Require Export GeoCoq.Tarski_dev.Ch12_parallel.
Require Export GeoCoq.Tarski_dev.Annexes.suma.
(** This development is inspired by The Foundations of Geometry and the Non-Euclidean Plane, by George E Martin, chapters 21 and 22 *)
Section Saccheri.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

(* Specifications: *)
Lemma sac_perm : forall A B C D, Saccheri A B C D -> Saccheri D C B A.
Lemma sac_distincts : forall A B C D,
Saccheri A B C D ->
A <> B /\ B <> C /\ C <> D /\ A <> D /\ A <> C /\ B <> D.
Lemma lam_perm : forall A B C D, Lambert A B C D -> Lambert A D C B.
(** The two following lemmas come from Theorem 21.10 *)
Lemma sac__cong : forall A B C D, Saccheri A B C D -> Cong A C B D.
Lemma sac__conga : forall A B C D, Saccheri A B C D -> CongA A B C B C D.
Lemma lam__os : forall A B C D, Lambert A B C D -> OS A B C D.
Lemma per2_os__ncol123 : forall A B C D, Per B A D -> Per A D C -> OS A D B C ->
~ Col A B C.
Lemma per2_os__ncol234 : forall A B C D,
Per B A D -> Per A D C ->
OS A D B C ->
~ Col B C D.
Lemma sac__ncol123 : forall A B C D, Saccheri A B C D -> ~ Col A B C.
Lemma sac__ncol124 : forall A B C D, Saccheri A B C D -> ~ Col A B D.
Lemma sac__ncol134 : forall A B C D, Saccheri A B C D -> ~ Col A C D.
Lemma sac__ncol234 : forall A B C D, Saccheri A B C D -> ~ Col B C D.
Lemma sac__coplanar : forall A B C D, Saccheri A B C D -> Coplanar A B C D.
(** The five following lemmas come from Theorem 21.8 *)
Lemma lt_os_per2__lta : forall A B C D,
Per B A D -> Per A D C ->
OS A D B C ->
Lt A B C D ->
LtA B C D A B C.
Lemma lt4321_os_per2__lta : forall A B C D,
Per B A D -> Per A D C ->
OS A D B C -> Lt D C B A ->
LtA A B C B C D.
Lemma lta_os_per2__lt : forall A B C D,
Per B A D -> Per A D C ->
OS A D B C -> LtA B C D A B C ->
Lt A B C D.
Lemma lta123234_os_per2__lt : forall A B C D,
Per B A D -> Per A D C ->
OS A D B C -> LtA A B C B C D ->
Lt D C B A.
Lemma conga_per2_os__cong : forall A B C D,
Per B A D -> Per A D C ->
OS A D B C -> CongA B C D A B C ->
Cong A B C D.