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
| H:TS ?A ?B ?C ?D |- _ =>
let h := fresh in
not_exist_hyp6 A B A C A D B C B D C D;
assert (h := ts_distincts A B C D H);decompose [and] h;clear h;clean_reap_hyps
| H:OS ?A ?B ?C ?D |- _ =>
let h := fresh in
not_exist_hyp5 A B A C A D B C B D;
assert (h := os_distincts A B C D H);decompose [and] h;clear h;clean_reap_hyps
| H:~ Coplanar ?A ?B ?C ?D |- _ =>
let h := fresh in
not_exist_hyp6 A B A C A D B C B D C D;
assert (h := ncop_distincts A B C D H);decompose [and] h;clear h;clean_reap_hyps
| H:CongA ?A ?B ?C ?A' ?B' ?C' |- _ =>
let T:= fresh in (not_exist_hyp_comm A B);
assert (T:= conga_diff1 A B C A' B' C' H);clean_reap_hyps
| H:CongA ?A ?B ?C ?A' ?B' ?C' |- _ =>
let T:= fresh in (not_exist_hyp_comm B C);
assert (T:= conga_diff2 A B C A' B' C' H);clean_reap_hyps
| H:CongA ?A ?B ?C ?A' ?B' ?C' |- _ =>
let T:= fresh in (not_exist_hyp_comm A' B');
assert (T:= conga_diff45 A B C A' B' C' H);clean_reap_hyps
| H:CongA ?A ?B ?C ?A' ?B' ?C' |- _ =>
let T:= fresh in (not_exist_hyp_comm B' C');
assert (T:= conga_diff56 A B C A' B' C' H);clean_reap_hyps
| H:(Orth_at ?X ?A ?B ?C ?U ?V) |- _ =>
let h := fresh in
not_exist_hyp4 A B A C B C U V;
assert (h := orth_at_distincts A B C U V X H);decompose [and] h;clear h;clean_reap_hyps
| H:(Orth ?A ?B ?C ?U ?V) |- _ =>
let h := fresh in
not_exist_hyp4 A B A C B C U V;
assert (h := orth_distincts A B C U V H);decompose [and] h;clear h;clean_reap_hyps
| H:SumA ?A ?B ?C ?D ?E ?F ?G ?I ?J |- _ =>
let h := fresh in
not_exist_hyp6 A B B C D E E F G I I J;
assert (h := suma_distincts A B C D E F G I J H);decompose [and] h;clear h;clean_reap_hyps
| H: TriSumA ?A ?B ?C ?D ?E ?F |- _ =>
let h := fresh in
not_exist_hyp5 A B B C A C D E E F;
assert (h := trisuma_distincts A B C D E F H);decompose [and] h;clear h; clean_reap_hyps
| H:SAMS ?A ?B ?C ?D ?E ?F |- _ =>
let h := fresh in
not_exist_hyp4 A B B C D E E F;
assert (h := sams_distincts A B C D E F H);decompose [and] h;clear h;clean_reap_hyps
| H:(InAngle ?P ?A ?B ?C) |- _ =>
let h := fresh in
not_exist_hyp3 A B C B P B;
assert (h := inangle_distincts A B C P H);decompose [and] h;clear h;clean_reap_hyps
| H:LeA ?A ?B ?C ?D ?E ?F |- _ =>
let h := fresh in
not_exist_hyp4 A B B C D E E F;
assert (h := lea_distincts A B C D E F H);decompose [and] h;clear h;clean_reap_hyps
| H:LtA ?A ?B ?C ?D ?E ?F |- _ =>
let h := fresh in
not_exist_hyp4 A B B C D E E F;
assert (h := lta_distincts A B C D E F H);decompose [and] h;clear h;clean_reap_hyps
| H:(Acute ?A ?B ?C) |- _ =>
let h := fresh in
not_exist_hyp2 A B B C;
assert (h := acute_distincts A B C H);decompose [and] h;clear h;clean_reap_hyps
| H:(Obtuse ?A ?B ?C) |- _ =>
let h := fresh in
not_exist_hyp2 A B B C;
assert (h := obtuse_distincts A B C H);decompose [and] h;clear h;clean_reap_hyps
| H:SuppA ?A ?B ?C ?D ?E ?F |- _ =>
let h := fresh in
not_exist_hyp4 A B B C D E E F;
assert (h := suppa_distincts A B C D E F H);decompose [and] h;clear h;clean_reap_hyps
end.
Hint Resolve suma_sym suma_left_comm suma_middle_comm suma_right_comm
suma_comm ts__suma inangle__suma bet__suma
sams_right_comm sams_comm sams_left_comm sams_sym
out213__sams out546__sams bet__sams suppa__sams inangle__sams : suma.
Ltac SumA := eauto with suma.
Section Suma_2.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
(** ABC <= ABC + DEF. *)

Lemma sams_suma__lea123789 : forall A B C D E F G H I, SumA A B C D E F G H I ->
SAMS A B C D E F -> LeA A B C G H I.