(***************************************************************************)
(* Formalization of the Chou, Gao and Zhang's decision procedure.          *)
(* Julien Narboux (Julien@narboux.fr)                                      *)
(* LIX/INRIA FUTURS 2004-2006                                              *)
(* University of Strasbourg 2008-2009                                       *)
(***************************************************************************)
Require Export area_coords_constructions.
(* Those lemma are not proved as they are used only in a optimisation of the method. *)

(* Specifications: *)
Lemma elim_py_a_ratio_right : forall Y O U V A B ro ru rv,
a_ratio Y O U V ro ru rv ->
Py A B Y = ro * Py A B O + ru * Py A B U + rv * Py A B V.
Lemma elim_signed_area_a_ratio : forall Y O U V A B ro ru rv,
a_ratio Y O U V ro ru rv ->
S A B Y = ro * S A B O + ru * S A B U + rv * S A B V.
Lemma elim_py_a_ratio_middle : forall Y O U V A B ro ru rv,
a_ratio Y O U V ro ru rv ->
Py A Y B = ro * Py A O B + ru * Py A U B + rv * Py A V B
- ro*ru*Py O U O - ro*ru*Py O V O - ru*rv * Py U V U.
Lemma elim_py_a_ratio_left_right : forall Y O U V A ro ru rv,
a_ratio Y O U V ro ru rv ->
Py Y A Y = ro * (ro * Py O A O + ru * Py O A U + rv * Py O A V) +
ru * (ro * Py U A O + ru * Py U A U + rv * Py U A V) +
rv * (ro * Py V A O + ru * Py V A U + rv * Py V A V).