(***************************************************************************)
(* 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 freepoints.
(* Exprime un triplet D E F suivant les coordonnées sur la base A B C *)
Ltac freepoints_independant A B C H D E F :=
rewrite (free_points_area_elimination A B C D E F H) in *.
(* Exprime tout suivant une base A B C *)
Ltac iter_coord_expr A B C H E :=
match constr:(E) with
| (?X1 = ?X2) =>
iter_coord_expr A B C H X1; iter_coord_expr A B C H X2
| (?X1 + ?X2) =>
iter_coord_expr A B C H X1; iter_coord_expr A B C H X2
| (?X1 * ?X2) =>
iter_coord_expr A B C H X1; iter_coord_expr A B C H X2
| (?X1 / ?X2) =>
iter_coord_expr A B C H X1; iter_coord_expr A B C H X2
| (?X1 - ?X2) =>
iter_coord_expr A B C H X1; iter_coord_expr A B C H X2
| (- ?X1) => iter_coord_expr A B C H X1
| (/ ?X1) => iter_coord_expr A B C H X1
| ?X5 =>
match constr:(X5) with
| (S ?X1 ?X2 ?X3) =>
match constr:(X1) with
| A =>
match constr:(X2) with
| B => idtac
| C => idtac
| _ =>
match constr:(X3) with
| B => idtac
| C => idtac
| _ => freepoints_independant A B C H X1 X2 X3
end
end
| B =>
match constr:(X2) with
| A => idtac
| _ =>
match constr:(X3) with
| A => idtac
| _ => freepoints_independant A B C H X1 X2 X3
end
end
| C =>
match constr:(X2) with
| A => idtac
| _ =>
match constr:(X3) with
| A => idtac
| _ => freepoints_independant A B C H X1 X2 X3
end
end
| _ =>
match constr:(X2) with
| A =>
match constr:(X3) with
| C => idtac
| B => idtac
| _ => freepoints_independant A B C H X1 X2 X3
end
| B =>
match constr:(X3) with
| A => idtac
| _ => freepoints_independant A B C H X1 X2 X3
end
| C =>
match constr:(X3) with
| A => idtac
| _ => freepoints_independant A B C H X1 X2 X3
end
| _ => freepoints_independant A B C H X1 X2 X3
end
end
| _ => idtac
end
end.
Ltac only_use_area_coordinates :=
unfold Det3,Col in *;
match goal with
| H:(S ?X1 ?X2 ?X3 <> 0) |- ?X4 =>
(** If there are three non collinear points in the context we use them *)
iter_coord_expr X1 X2 X3 H X4; unfold Det3 in *; basic_simpl;
uniformize_signed_areas; basic_simpl
| A:Point, B:Point, H:?A<>?B |- ?X4 =>
(** Othewise we build a point which is not collinear to two existing points *)
let T := fresh in
(assert (T:=(build_point_not_collinear_1 A B H));
decompose [ex] T;clear T);only_use_area_coordinates
| A:Point |- ?X4 =>
let T := fresh in
(assert (T:=(build_point_not_collinear_2 A));
decompose [ex] T;clear T);only_use_area_coordinates
end.
(** A few tests *)

Lemma test_only_use_area_coordinates_3 : forall A B C D,   S D B C + S A D C  + S A B D = S A B C.