(** The following lemmas express the triangle inequality only with predicates from Tarski : AC <= AB + BC, and AC = AB + BC <-> Bet A B C *)

(* Specifications: *)
Lemma triangle_strict_inequality : forall A B C D, Bet A B D -> Cong B C B D -> ~ Bet A B C ->
Lt A C A D.
Lemma triangle_inequality : forall A B C D, Bet A B D -> Cong B C B D -> Le A C A D.
Lemma triangle_strict_inequality_2 : forall A B C A' B' C',
Bet A' B' C' -> Cong A B A' B' -> Cong B C B' C' -> ~ Bet A B C ->
Lt A C A' C'.
Lemma triangle_inequality_2 : forall A B C A' B' C',
Bet A' B' C' -> Cong A B A' B' -> Cong B C B' C' ->
Le A C A' C'.