(***************************************************************************)
(* Formalization of the Chou, Gao and Zhang's decision procedure.          *)
(* Julien Narboux (Julien@narboux.fr)                                      *)
(* LIX/INRIA FUTURS 2004-2006                                              *)
(* University of Strasbourg 2008                                           *)
(***************************************************************************)
Require Export construction_lemmas.
Definition weak_parallelogram (A B C D : Point) : Prop :=
A<>C /\ B<>D /\
exists O, mid_point O A C /\ mid_point O B D.
Proof with Geometry.
intros.
unfold weak_parallelogram in H0.
DecompAndAll.
DecompExAnd H4 X.
unfold parallelogram.
assert (parallel A B C D).
eapply diag_mid_point_parallel;eauto.
repeat split;try assumption.
cut (parallel C B A D).
Geometry.
eapply diag_mid_point_parallel.
2:apply H5.
unfold mid_point in *;DecompAndAll;Geometry.
Qed.
Proof with Geometry.
intros.
unfold weak_parallelogram in H.
DecompAndAll.
DecompExAnd H3 X.
unfold mid_point in *.
DecompAndAll.
assert (S A P Q + S C P Q = 2 * S X P Q).
apply mid_point_equation; Geometry.
assert (S B P Q + S D P Q = 2 * S X P Q).
apply mid_point_equation; Geometry.
congruence.
Qed.
Definition weak_2_parallelogram (A B C D : Point) : Prop :=
(A=C <-> B=D) /\
exists O, mid_point O A C /\ mid_point O B D.
(** We choose the most general notion such that l2_11 is still valid *)
Definition  weak_3_parallelogram (A B C D : Point) : Prop :=
exists O, mid_point O A C /\ mid_point O B D.
Hint Resolve weak_para_1 : Geom.

Theorem l2_11a_strong_strong_strong_aux :
forall A B C D P Q : Point, A=C ->
weak_3_parallelogram A B C D -> S A P Q + S C P Q = S B P Q + S D P Q.