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(* Formalization of the Chou, Gao and Zhang's decision procedure.*)
(* Julien Narboux (Julien.Narboux@inria.fr)                                     *)
(* LIX/INRIA FUTURS 2004-2008                                                     *)
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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).

Lemma freepoint_elimination_aux : forall O U V B Y:Point, ~ Col O U V ->
S O B Y = 1/ (S O U V) * (S O B V * S O U Y + S O B U * S O Y V).