Require Export GeoCoq.Axioms.parallel_postulates.
Require Export GeoCoq.Meta_theory.Parallel_postulates.SPP_tarski.
Require Export GeoCoq.Meta_theory.Parallel_postulates.alternate_interior_angles_consecutive_interior_angles.
Require Export GeoCoq.Meta_theory.Parallel_postulates.alternate_interior_angles_playfair_bis.
Require Export GeoCoq.Meta_theory.Parallel_postulates.alternate_interior_angles_triangle.
Require Export GeoCoq.Meta_theory.Parallel_postulates.alternate_interior_angles_proclus.
Require Export GeoCoq.Meta_theory.Parallel_postulates.bachmann_s_lotschnittaxiom_legendre_s_parallel_postulate.
Require Export GeoCoq.Meta_theory.Parallel_postulates.bachmann_s_lotschnittaxiom_weak_inverse_projection_postulate.
Require Export GeoCoq.Meta_theory.Parallel_postulates.bachmann_s_lotschnittaxiom_weak_triangle_circumscription_principle.
Require Export GeoCoq.Meta_theory.Parallel_postulates.consecutive_interior_angles_alternate_interior_angles.
Require Export GeoCoq.Meta_theory.Parallel_postulates.existential_playfair_rah.
Require Export GeoCoq.Meta_theory.Parallel_postulates.existential_saccheri_rah.
Require Export GeoCoq.Meta_theory.Parallel_postulates.inverse_projection_postulate_proclus_bis.
Require Export GeoCoq.Meta_theory.Parallel_postulates.legendre.
Require Export GeoCoq.Meta_theory.Parallel_postulates.midpoint_playfair.
Require Export GeoCoq.Meta_theory.Parallel_postulates.original_euclid_original_spp.
Require Export GeoCoq.Meta_theory.Parallel_postulates.original_spp_inverse_projection_postulate.
Require Export GeoCoq.Meta_theory.Parallel_postulates.par_perp_2_par_par_perp_perp.
Require Export GeoCoq.Meta_theory.Parallel_postulates.par_perp_perp_par_perp_2_par.
Require Export GeoCoq.Meta_theory.Parallel_postulates.par_perp_perp_playfair.
Require Export GeoCoq.Meta_theory.Parallel_postulates.par_trans_playfair.
Require Export GeoCoq.Meta_theory.Parallel_postulates.playfair_existential_playfair.
Require Export GeoCoq.Meta_theory.Parallel_postulates.playfair_midpoint.
Require Export GeoCoq.Meta_theory.Parallel_postulates.playfair_par_trans.
Require Export GeoCoq.Meta_theory.Parallel_postulates.posidonius_postulate_rah.
Require Export GeoCoq.Meta_theory.Parallel_postulates.proclus_aristotle.
Require Export GeoCoq.Meta_theory.Parallel_postulates.proclus_bis_proclus.
Require Export GeoCoq.Meta_theory.Parallel_postulates.rah_existential_saccheri.
Require Export GeoCoq.Meta_theory.Parallel_postulates.rah_rectangle_principle.
Require Export GeoCoq.Meta_theory.Parallel_postulates.rah_similar.
Require Export GeoCoq.Meta_theory.Parallel_postulates.rah_thales_postulate.
Require Export GeoCoq.Meta_theory.Parallel_postulates.rah_posidonius_postulate.
Require Export GeoCoq.Meta_theory.Parallel_postulates.rectangle_existence_rah.
Require Export GeoCoq.Meta_theory.Parallel_postulates.rectangle_principle_rectangle_existence.
Require Export GeoCoq.Meta_theory.Parallel_postulates.similar_rah.
Require Export GeoCoq.Meta_theory.Parallel_postulates.tarski_playfair.
Require Export GeoCoq.Meta_theory.Parallel_postulates.thales_converse_postulate_thales_existence.
Require Export GeoCoq.Meta_theory.Parallel_postulates.thales_converse_postulate_weak_triangle_circumscription_principle.
Require Export GeoCoq.Meta_theory.Parallel_postulates.thales_existence_rah.
Require Export GeoCoq.Meta_theory.Parallel_postulates.thales_postulate_thales_converse_postulate.
Require Export GeoCoq.Meta_theory.Parallel_postulates.weak_inverse_projection_postulate_bachmann_s_lotschnittaxiom.
Require Export GeoCoq.Meta_theory.Parallel_postulates.weak_inverse_projection_postulate_weak_tarski_s_parallel_postulate.
Require Export GeoCoq.Meta_theory.Parallel_postulates.weak_tarski_s_parallel_postulate_weak_inverse_projection_postulate.
