Require Export GeoCoq.Tarski_dev.Ch08_orthogonality.
Require Export GeoCoq.Tarski_dev.Annexes.coplanar.
Ltac clean_reap_hyps :=
clean_duplicated_hyps;
repeat
match goal with
| H:(Midpoint ?A ?B ?C), H2 : Midpoint ?A ?C ?B |- _ => clear H2
| 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:(Perp ?A ?B ?C ?D), H2 : Perp ?A ?B ?D ?C |- _ => clear H2
| H:(Perp ?A ?B ?C ?D), H2 : Perp ?C ?D ?A ?B |- _ => clear H2
| H:(Perp ?A ?B ?C ?D), H2 : Perp ?C ?D ?B ?A |- _ => clear H2
| H:(Perp ?A ?B ?C ?D), H2 : Perp ?D ?C ?B ?A |- _ => clear H2
| H:(Perp ?A ?B ?C ?D), H2 : Perp ?D ?C ?A ?B |- _ => clear H2
| H:(Perp ?A ?B ?C ?D), H2 : Perp ?B ?A ?C ?D |- _ => clear H2
| H:(Perp ?A ?B ?C ?D), H2 : Perp ?B ?A ?D ?C |- _ => clear H2
| H:(?A<>?B), H2 : (?B<>?A) |- _ => clear H2
| H:(Per ?A ?D ?C), H2 : (Per ?C ?D ?A) |- _ => clear H2
| H:(Perp_at ?X ?A ?B ?C ?D), H2 : Perp_at ?X ?A ?B ?D ?C |- _ => clear H2
| H:(Perp_at ?X ?A ?B ?C ?D), H2 : Perp_at ?X ?C ?D ?A ?B |- _ => clear H2
| H:(Perp_at ?X ?A ?B ?C ?D), H2 : Perp_at ?X ?C ?D ?B ?A |- _ => clear H2
| H:(Perp_at ?X ?A ?B ?C ?D), H2 : Perp_at ?X ?D ?C ?B ?A |- _ => clear H2
| H:(Perp_at ?X ?A ?B ?C ?D), H2 : Perp_at ?X ?D ?C ?A ?B |- _ => clear H2
| H:(Perp_at ?X ?A ?B ?C ?D), H2 : Perp_at ?X ?B ?A ?C ?D |- _ => clear H2
| H:(Perp_at ?X ?A ?B ?C ?D), H2 : Perp_at ?X ?B ?A ?D ?C |- _ => clear H2
end.
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:Le ?A ?B ?C ?D, H2 : ?B <> ?A |-_ =>
let T:= fresh in (not_exist_hyp_comm C D);
assert (T:= le_diff A B C D (swap_diff B A 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:Midpoint ?I ?A ?B, H2 : ?A<>?B |- _ =>
let T:= fresh in (not_exist_hyp2 I B I A);
assert (T:= midpoint_distinct_1 I A B H2 H);
decompose [and] T;clear T;clean_reap_hyps
| H:Midpoint ?I ?A ?B, H2 : ?B<>?A |- _ =>
let T:= fresh in (not_exist_hyp2 I B I A);
assert (T:= midpoint_distinct_1 I A B (swap_diff B A H2) H);
decompose [and] T;clear T;clean_reap_hyps
| H:Midpoint ?I ?A ?B, H2 : ?I<>?A |- _ =>
let T:= fresh in (not_exist_hyp2 I B A B);
assert (T:= midpoint_distinct_2 I A B H2 H);
decompose [and] T;clear T;clean_reap_hyps
| H:Midpoint ?I ?A ?B, H2 : ?A<>?I |- _ =>
let T:= fresh in (not_exist_hyp2 I B A B);
assert (T:= midpoint_distinct_2 I A B (swap_diff A I H2) H);
decompose [and] T;clear T;clean_reap_hyps
| H:Midpoint ?I ?A ?B, H2 : ?I<>?B |- _ =>
let T:= fresh in (not_exist_hyp2 I A A B);
assert (T:= midpoint_distinct_3 I A B H2 H);
decompose [and] T;clear T;clean_reap_hyps
| H:Midpoint ?I ?A ?B, H2 : ?B<>?I |- _ =>
let T:= fresh in (not_exist_hyp2 I A A B);
assert (T:= midpoint_distinct_3 I A B (swap_diff B I H2) H);
decompose [and] T;clear T;clean_reap_hyps
| H:Per ?A ?B ?C, H2 : ?A<>?B |- _ =>
let T:= fresh in (not_exist_hyp_comm A C);
assert (T:= per_distinct A B C H H2); clean_reap_hyps
| H:Per ?A ?B ?C, H2 : ?B<>?A |- _ =>
let T:= fresh in (not_exist_hyp_comm A C);
assert (T:= per_distinct A B C H (swap_diff B A H2)); clean_reap_hyps
| H:Per ?A ?B ?C, H2 : ?B<>?C |- _ =>
let T:= fresh in (not_exist_hyp_comm A C);
assert (T:= per_distinct_1 A B C H H2); clean_reap_hyps
| H:Per ?A ?B ?C, H2 : ?C<>?B |- _ =>
let T:= fresh in (not_exist_hyp_comm A C);
assert (T:= per_distinct_1 A B C H (swap_diff C B H2)); clean_reap_hyps
| H:Perp ?A ?B ?C ?D |- _ =>
let T:= fresh in (not_exist_hyp2 A B C D);
assert (T:= perp_distinct A B C D H);
decompose [and] T;clear T;clean_reap_hyps
| H:Perp_at ?X ?A ?B ?C ?D |- _ =>
let T:= fresh in (not_exist_hyp2 A B C D);
assert (T:= perp_in_distinct X A B C D H);
decompose [and] T;clear T;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 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
| H:(Per ?X1 ?X2 ?X2)     |- _ => clear H
| H:(Per ?X1 ?X1 ?X2)     |- _ => clear H
| H:(Midpoint ?X1 ?X1 ?X1) |- _ => clear H
end.
Ltac clean := clean_trivial_hyps;clean_reap_hyps.
Section T9.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

Lemma l9_2 : forall A B P Q, TS A B P Q -> TS A B Q P.