Section Vectors.
Context `{TE:Tarski_euclidean}.
intros.
assert(HH:= vector_construction C D B).
ex_and HH F.
exists A.
exists F.
unfold SumV.
intros.
assert(D' = F).
apply (vector_construction_uniqueness C D B); auto.
subst D'.
apply eqv_refl.
Qed.
intros.
unfold Same_dir.
induction (eq_dec_points A B).
left.
tauto.
right.
exists B.
split.
apply out_trivial.
auto.
apply eqv_refl.
Qed.
intros.
unfold Same_dir in H.
induction H.
right.
auto.
ex_and H D'.
unfold EqV in H0.
induction H0.
induction H0.
apply plgs_par_strict in H0.
spliter.
apply False_ind.
apply H1.
exists B.
split; Col.
apply plgf_permut in H0.
apply plgf_sym in H0.
apply plgf_trivial_neq in H0.
spliter.
subst D'.
left.
apply l6_6.
assumption.
spliter.
subst D'.
subst B.
unfold Out in H.
tauto.
Qed.
intros.
unfold Same_dir in H.
induction H.
right.
spliter.
subst B.
tauto.
ex_and H D'.
unfold EqV in H0.
induction H0.
induction H0.
apply plgs_par_strict in H0.
spliter.
apply False_ind.
apply H1.
exists B.
split; Col.
unfold Parallelogram_flat in H0.
spliter.
assert(A = D' \/ Midpoint B A D').
apply l7_20.
Col.
Cong.
induction H5.
subst D'.
tauto.
left.
unfold Midpoint in H5.
spliter.
unfold Out.
repeat split.
intro.
subst B.
apply cong_symmetry in H6.
apply cong_identity in H6.
induction H4; tauto.
intro.
subst C.
unfold Out in H.
spliter.
induction H8.
apply H.
apply (between_equality _ _ D');
Between.
apply H7.
apply (between_equality _ _ A);
Between.
induction H.
spliter.
induction H8.
left.
apply (between_inner_transitivity _ _ _ D').
apply H5.
apply H8.
left.
apply (outer_transitivity_between _ _  D').
apply H5.
auto.
auto.
spliter.
subst B.
subst D'.
unfold Out in H.
tauto.
Qed.
intros.
unfold Same_dir in H.
induction H.
tauto.
ex_and H D.
apply null_vector in H0.
subst D.
unfold Out in H.
tauto.
Qed.
intros.
induction (eq_dec_points A B).
subst B.
apply same_dir_null in H.
subst D.
left.
tauto.
unfold Same_dir in *.
induction H.
left.
tauto.
ex_and H D'.
right.
assert(HH:=vector_construction C D A).
ex_and HH B'.
exists B'.
split.
unfold EqV in H1.
unfold EqV in H2.
unfold Out in *.
spliter.
induction H1; induction H2.
repeat split.
auto.
intro.
subst B'.
induction H2.
apply H2.
Col.
apply plgf_sym in H2.
apply plgf_trivial_neq in H2.
spliter.
auto.
induction H4.
right.
apply (plg_plg_bet C D _ _ D').
apply H2.
apply H4.
apply plg_sym.
apply plg_comm2.
assumption.
left.
apply (plg_plg_bet C D' _ _ D).
apply plg_sym.
apply plg_comm2.
apply H1.
apply H4.
assumption.
spliter.
subst D.
tauto.
spliter.
subst B.
tauto.
spliter.
subst D.
tauto.
assumption.
Qed.
intros.
unfold Same_dir in *.
induction H; induction H0; spliter.
left.
tauto.
ex_and H0 F'.
subst B.
subst D.
apply null_vector in H2.
subst F'.
unfold Out in H0.
tauto.
ex_and H D'.
subst D.
subst F.
unfold Out in H.
tauto.
ex_and H D'.
ex_and H0 F'.
right.
induction(eq_dec_points A B).
subst B.
apply null_vector in H1.
subst D'.
unfold Out in H.
tauto.
assert(HH:=vector_construction A B E).
ex_and HH F''.
exists F''.
split; trivial.
assert(C <> D /\ C <> D' /\ E <> F /\ E <> F').
unfold Out in *.
spliter.
repeat split;
auto.
spliter.
unfold EqV in *.
induction H1; induction H2; induction H4.
unfold Out in *.
spliter.
induction H10; induction H12.
repeat split.
auto.
intro.
subst F''.
induction H4.
apply H4.
Col.
apply plgf_sym in H4.
apply plgf_trivial_neq in H4.
tauto.
left.
assert(Bet E F' F'').
apply (plg_plg_bet C D _ _ D').
apply H2.
apply H12.
assert(Parallelogram C D' F'' E \/ C = D' /\ B = A /\ E = F'' /\ C = E).
apply(plg_pseudo_trans C D' B A E F'').
apply plg_sym.
apply plg_comm2.
auto.
apply plg_comm2.
auto.
induction H13.
assumption.
tauto.
apply (between_exchange4 _ _ F').
apply H10.
auto.
repeat split.
auto.
intro.
subst F''.
induction H4.
apply H4.