Require Export GeoCoq.Meta_theory.Parallel_postulates.weak_triangle_circumscription_principle_bachmann_s_lotschnittaxiom.
Require Export GeoCoq.Tarski_dev.Annexes.saccheri.
Require Export GeoCoq.Tarski_dev.Annexes.perp_bisect.
Require Export GeoCoq.Tarski_dev.Annexes.quadrilaterals.
Require Export GeoCoq.Tarski_dev.Ch13_1.
Section Euclid.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

(* Specifications: *)
Theorem equivalent_postulates_without_decidability_of_intersection_of_lines :
all_equiv
(alternate_interior_angles_postulate::
alternative_playfair_s_postulate::
consecutive_interior_angles_postulate::
midpoint_converse_postulate::
perpendicular_transversal_postulate::
playfair_s_postulate::
universal_posidonius_postulate::
postulate_of_parallelism_of_perpendicular_transversals::
postulate_of_transitivity_of_parallelism::
nil).
Theorem equivalent_postulates_without_any_continuity :
all_equiv
(existential_thales_postulate::
posidonius_postulate::
postulate_of_existence_of_a_right_lambert_quadrilateral::
postulate_of_existence_of_a_right_saccheri_quadrilateral::
postulate_of_existence_of_a_triangle_whose_angles_sum_to_two_rights::
postulate_of_existence_of_similar_triangles::
postulate_of_right_lambert_quadrilaterals::
postulate_of_right_saccheri_quadrilaterals::
thales_postulate::
thales_converse_postulate::
triangle_postulate::
nil).
Theorem equivalent_postulates_with_decidability_of_intersection_of_lines :
decidability_of_intersection ->
all_equiv
(alternate_interior_angles_postulate::
alternative_playfair_s_postulate::
alternative_proclus_postulate::
alternative_strong_parallel_postulate::
consecutive_interior_angles_postulate::
euclid_5::
euclid_s_parallel_postulate::
inverse_projection_postulate::
midpoint_converse_postulate::
perpendicular_transversal_postulate::
playfair_s_postulate::
universal_posidonius_postulate::
postulate_of_parallelism_of_perpendicular_transversals::
postulate_of_transitivity_of_parallelism::
proclus_postulate::
strong_parallel_postulate::
tarski_s_parallel_postulate::
triangle_circumscription_principle::
nil).
(** Every postulate which states the existence of a point is equivalent *)
Theorem equivalent_postulates_without_decidability_of_intersection_of_lines_bis :
all_equiv
(alternative_strong_parallel_postulate::
alternative_proclus_postulate::
euclid_5::
euclid_s_parallel_postulate::
inverse_projection_postulate::
proclus_postulate::
strong_parallel_postulate::
tarski_s_parallel_postulate::
triangle_circumscription_principle::
nil).
Theorem stronger_postulates :
stronger
(alternative_strong_parallel_postulate::
alternative_proclus_postulate::
euclid_5::
euclid_s_parallel_postulate::
inverse_projection_postulate::
proclus_postulate::
strong_parallel_postulate::
tarski_s_parallel_postulate::
triangle_circumscription_principle::
nil)
(alternate_interior_angles_postulate::
alternative_playfair_s_postulate::
consecutive_interior_angles_postulate::
midpoint_converse_postulate::
perpendicular_transversal_postulate::
playfair_s_postulate::
universal_posidonius_postulate::
postulate_of_parallelism_of_perpendicular_transversals::
postulate_of_transitivity_of_parallelism::
nil).
Theorem stronger_postulates_bis :
stronger
(alternate_interior_angles_postulate::
alternative_playfair_s_postulate::
consecutive_interior_angles_postulate::
midpoint_converse_postulate::
perpendicular_transversal_postulate::
playfair_s_postulate::
universal_posidonius_postulate::
postulate_of_parallelism_of_perpendicular_transversals::
postulate_of_transitivity_of_parallelism::
nil)
(existential_thales_postulate::
posidonius_postulate::
postulate_of_existence_of_a_right_lambert_quadrilateral::
postulate_of_existence_of_a_right_saccheri_quadrilateral::
postulate_of_existence_of_a_triangle_whose_angles_sum_to_two_rights::
postulate_of_existence_of_similar_triangles::
postulate_of_right_lambert_quadrilaterals::
postulate_of_right_saccheri_quadrilaterals::
thales_postulate::
thales_converse_postulate::
triangle_postulate::
nil).
