Ltac permut :=
match goal with
   |H : (Col ?X ?Y ?Z) |- Col ?X ?Y ?Z => assumption
   |H : (Col ?X ?Y ?Z) |- Col ?Y ?Z ?X => apply col_permutation_1; assumption
   |H : (Col ?X ?Y ?Z) |- Col ?Z ?X ?Y => apply col_permutation_2; assumption
   |H : (Col ?X ?Y ?Z) |- Col ?X ?Z ?Y => apply col_permutation_5; assumption
   |H : (Col ?X ?Y ?Z) |- Col ?Y ?X ?Z => apply col_permutation_4; assumption
   |H : (Col ?X ?Y ?Z) |- Col ?Z ?Y ?X => apply col_permutation_3; assumption
   |_ : _ |- _ => idtac
end.Ltac bet :=
match goal with
   |H0 : Bet ?A ?B ?C |- Bet ?A ?B ?C => assumption
   |H0 : Bet ?A ?B ?C, H1 : Bet ?B ?C ?D , H : ?B <> ?C |- Bet ?A ?B ?D => apply (outer_transitivity_between _ _ _ _ H0 H1 H)
   |H0 : Bet ?A ?B ?C, H1 : Bet ?B ?C ?D , H : ?B <> ?C |- Bet ?A ?C ?D => apply (outer_transitivity_between2 _ _ _ _ H0 H1 H)
   |H0 : Bet ?A ?B ?D, H1 : Bet ?B ?C ?D  |- Bet ?A ?B ?C => apply (between_inner_transitivity _ _ _ _ H0 H1)
   |H0 : Bet ?A ?B ?C, H1 : Bet ?A ?C ?D  |- Bet ?B ?C ?D => apply (between_exchange3 _ _ _ _ H0 H1)
   |H0 : Bet ?A ?B ?D, H1 : Bet ?B ?C ?D  |- Bet ?A ?C ?D => apply (between_exchange2 _ _ _ _ H0 H1)
   |H0 : Bet ?A ?B ?C, H1 : Bet ?A ?C ?D  |- Bet ?A ?B ?D => apply (between_exchange4 _ _ _ _ H0 H1)

   |H0 : Bet ?A ?B ?C |- Bet ?A ?B ?C => assumption
   |H0 : Bet ?A ?B ?C, H1 : Bet ?B ?C ?D , H : ?B <> ?C |- Bet ?D ?B ?A => apply between_symmetry; apply (outer_transitivity_between _ _ _ _ H0 H1 H)
   |H0 : Bet ?A ?B ?C, H1 : Bet ?B ?C ?D , H : ?B <> ?C |- Bet ?D ?C ?A => apply (outer_transitivity_between2 _ _ _ _ H0 H1 H)
   |H0 : Bet ?A ?B ?D, H1 : Bet ?B ?C ?D  |- Bet ?C ?B ?A => apply between_symmetry; apply (between_inner_transitivity _ _ _ _ H0 H1)
   |H0 : Bet ?A ?B ?C, H1 : Bet ?A ?C ?D  |- Bet ?D ?C ?B => apply between_symmetry; apply (between_exchange3 _ _ _ _ H0 H1)
   |H0 : Bet ?A ?B ?D, H1 : Bet ?B ?C ?D  |- Bet ?D ?C ?A => apply between_symmetry; apply (between_exchange2 _ _ _ _ H0 H1)
   |H0 : Bet ?A ?B ?C, H1 : Bet ?A ?C ?D  |- Bet ?D ?B ?A => apply between_symmetry; apply (between_exchange4 _ _ _ _ H0 H1)
   |H0 : Bet ?A ?B ?C |- Bet ?C ?B ?A => apply (between_symmetry _ _ _  H0)

   |H0 : Bet ?A ?B ?C |- Bet ?C ?B ?A => apply (between_symmetry _ _ _  H0)
   |_ : _ |- Bet ?A ?B ?B => apply between_trivial
   |_ : _ |- Bet ?A ?A ?B => apply between_trivial2
   |_ : _ |- _ => idtac
end.Ltac cong :=
match goal with
   |_ :  Cong ?A ?B ?C ?BD |- Cong ?A ?B ?C ?D => assumption
   |_ : _ |- Cong ?A ?B ?A ?B => apply cong_reflexivity
   |_ : _ |- Cong ?A ?A ?B ?B => apply cong_trivial_identity

   |H0 : Cong ?A ?B ?C ?D |- Cong ?A ?B ?C ?C => assumption
   |H0 : Cong ?A ?B ?C ?D |- Cong ?A ?B ?D ?C => apply (cong_right_commutativity _ _ _ _ H0)
   |H0 : Cong ?A ?B ?C ?D |- Cong ?B ?A ?D ?C => apply (cong_commutativity _ _ _ _ H0)
   |H0 : Cong ?A ?B ?C ?D |- Cong ?B ?A ?C ?D => apply (cong_left_commutativity _ _ _ _ H0)

   |H0 : Cong ?A ?B ?C ?D |- Cong ?C ?D ?A ?B => apply (cong_symmetry _ _ _ _ H0)
   |H0 : Cong ?A ?B ?C ?D |- Cong ?C ?D ?B ?A => apply (cong_symmetry _ _ _ _ (cong_left_commutativity _ _ _ _ H0))
   |H0 : Cong ?A ?B ?C ?D |- Cong ?D ?C ?B ?B => apply (cong_symmetry _ _ _ _ (cong_commutativity _ _ _ _ H0))
   |H0 : Cong ?A ?B ?C ?D |- Cong ?D ?C ?A ?B => apply (cong_symmetry _ _ _ _ (cong_right_commutativity _ _ _ _ H0))
   |_ : _ |- _ => idtac
end.

Lemma triangle_inequality : forall A B C D, Bet A B D -> Cong B C B D -> Le A C A D.