(***************************************************************************)
(* Formalization of the Chou, Gao and Zhang's decision procedure.*)
(* Julien Narboux (Julien.Narboux@inria.fr)                                     *)
(* LIX/INRIA FUTURS 2004-2008                                                     *)
(***************************************************************************)
Require Export area_elimination_lemmas.
(** For the first time, we need decidability of the parallel predicate *)
Ltac cases_parallel A B C D := elim (parallel_dec A B C D);intros.
Definition Det3 (x1 x2 x3 y1 y2 y3 z1 z2 z3 : F) : F :=
x1 * (y2 * z3) - x1 * (y3 * z2) - x2 * (y1 * z3) + x2 * (y3 * z1) +
x3 * (y1 * z2) - x3 * (y2 * z1).
Proof with Geometry.
intros.
cases_parallel U V O Y.
rename H into T.
assert (O**Y / U**V = S U O Y / S O V U).
apply l2_15...
assert (S O B Y = S O B O + (O**Y/U**V) * S4 O U B V).
apply elim_area_on_parallel.
unfold on_parallel;repeat split...
eauto with Geom.
rewrite H in H1.
basic_simpl.
unfold S4 in *.
assert (parallel O Y U V).
Geometry.
unfold parallel, S4 in *.
IsoleVarRing (S O Y V) H2.
rewrite H2.
rewrite H1.
uniformize_signed_areas.
field.
split...
assert (exists W, inter_ll W U V O Y).
apply inter_ll_ex...
DecompEx H1 W.
cases_equality W O.
subst W.
unfold inter_ll in *.
intuition.
rename H1 into Hdiff.
assert (S B O W = 1 / S4 U O V Y * (S U O Y * S B O V + S V Y O * S B O U)).
apply  elim_area_inter_ll...
replace (S4 U O V Y) with (- S4 O U Y V) in H1.
2:Geometry.
replace ((S U O Y * S B O V + S V Y O * S B O U)) with
(S O B V * S O U Y + S O B U * S O Y V) in H1.
2:uniformize_signed_areas;field.
unfold inter_ll in *;DecompAndAll.
assert (O ** W / O ** Y = S O U V / S4 O U Y V).
apply co_side_ter...
cases_col B O W.
cases_equality V W.
subst W.
assert (Col O Y B).
eapply col_trans_1 with (B:=V)...
unfold Col in *.
uniformize_signed_areas.
rewrite H4.
rewrite H5.
rewrite H7.
ring.
cases_equality O B.
subst B.
basic_simpl.
ring.
cases_col  V O B.
assert (Col O Y B).
eapply col_trans_1 with (B:= W)...
assert (Col O Y V).
eapply col_trans_1 with (B:= B)...
unfold Col in *.
uniformize_signed_areas.
rewrite H9.
rewrite H10.
rewrite H11.
ring.
assert (U**W/V**W = S U O B / S V O B).
apply co_side...
cases_col V O Y.
cases_equality Y O.
subst Y.
basic_simpl.
ring.
assert (Col Y W V).
eapply col_trans_1 with (B:= O)...
assert (Col W U Y).
eapply col_trans_1 with (B:= V)...
cases_equality Y W.
Unset Regular Subst Tactic.
subst W.
Set Regular Subst Tactic.
clear H13 H12 H5.
assert (Col O V B).
eapply col_trans_1 with (B:= Y)...
unfold Col in *.
uniformize_signed_areas.
rewrite H4.
replace (S V O B) with (-0).
rewrite H11.
ring.
rewrite <- H5.
ring.
assert (Col Y U O).
eapply col_trans_1 with (B:= W)...
assert (Col O U V).
eapply col_trans_1 with (B:= Y)...
intuition.
assert (U**W/V**W = S U O Y / S V O Y).
apply co_side...
rewrite H10 in H12.
assert (Col O B Y).
eapply col_trans_1 with (B:= W)...
rewrite H13.
uniformize_signed_areas.
RewriteVar (S O B U) H12.
2:Geometry.
field.
split...
(** Second case *)
assert (O**Y / O**W = S B O Y / S B O W).
apply A6...
assert (O**Y / O**W = S4 O U Y V / S O U V).
replace (O ** Y / O ** W) with (1/(O ** W / O ** Y)).
rewrite H2.
field.
Geometry.
repeat apply nonzeromult...
field.
split...
unfold not;intro.
assert (O=Y).
Geometry.
subst Y.
assert (parallel U V O O)...
rewrite H7 in H8.
set (S O B V * S O U Y + S O B U * S O Y V) in *.
rewrite H1 in H8.
replace (S B O Y) with (-S O B Y) in H8.
2:Geometry.
RewriteVar (S O B Y) H8.
field;split...
apply nonzeromult...
apply nonzerodiv...
unfold not;intro.
rewrite H9 in H1.
basic_simpl.
assert (Col O Y B).
eapply col_trans_1 with (B:=W)...
Geometry.
Qed.
Proof with Geometry.
intros.
unfold Det3.
replace (S A B Y) with (S A B O + S A O Y + S O B Y)...
replace (S A O Y) with (- S O A Y).
2:symmetry;Geometry.
replace (S O A Y) with (1/ (S O U V) * (S O A V * S O U Y + S O A U * S O Y V)).
2:symmetry;eapply freepoint_elimination_aux...
replace (S O B Y) with (1/ (S O U V) * (S O B V * S O U Y + S O B U * S O Y V)).
2:symmetry;eapply freepoint_elimination_aux...
replace (S A B O) with (S O A B).
2:Geometry.
replace (S O A B) with (1/ (S O U V) * (S O A V * S O U B + S O A U * S O B V)).
2:symmetry;eapply freepoint_elimination_aux...
uniformize_signed_areas.
field...
Qed.
intros.
assert (S O V U <> 0).
intro; assert (Col O U V) by auto with Geom;intuition.
rewrite (free_points_ratio_elimination_3 O V U A B C D) by auto.
reflexivity.
Qed.
(* set (X_C := S O U C). set (X_A := S O U A). set (X_B := S O U B). set (X_D := S O U D). set (Y_C := S O V C). set (Y_A := S O V A). set (Y_B := S O V B). set (Y_D := S O V D). *)

Lemma free_points_ratio_elimination_7_non_zero_denom :
forall O U V A B C D: Point,
parallel A B C D ->
C<>D ->
~ Col O U V ->
~ Col U A C ->
Col A C D ->
(S O U V * (S O U D - S O U C) - S O U C * S O V D + S O U D * S O V C) <> 0.