Heq2, Heq4, Heq5, Heq7, Heq8.
apply mult_reg_l with d''.
rewrite (mult_comm (S p5)).
rewrite mult_plus_distr_r.
repeat
(rewrite <- mult_assoc; try rewrite (mult_comm (S p5));
try rewrite (mult_comm (S q5))).
repeat rewrite (mult_comm (S d'')); rewrite mult_plus_distr_r;
repeat rewrite <- mult_assoc; rewrite <- Heq7; rewrite <- Heq8.
repeat
(repeat rewrite (mult_comm (S p4)); repeat rewrite (mult_comm (S q4));
rewrite <- mult_assoc).
apply mult_reg_l with d'; repeat rewrite (mult_comm (S d'));
rewrite mult_plus_distr_r; repeat rewrite <- mult_assoc.
rewrite <- Heq4; rewrite <- Heq5.
apply mult_reg_l with d; repeat rewrite (mult_comm (S d));
rewrite mult_plus_distr_r; repeat rewrite (mult_comm (S q3));
rewrite (mult_comm (S p3)); repeat rewrite <- mult_assoc.
rewrite <- Heq1; rewrite <- Heq2.
ring.
simpl in |- *; auto with arith.
simpl in |- *; auto with arith.
auto.
simpl in |- *; auto with arith.
simpl in |- *; auto with arith.
auto.
simpl in |- *; auto with arith.
simpl in |- *; auto with arith.
auto.
Qed.

Theorem Qpositive_mult_inv :
forall w : Qpositive, Qpositive_mult w (Qpositive_inv w) = One.