Theorem equivalence_of_aristotle_greenberg_and_decidability_of_intersection :
all_equiv_under
(alternate_interior_angles_postulate::
alternative_playfair_s_postulate::
alternative_proclus_postulate::
alternative_strong_parallel_postulate::
consecutive_interior_angles_postulate::
euclid_5::
euclid_s_parallel_postulate::
inverse_projection_postulate::
midpoint_converse_postulate::
perpendicular_transversal_postulate::
playfair_s_postulate::
universal_posidonius_postulate::
postulate_of_parallelism_of_perpendicular_transversals::
postulate_of_transitivity_of_parallelism::
proclus_postulate::
strong_parallel_postulate::
tarski_s_parallel_postulate::
triangle_circumscription_principle::
nil)
(aristotle_s_axiom::
greenberg_s_axiom::
decidability_of_intersection::
nil).
Theorem equivalent_postulates_assuming_greenberg_s_axiom :
greenberg_s_axiom ->
all_equiv
(alternate_interior_angles_postulate::
alternative_playfair_s_postulate::
alternative_proclus_postulate::
alternative_strong_parallel_postulate::
consecutive_interior_angles_postulate::
euclid_5::
euclid_s_parallel_postulate::
existential_playfair_s_postulate::
existential_thales_postulate::
inverse_projection_postulate::
midpoint_converse_postulate::
perpendicular_transversal_postulate::
postulate_of_transitivity_of_parallelism::
playfair_s_postulate::
posidonius_postulate::
universal_posidonius_postulate::
postulate_of_existence_of_a_right_lambert_quadrilateral::
postulate_of_existence_of_a_right_saccheri_quadrilateral::
postulate_of_existence_of_a_triangle_whose_angles_sum_to_two_rights::
postulate_of_existence_of_similar_triangles::
postulate_of_parallelism_of_perpendicular_transversals::
postulate_of_right_lambert_quadrilaterals::
postulate_of_right_saccheri_quadrilaterals::
postulate_of_transitivity_of_parallelism::
proclus_postulate::
strong_parallel_postulate::
tarski_s_parallel_postulate::
thales_postulate::
thales_converse_postulate::
triangle_circumscription_principle::
triangle_postulate::
nil).
Theorem equivalent_postulates_without_any_continuity_bis :
all_equiv
(bachmann_s_lotschnittaxiom::
weak_inverse_projection_postulate::
weak_tarski_s_parallel_postulate::
weak_triangle_circumscription_principle::
nil).
Theorem stronger_postulates_ter :
stronger
(existential_thales_postulate::
posidonius_postulate::
postulate_of_existence_of_a_right_lambert_quadrilateral::
postulate_of_existence_of_a_right_saccheri_quadrilateral::
postulate_of_existence_of_a_triangle_whose_angles_sum_to_two_rights::
postulate_of_existence_of_similar_triangles::
postulate_of_right_lambert_quadrilaterals::
postulate_of_right_saccheri_quadrilaterals::
thales_postulate::
thales_converse_postulate::
triangle_postulate::
nil)
(legendre_s_parallel_postulate::
bachmann_s_lotschnittaxiom::
weak_inverse_projection_postulate::
weak_tarski_s_parallel_postulate::
weak_triangle_circumscription_principle::
nil).
Theorem equivalent_postulates_assuming_archimedes_axiom :
archimedes_axiom ->
all_equiv
(alternate_interior_angles_postulate::
alternative_playfair_s_postulate::
alternative_proclus_postulate::
alternative_strong_parallel_postulate::
bachmann_s_lotschnittaxiom::
consecutive_interior_angles_postulate::
euclid_5::
euclid_s_parallel_postulate::
existential_playfair_s_postulate::
existential_thales_postulate::
inverse_projection_postulate::
legendre_s_parallel_postulate::
midpoint_converse_postulate::
perpendicular_transversal_postulate::
postulate_of_transitivity_of_parallelism::
playfair_s_postulate::
posidonius_postulate::
universal_posidonius_postulate::
postulate_of_existence_of_a_right_lambert_quadrilateral::
postulate_of_existence_of_a_right_saccheri_quadrilateral::
postulate_of_existence_of_a_triangle_whose_angles_sum_to_two_rights::
postulate_of_existence_of_similar_triangles::
postulate_of_parallelism_of_perpendicular_transversals::
postulate_of_right_lambert_quadrilaterals::
postulate_of_right_saccheri_quadrilaterals::
postulate_of_transitivity_of_parallelism::
proclus_postulate::
strong_parallel_postulate::
tarski_s_parallel_postulate::
thales_postulate::
thales_converse_postulate::
triangle_circumscription_principle::
triangle_postulate::
weak_inverse_projection_postulate::
weak_tarski_s_parallel_postulate::
weak_triangle_circumscription_principle::
nil).