(** Some tactics *)

Ltac not_exist_hyp_perm_col A B C := not_exist_hyp (Col A B C); not_exist_hyp (Col A C B);
                                 not_exist_hyp (Col B A C); not_exist_hyp (Col B C A);
                                 not_exist_hyp (Col C A B); not_exist_hyp (Col C B A).Ltac assert_cols :=
repeat
 match goal with
      | H:Bet ?X1 ?X2 ?X3 |- _ =>
     not_exist_hyp_perm_col X1 X2 X3;assert (Col X1 X2 X3) by (apply bet_col;apply H)
 end.Ltac clean_trivial_hyps :=
  repeat
  match goal with
   | H:(Cong ?X1 ?X1 ?X2 ?X2) |- _ => clear H
   | H:(Cong ?X1 ?X2 ?X2 ?X1) |- _ => clear H
   | H:(Cong ?X1 ?X2 ?X1 ?X2) |- _ => clear H
   | H:(Bet ?X1 ?X1 ?X2) |- _ => clear H
   | H:(Bet ?X2 ?X1 ?X1) |- _ => clear H
   | H:(Col ?X1 ?X1 ?X2) |- _ => clear H
   | H:(Col ?X2 ?X1 ?X1) |- _ => clear H
   | H:(Col ?X1 ?X2 ?X1) |- _ => clear H
end.Ltac clean_reap_hyps :=
  clean_duplicated_hyps;
  repeat
  match goal with
   | H:(Col ?A ?B ?C), H2 : Col ?A ?C ?B |- _ => clear H2
   | H:(Col ?A ?B ?C), H2 : Col ?B ?A ?C |- _ => clear H2
   | H:(Col ?A ?B ?C), H2 : Col ?B ?C ?A |- _ => clear H2
   | H:(Col ?A ?B ?C), H2 : Col ?C ?B ?A |- _ => clear H2
   | H:(Col ?A ?B ?C), H2 : Col ?C ?A ?B |- _ => clear H2
   | H:(Bet ?A ?B ?C), H2 : Bet ?C ?B ?A |- _ => clear H2
   | H:(Cong ?A ?B ?C ?D), H2 : Cong ?A ?B ?D ?C |- _ => clear H2
   | H:(Cong ?A ?B ?C ?D), H2 : Cong ?C ?D ?A ?B |- _ => clear H2
   | H:(Cong ?A ?B ?C ?D), H2 : Cong ?C ?D ?B ?A |- _ => clear H2
   | H:(Cong ?A ?B ?C ?D), H2 : Cong ?D ?C ?B ?A |- _ => clear H2
   | H:(Cong ?A ?B ?C ?D), H2 : Cong ?D ?C ?A ?B |- _ => clear H2
   | H:(Cong ?A ?B ?C ?D), H2 : Cong ?B ?A ?C ?D |- _ => clear H2
   | H:(Cong ?A ?B ?C ?D), H2 : Cong ?B ?A ?D ?C |- _ => clear H2
   | H:(?A<>?B), H2 : (?B<>?A) |- _ => clear H2
end.Ltac clean := clean_trivial_hyps;clean_reap_hyps.Ltac smart_subst X := subst X;clean.Ltac treat_equalities_aux :=
  repeat
  match goal with
   | H:(?X1 = ?X2) |- _ => smart_subst X2
end.Ltac treat_equalities :=
try treat_equalities_aux;
repeat
  match goal with
   | H:(Cong ?X3 ?X3 ?X1 ?X2) |- _ =>
      apply cong_symmetry in H; apply cong_identity in H;smart_subst X2
   | H:(Cong ?X1 ?X2 ?X3 ?X3) |- _ =>
      apply cong_identity in H;smart_subst X2
   | H:(Bet ?X1 ?X2 ?X1) |- _ => apply  between_identity in H;smart_subst X2
end.Ltac show_distinct X Y := assert (X<>Y);[unfold not;intro;treat_equalities|idtac].Ltac eBetween := treat_equalities;eauto with between.Ltac Between := treat_equalities;auto with between between_no_eauto.

Lemma between_inner_transitivity : forall A B C D, Bet A B D -> Bet B C D -> Bet A B C.