Section TCP_tarski.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

(* Specifications: *)
Lemma impossible_case_1 :
forall A B C D T x y,
A <> B ->
A <> C ->
A <> D ->
A <> T ->
B <> C ->
B <> D ->
B <> T ->
C <> D ->
C <> T ->
D <> T ->
x <> y ->
~ Col A B C ->
Bet A D T ->
~ Col B C T ->
Bet B D C ->
Bet A B x ->
Bet C y A ->
Bet x T y ->
Par_strict B C x y ->
False.
Lemma impossible_case_2 :
forall A B C D T x y,
A <> B ->
A <> C ->
A <> D ->
A <> T ->
B <> C ->
B <> D ->
B <> T ->
C <> D ->
C <> T ->
D <> T ->
x <> y ->
~ Col A B C ->
Col A B x ->
Bet A D T ->
~ Col B C T ->
Bet B D C ->
Bet y A C ->
Bet x T y ->
False.
Lemma impossible_case_3 :
forall A B C D T x y,
A <> B ->
A <> C ->
A <> D ->
A <> T ->
B <> C ->
B <> D ->
B <> T ->
C <> D ->
C <> T ->
D <> T ->
x <> y ->
~ Col A B C ->
Bet A D T ->
~ Col B C T ->
Bet B D C ->
Bet B x A ->
Bet x T y ->
Par_strict B C x y ->
False.
Lemma impossible_case_4_1 :
forall A B C D T x y,
A <> B ->
A <> C ->
A <> D ->
A <> T ->
B <> C ->
B <> D ->
B <> T ->
C <> D ->
C <> T ->
D <> T ->
x <> y ->
~ Col A B C ->
Col A C y ->
Bet A D T ->
~ Col B C T ->
Bet B D C ->
Bet A x B \/ Bet A B x ->
Bet T y x ->
False.
Lemma impossible_case_4_2 :
forall A B C D T x y,
A <> B ->
A <> C ->
A <> D ->
A <> T ->
B <> C ->
B <> D ->
B <> T ->
C <> D ->
C <> T ->
D <> T ->
x <> y ->
~ Col A B C ->
Col A C y ->
Bet A D T ->
~ Col B C T ->
Bet B D C ->
Bet B A x ->
Bet T y x ->
Par_strict B C x y ->
False.
Lemma impossible_case_4 :
forall A B C D T x y,
A <> B ->
A <> C ->
A <> D ->
A <> T ->
B <> C ->
B <> D ->
B <> T ->
C <> D ->
C <> T ->
D <> T ->
x <> y ->
~ Col A B C ->
Col A C y ->
Bet A D T ->
~ Col B C T ->
Bet B D C ->
Col A B x ->
Bet T y x ->
Par_strict B C x y ->
False.
Lemma impossible_two_sides_not_col : forall A B C D T Y,
A <> B ->
A <> C ->
A <> D ->
A <> T ->
A <> Y ->
B <> C ->
B <> D ->
B <> T ->
B <> Y ->
C <> D ->
C <> T ->
C <> Y ->
D <> T ->
T <> Y ->
~ Col A B C ->
Bet A D T ->
~ Col B C T ->
Bet B D C ->
Bet B Y T ->
~ Col A C Y.