Col.
apply plgf_sym in H4.
apply plgf_trivial_neq in H4.
tauto.
assert(Bet E F'' F').
apply (plg_plg_bet C D' _ _ D); trivial.
assert(Parallelogram C D' F'' E \/ C = D' /\ B = A /\ E = F'' /\ C = E).
apply plg_pseudo_trans.
apply plg_sym.
apply plg_comm2.
auto.
apply plg_comm2.
auto.
induction H13.
assumption.
tauto.
apply (l5_3 _ _ _ F').
apply H10.
assumption.
repeat split.
auto.
intro.
subst F''.
induction H4.
apply H4.
Col.
apply plgf_sym in H4.
apply plgf_trivial_neq in H4.
tauto.
assert(Bet E F' F'').
apply (plg_plg_bet C D _ _ D').
apply H2.
apply H12.
assert(Parallelogram C D' F'' E \/ C = D' /\ B = A /\ E = F'' /\ C = E).
apply plg_pseudo_trans.
apply plg_sym.
apply plg_comm2.
auto.
apply plg_comm2.
auto.
induction H13.
assumption.
tauto.
apply (l5_1 _ F').
apply H8.
auto.
auto.
repeat split.
auto.
intro.
subst F''.
induction H4.
apply H4.
Col.
apply plgf_sym in H4.
apply plgf_trivial_neq in H4.
tauto.
assert(Bet E F'' F').
apply (plg_plg_bet C D' _ _ D); trivial.
assert(Parallelogram C D' F'' E \/ C = D' /\ B = A /\ E = F'' /\ C = E).
apply plg_pseudo_trans.
apply plg_sym.
apply plg_comm2.
auto.
apply plg_comm2.
auto.
induction H13.
auto.
tauto.
right.
apply (between_exchange4 _ _ F').
apply H13.
auto.
tauto.
tauto.
tauto.
tauto.
tauto.
tauto.
tauto.
Qed.
intros.
unfold Same_dir in *.
induction H.
left.
auto.
spliter.
split; auto.
right.
ex_and H D'.
assert(A <> B).
intro.
subst B.
apply null_vector in H0.
unfold Out in H.
spliter.
auto.
assert(HH:=vector_construction B A D).
ex_and HH C'.
exists C'.
split; trivial.
unfold Out in *.
spliter.
unfold EqV in *.
induction H4.
repeat split.
auto.
intro.
subst C'.
apply eqv_sym in H2.
apply null_vector in H2.
subst B.
tauto.
left.
induction H0; induction H2;try tauto.
assert(Parallelogram C D' D C' \/ C = D' /\ B = A /\ C' = D /\ C = C').
apply(plg_pseudo_trans C D' B A C' D).
apply plg_sym.
apply plg_comm2.
assumption.
assumption.
induction H5.
assert(Parallelogram_flat C D' D C').
induction H5.
apply False_ind.
apply plgs_par_strict in H5.
spliter.
apply H5.
exists D.
apply bet_col in H4.
split; Col.
assumption.
unfold Parallelogram_flat in H6.
spliter.
apply (col_cong2_bet1 D').
Col.
Between.
Cong.
Cong.
spliter.
subst B.
tauto.
spliter.
subst B.
tauto.
repeat split.
auto.
intro.
subst C'.
induction H2.
induction H2.
apply H2.
Col.
apply plgf_sym in H2.
apply plgf_trivial_neq in H2.
spliter.
auto.
spliter.
auto.
induction H0; induction H2;try tauto.
induction(eq_dec_points C C').
subst C'.
left.
Between.
assert(Parallelogram C D' D C' \/ C = D' /\ B = A /\ C' = D /\ C = C').
apply(plg_pseudo_trans C D' B A C' D).
apply plg_sym.
apply plg_comm2.
assumption.
assumption.
induction H6.
assert(Parallelogram_flat C D' D C').
induction H6.
apply False_ind.
apply plgs_par_strict in H6.
spliter.
apply H6.
exists D.
apply bet_col in H4.
split; Col.
assumption.
right.
assert(HH:= H7).
unfold Parallelogram_flat in H7.
spliter.
apply plgf_bet in HH.
induction HH.
spliter.
apply False_ind.
apply H3.
apply (between_equality _ _ D).
apply between_symmetry.
apply H13.
assumption.
induction H12.
spliter.
assert(D = D').
apply (between_equality _ _ C).
Between.
Between.
subst D'.
apply cong_identity in H10.
contradiction.
induction H12.
spliter.
eBetween.
spliter.
eBetween.
spliter.
subst B.
tauto.
spliter.
subst B.
tauto.
Qed.
intros.
unfold Same_dir.
right.
exists B.
split.
unfold Out.
repeat split.
intro.
subst C.
apply between_identity in H1.
tauto.
auto.
right.
assumption.
apply eqv_refl.
