Ltac not_exist_hyp_comm A B := not_exist_hyp (A<>B);not_exist_hyp (B<>A).Ltac not_exist_hyp2 A B C D := first [not_exist_hyp_comm A B | not_exist_hyp_comm C D].Ltac not_exist_hyp3 A B C D E F := first [not_exist_hyp_comm A B | not_exist_hyp_comm C D | not_exist_hyp_comm E F].Ltac assert_diffs :=
repeat
 match goal with
      | H:(~Col ?X1 ?X2 ?X3) |- _ =>
      let h := fresh in
      not_exist_hyp3 X1 X2 X1 X3 X2 X3;
      assert (h := not_col_distincts X1 X2 X3 H);decompose [and] h;clear h;clean_reap_hyps

      | H:(~Bet ?X1 ?X2 ?X3) |- _ =>
      let h := fresh in
      not_exist_hyp2 X1 X2 X2 X3;
      assert (h := not_bet_distincts X1 X2 X3 H);decompose [and] h;clear h;clean_reap_hyps
      | H:Bet ?A ?B ?C, H2 : ?A <> ?B |-_ =>
      let T:= fresh in (not_exist_hyp_comm A C);
        assert (T:= bet_neq12__neq A B C H H2);clean_reap_hyps
      | H:Bet ?A ?B ?C, H2 : ?B <> ?A |-_ =>
      let T:= fresh in (not_exist_hyp_comm A C);
        assert (T:= bet_neq21__neq A B C H H2);clean_reap_hyps
      | H:Bet ?A ?B ?C, H2 : ?B <> ?C |-_ =>
      let T:= fresh in (not_exist_hyp_comm A C);
        assert (T:= bet_neq23__neq A B C H H2);clean_reap_hyps
      | H:Bet ?A ?B ?C, H2 : ?C <> ?B |-_ =>
      let T:= fresh in (not_exist_hyp_comm A C);
        assert (T:= bet_neq32__neq A B C H H2);clean_reap_hyps

      | H:Cong ?A ?B ?C ?D, H2 : ?A <> ?B |-_ =>
      let T:= fresh in (not_exist_hyp_comm C D);
        assert (T:= cong_diff A B C D H2 H);clean_reap_hyps
      | H:Cong ?A ?B ?C ?D, H2 : ?B <> ?A |-_ =>
      let T:= fresh in (not_exist_hyp_comm C D);
        assert (T:= cong_diff_2 A B C D H2 H);clean_reap_hyps
      | H:Cong ?A ?B ?C ?D, H2 : ?C <> ?D |-_ =>
      let T:= fresh in (not_exist_hyp_comm A B);
        assert (T:= cong_diff_3 A B C D H2 H);clean_reap_hyps
      | H:Cong ?A ?B ?C ?D, H2 : ?D <> ?C |-_ =>
      let T:= fresh in (not_exist_hyp_comm A B);
        assert (T:= cong_diff_4 A B C D H2 H);clean_reap_hyps

      | H:Le ?A ?B ?C ?D, H2 : ?A <> ?B |-_ =>
      let T:= fresh in (not_exist_hyp_comm C D);
        assert (T:= le_diff A B C D H2 H);clean_reap_hyps
      | H:Lt ?A ?B ?C ?D |-_ =>
      let T:= fresh in (not_exist_hyp_comm C D);
        assert (T:= lt_diff A B C D H);clean_reap_hyps

      | H:Out ?A ?B ?C |- _ =>
      let T:= fresh in (not_exist_hyp2 A B A C);
       assert (T:= out_distinct A B C H);
       decompose [and] T;clear T;clean_reap_hyps
 end.Ltac ColR :=
 let tpoint := constr:(Tpoint) in
 let col := constr:(Col) in
   treat_equalities; assert_cols; assert_diffs; try (solve [Col]); Col_refl tpoint col.

Lemma midpoint_dec : forall I A B, Midpoint I A B \/ ~ Midpoint I A B.