(** This development is inspired by The Foundations of Geometry and the Non-Euclidean Plane, by George E Martin, chapter 22 *)
Section Archimedes.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

(* Specifications: *)
(** For every m, there exists n such that A0Dm = A0An - E0En = n(A0A1 - E0E1) (m=n) *)
Lemma t22_18_aux1 : forall A0 A1 E0 E1 D1 D,
Bet A0 D1 A1 -> Cong E0 E1 A1 D1 ->
Grad A0 D1 D ->
exists A E, Grad2 A0 A1 A E0 E1 E /\ Cong E0 E A D /\ Bet A0 D A.
(** For every n, B0Bn is lower than or equal to n times B0B1 *)
Lemma t22_18_aux2 : forall A0 A1 B0 B1 A B E,
Saccheri A0 B0 B1 A1 ->
Grad2 A0 A1 A B0 B1 E -> Saccheri A0 B0 B A -> Le B0 B B0 E.
Lemma t22_18 :
archimedes_axiom ->
forall A0 B0 B1 A1, Saccheri A0 B0 B1 A1 -> ~ Lt B0 B1 A1 A0.
Lemma t22_19 :
archimedes_axiom ->
forall A B C D, Saccheri A B C D -> ~ Obtuse A B C.
Lemma archi__obtuse_case_elimination :
archimedes_axiom ->
~ hypothesis_of_obtuse_saccheri_quadrilaterals.
Lemma t22_23_aux :
forall A B C M N L,
~ Col A M N -> Per B C A -> A <> C -> Midpoint M A B ->
Per M N A -> Col A C N ->
Midpoint M N L -> Bet A N C /\ Lambert N L B C /\ Cong B L A N.
Lemma t22_23 :
~ hypothesis_of_obtuse_saccheri_quadrilaterals ->
forall A B C M N L,
~ Col A M N -> Per B C A -> A <> C -> Midpoint M A B ->
Per M N A -> Col A C N -> Midpoint M N L ->
Bet A N C /\ Le N C A N /\ Le L N B C.
(** For every n, 2^n times B0C0 is lower than or equal to BnCn *)
(** B0 is introduced twice for the induction tactic to work properly *)
Lemma t22_24_aux :
~ hypothesis_of_obtuse_saccheri_quadrilaterals ->
forall A B0 B00 C0 B C E,
~ Col A B0 C0 -> Perp A C0 B0 C0 -> B0 = B00 ->
GradExp2 A B0 B B00 C0 E -> Perp A C0 B C -> Col A C0 C ->
Le B0 E B C.
(** For every n, it is possible to get Bn and Cn *)
Lemma t22_24_aux1 : forall A B0 C0 E,
~ Col A B0 C0 -> Perp A C0 B0 C0 -> GradExp B0 C0 E ->
exists B C, GradExp2 A B0 B B0 C0 E /\ Perp A C0 B C /\ Col A C0 C.
Lemma t22_24 : archimedes_axiom -> aristotle_s_axiom.
End Archimedes.