Qed.
intros.
unfold Same_dir.
right.
assert(HH:=vector_construction A B B).
ex_and HH C'.
exists C'.
split; trivial.
unfold EqV in H2.
induction H2; [|tauto].
induction H2.
apply plgs_par_strict in H2.
spliter.
apply False_ind.
apply H3.
exists B.
split; Col.
assert(HH:= H2).
unfold Parallelogram_flat in HH.
apply plgf_bet in H2.
spliter.
unfold Out.
repeat split.
auto.
intro.
subst C'.
apply cong_identity in H5.
contradiction.
induction H2.
assert(Bet B A B).
spliter.
apply (outer_transitivity_between2 _ C').
Between.
auto.
induction H7.
auto.
tauto.
apply between_identity in H8.
subst B.
tauto.
induction H2.
assert(Bet B A B).
spliter.
apply (outer_transitivity_between _ _ C').
apply H2.
auto.
induction H7.
auto.
tauto.
apply between_identity in H8.
subst B.
tauto.
induction H2.
assert( A = C' \/ Midpoint B A C').
apply l7_20.
Col.
Cong.
induction H8.
induction H7.
tauto.
tauto.
unfold Midpoint in H8.
spliter.
apply (l5_2 A); auto.
assert( A = C' \/ Midpoint B A C').
apply l7_20.
Col.
Cong.
induction H8.
induction H7.
tauto.
tauto.
unfold Midpoint in H8.
spliter.
apply (l5_2 A); auto.
Qed.
intros.
induction(eq_dec_points A B).
subst B.
induction H.
apply False_ind.
apply plgs_sym in H.
apply H.
Col.
apply plgf_trivial_neq in H.
spliter.
subst D.
left.
tauto.
unfold Same_dir.
right.
exists C.
split.
apply out_trivial.
intro.
subst D.
induction H.
apply H.
Col.
apply plgf_sym in H.
apply plgf_trivial_neq in H.
tauto.
unfold EqV.
left.
assumption.
Qed.
intros.
induction (same_dir_dec A B C D).
left.
assumption.
right.
unfold Opp_dir.
unfold Same_dir.
right.
assert(HH:= vector_construction A B D).
ex_and HH C'.
exists C'.
split; trivial.
unfold EqV in H1.
induction (eq_dec_points B C').
subst C'.
induction H1.
induction H1.
apply False_ind.
apply plgs_permut in H1.
apply plgs_sym in H1.
apply H1.
Col.
apply plgf_permut in H1.
apply plgf_trivial_neq in H1.
spliter.
subst D.
induction H.
apply False_ind.
apply H.
exists A.
split; Col.
spliter.
induction H4.
unfold Out.
repeat split; auto.
left.
Between.
induction H4.
unfold Out.
repeat split; auto.
apply False_ind.
apply H0.
apply same_dir_sym.
apply bet_same_dir2; auto.
unfold Out.
repeat split.
auto.
auto.
right.
assumption.
spliter.
subst A.
subst D.
apply par_distinct in H.
tauto.
induction H1.
assert(Col C' C D /\ Col D C D).
apply plg_par in H1.
spliter.
apply(parallel_uniqueness A B C D C' D D); Col.
apply par_distinct in H.
tauto.
assumption.
spliter.
clear H4.
unfold Out.
repeat split.
apply par_distinct in H.
tauto.
intro.
subst C'.
induction H1.
apply H1.
Col.
apply plgf_sym in H1.
apply plgf_trivial_neq in H1.
spliter.
apply par_distinct in H.
tauto.
induction H3.
left.
Between.
induction H3.
apply False_ind.
assert(Same_dir A B D C').
apply plg_opp_dir.
assumption.
assert(Same_dir C D D C').
apply bet_same_dir2.
apply par_distinct in H.
spliter.
auto.
intro.
subst C'.
induction H1.
apply H1.
Col.
apply plgf_sym in H1.
apply plgf_trivial_neq in H1.
spliter.
apply par_distinct in H.
tauto.
assumption.
apply False_ind.
apply H0.
apply (same_dir_trans _ _ D C').
apply H4.
apply same_dir_sym.
auto.
right.
assumption.
apply par_distinct in H.
tauto.
Qed.
intros.
unfold Same_dir in H.
induction H.
tauto.
ex_and H C.
apply eqv_mid in H0.
unfold Midpoint in H0.
unfold Out in H.
spliter.
induction H3.
apply (between_equality _ _ C).
apply H0.
assumption.
apply False_ind.
apply H2.
apply (between_equality _ _ A).
Between.
assumption.
Qed.
intros.
unfold Opp_dir in H.
apply same_dir_id in H.
auto.
Qed.
intros.
assert(Same_dir C D D C).
apply (same_dir_trans _ _ A B).
apply same_dir_sym.
apply H.
assumption.
apply same_dir_id in H1.
subst D.
apply same_dir_sym in H.
apply same_dir_null in H.
subst B.
tauto.
Qed.
unfold Opp_dir.
intros.
apply same_dir_to_null; auto.
Qed.
intros.
intro.
apply same_dir_to_null in H0.
tauto.
assumption.
Qed.

Lemma opp_not_same_dir : forall A B C D, A <> B -> Opp_dir A B C D -> ~ Same_dir A B C D.