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Droite_espace.v
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(* This program is free software; you can redistribute it and/or *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1 *)
(* of the License, or (at your option) any later version. *)
(* *)
(* This program is distributed in the hope that it will be useful, *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
(* GNU General Public License for more details. *)
(* *)
(* You should have received a copy of the GNU Lesser General Public *)
(* License along with this program; if not, write to the Free *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA *)
Require Export parallelisme_concours.
Require Export coplanarite.
Set Implicit Arguments.
Unset Strict Implicit.
Lemma paralleles_coplanaires :
forall A B C D : PO,
A <> B ->
C <> D -> paralleles (droite A B) (droite C D) -> coplanaires A B C D.
unfold coplanaires in |- *; intros A B C D H H10 H0.
elim def_paralleles2 with (3 := H0); auto.
intros k H1.
elim (classic (k = 0)); intros.
rewrite H2 in H1.
absurd (A = B); auto.
apply conversion_PP with (a := 1) (b := 1); auto with *.
RingPP1 H1.
RingPP.
right; try assumption.
exists (- / k).
exists (/ k).
apply mult_PP_regulier with k; auto.
replace
(mult_PP k
(add_PP (cons (- / k) A)
(add_PP (cons (/ k) B) (cons (1 + - (- / k + / k)) C)))) with
(add_PP (cons (-1) A) (add_PP (cons 1 B) (cons k C))).
RingPP1 H1.
RingPP.
FieldPP k.
Qed.
Lemma concours_coplanaires :
forall A B C D : PO,
A <> B :>PO ->
C <> D :>PO -> concours (droite A B) (droite C D) -> coplanaires A B D C.
unfold coplanaires in |- *; intros A B C D H20 H21 H11.
elim def_concours2 with (A := A) (B := B) (C := C) (D := D);
[ intros I H0; elim H0; intros H1 H2; try clear H0; try exact H2
| auto
| auto
| auto ].
hPPalignes H2 ipattern:(k).
hPPalignes H1 ipattern:(k0).
cut
(add_PP (cons k C) (cons (1 + - k) D) =
add_PP (cons k0 A) (cons (1 + - k0) B)); intros.
elim (classic (k = 0)); intros.
left; try assumption.
unfold alignes, alignes1 in |- *.
right; try assumption.
rewrite H0 in H.
exists k0.
VReplace (cons 1 D) (cons (1 + -0) D).
RingPP2 H; RingPP.
right; try assumption.
exists (/ k * k0).
exists (/ k * (1 + - k0)).
apply mult_PP_regulier with k; auto.
VReplace (mult_PP k (cons 1 C)) (cons k C).
RingPP1 H.
FieldPP k.
rewrite <- H2; trivial.
Qed.
Lemma droites_non_paralleles :
forall A B C D : PO,
A <> B :>PO ->
C <> D :>PO ->
~ paralleles (droite C D) (droite A B) ->
concours (droite C D) (droite A B) \/ ~ coplanaires A B C D.
intros A B C D H20 H21 H; try assumption.
elim (classic (coplanaires A B C D)); intros.
left; try assumption.
hPPcoplanaires H0 ipattern:(a) ipattern:(b).
apply def_concours with C; auto with geo.
cut (a + b <> 0); intros.
apply def_concours with (barycentre (cons a A) (cons b B)); auto.
replace (barycentre (cons a A) (cons b B)) with
(barycentre (cons (- (1 + - (a + b))) C) (cons 1 D)).
apply barycentre_alignes.
replace (- (1 + - (a + b)) + 1) with (a + b); auto.
ring.
apply conversion_PP with (a := - (1 + - (a + b)) + 1) (b := a + b).
rewrite <- add_PP_barycentre; auto.
rewrite <- add_PP_barycentre; auto.
rewrite H0.
RingPP.
replace (- (1 + - (a + b)) + 1) with (a + b); auto.
ring.
replace (- (1 + - (a + b)) + 1) with (a + b); auto.
ring.
ring.
apply barycentre_alignes; auto.
unfold not in |- *; intros; apply H.
apply def_paralleles with (- a); auto.
rewrite H0.
rewrite H1.
replace b with (- a + (a + b)); auto.
rewrite H1.
RingPP.
ring.
right; try assumption.
Qed.
Lemma concours_sym :
forall A B C D : PO,
A <> B :>PO ->
C <> D :>PO ->
concours (droite A B) (droite C D) -> concours (droite C D) (droite A B).
intros A B C D H10 H11 H; try assumption.
elim def_concours2 with (3 := H); auto; intros.
apply def_concours with x; auto.
elim H0; intros H1 H2; try clear H0; try exact H2.
elim H0; intros H1 H2; try clear H0; try exact H1.
Qed.
Hint Resolve concours_sym paralleles_sym paralleles_trans: geo.
Theorem position_relative_droites_espace :
forall A B C D : PO,
A <> B :>PO ->
C <> D :>PO ->
~ coplanaires A B C D \/
concours (droite A B) (droite C D) \/ paralleles (droite A B) (droite C D).
intros A B C D H10 H; try assumption.
elim (classic (paralleles (droite A B) (droite C D))); intros.
right; right; try assumption.
elim droites_non_paralleles with (A := A) (B := B) (C := C) (D := D);
auto with geo; intros.
Qed.
Theorem position_relative_droites_coplanaires :
forall A B C D : PO,
A <> B :>PO ->
C <> D :>PO ->
coplanaires A B C D ->
concours (droite A B) (droite C D) \/ paralleles (droite A B) (droite C D).
intros A B C D H10 H H0; try assumption.
elim
position_relative_droites_espace with (A := A) (B := B) (C := C) (D := D);
[ intros H1; try clear position_relative_droites_espace
| intros H1; try clear position_relative_droites_espace; auto
| auto
| auto ].
absurd (coplanaires A B C D); auto.
Qed.