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(* Formalization of the Chou, Gao and Zhang's decision procedure.          *)
(* Julien Narboux (Julien@narboux.fr)                                      *)
(* LIX/INRIA FUTURS 2004-2006                                              *)
(* University of Strasbourg 2008                                           *)
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Require Export field.
Ltac Geometry := auto with Geom field_hints.
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(** Axioms                                                             *)
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(** The set of Points *)
Parameter Point : Set.
(** The signed area *)
Parameter S : Point -> Point -> Point -> F.
(** The signed distance *)
Parameter DSeg : Point -> Point -> F.
Infix "**" := DSeg (left associativity, at level 20) : F_scope.
Definition Col (A B C : Point) : Prop := S A B C = 0.
Definition S4 (A B C D : Point) : F := S A B C + S A C D.
Definition parallel (A B C D : Point) : Prop := S4 A C B D = 0.

Axiom  A2b : forall (A B P P' : Point) (r : F),
A <> B ->
Col A B P ->
A ** P = r * A ** B -> Col A B P' -> A ** P' = r * A ** B -> P = P'.