Section Hilbert_2D_to_Tarski_2D.
Context `{Hi:Hilbert_neutral_dimensionless}.
(** Tarski's betweenness is not strict: *)
Definition Bet A B C := BetH A B C \/ A = B \/ B = C.

(* Specifications: *)
Lemma betH_distincts : forall A B C, BetH A B C -> A <> B /\ B <> C /\ A <> C.
(** Hilbert's congruence is 'defined' only for non degenerated segments, while Tarski's segment congruence allows the null segment. *)
Lemma congH_perm : forall A B, A<>B -> CongH A B B A.
Lemma congH_refl : forall A B, A<>B -> CongH A B A B.
Lemma congH_sym : forall A B C D, A<>B -> CongH A B C D -> CongH C D A B.
Global Instance IncidL_morphism' : Proper(eq ==> EqL ==> iff) IncidL.
Lemma colH_permut_312 : forall A B C, ColH A B C -> ColH C A B.
Lemma colH_permut_213 : forall A B C, ColH A B C -> ColH B A C.
Lemma colH_permut_132 : forall A B C, ColH A B C -> ColH A C B.
Lemma colH_permut_321 : forall A B C, ColH A B C -> ColH C B A.
Lemma other_point_exists : forall A: Point, exists B, A <> B.
Lemma colH_trivial111 : forall A, ColH A A A.
Lemma colH_trivial112 : forall A B, ColH A A B.
Lemma colH_trivial122 : forall A B, ColH A B B.
Lemma colH_trivial121 : forall A B, ColH A B A.
Hint Resolve colH_trivial121 colH_trivial122 colH_trivial112 colH_trivial111 colH_permut_231 colH_permut_312 colH_permut_321
colH_permut_213 colH_permut_132 colH_permut_231 : colH.