Require Export GeoCoq.Axioms.tarski_axioms.
Section segment_construction_independent.
Inductive Point :=
P0.
Definition Bet (A B C : Point) := False.
Definition Cong (A B C D : Point) := True.

(* Specifications: *)
Lemma between_identity : forall A B, Bet A B A -> A=B.
Lemma cong_pseudo_reflexivity : forall A B, Cong A B B A.
Lemma cong_identity : forall A B C, Cong A B C C -> A = B.
Lemma cong_inner_transitivity : forall A B C D E F,
Cong A B C D -> Cong A B E F -> Cong C D E F.
Lemma inner_pasch : forall A B C P Q,
Bet A P C -> Bet B Q C ->
exists x, Bet P x B /\ Bet Q x A.
Lemma five_segment : forall A A' B B' C C' D D',
Cong A B A' B' ->
Cong B C B' C' ->
Cong A D A' D' ->
Cong B D B' D' ->
Bet A B C -> Bet A' B' C' -> A <> B -> Cong C D C' D'.
Lemma not_segment_construction : ~ (forall A B C D,
exists E, Bet A B E /\ Cong B E C D).
Lemma lower_dim : exists A, exists B, exists C, ~ (Bet A B C \/ Bet B C A \/ Bet C A B).
Lemma upper_dim : forall A B C P Q ,
P <> Q -> Cong A P A Q -> Cong B P B Q -> Cong C P C Q ->
(Bet A B C \/ Bet B C A \/ Bet C A B).
Lemma euclid : forall A B C,
~ (Bet A B C \/ Bet B C A \/ Bet C A B) -> exists CC, Cong A CC B CC /\ Cong A CC C CC.