| name: GeoCoq_train | |
| num_files: 394 | |
| language: COQ | |
| few_shot_data_path_for_retrieval: null | |
| few_shot_metadata_filename_for_retrieval: null | |
| dfs_data_path_for_retrieval: null | |
| dfs_metadata_filename_for_retrieval: local.meta.json | |
| theorem_cnt: 4036 | |
| datasets: | |
| - project: <path-to-repo>/GeoCoq | |
| files: | |
| - path: theories/Coinc/Utils/arity.v | |
| theorems: | |
| - minus_n_0 | |
| - app_2_n | |
| - nthCircPerm2 | |
| - app_n_1_app_eq | |
| - app_app_2_n | |
| - circPermNIdFirst | |
| - app_2_n_app | |
| - app_1_n | |
| - PermOK | |
| - circPermNConsTlOK | |
| - fixLastCPOK | |
| - pred_conj | |
| - app_n_1 | |
| - lastTailOK | |
| - app | |
| - allButLastCPTl | |
| - CPToListTl1 | |
| - NoDupOK | |
| - tailCP | |
| - headCPbis | |
| - consHeadCPTl | |
| - minus_n1_n2_0 | |
| - pcaHdTl | |
| - circPermNCPOK | |
| - app_2_n_app_eq | |
| - nthCP01 | |
| - consHeadCPHd | |
| - lastCP | |
| - CPPair | |
| - CPLOK | |
| - nthCircPermNAny | |
| - lengthNilOK | |
| - circPermNId | |
| - nthFirst | |
| - plus_0_n | |
| - tailDefaultCP | |
| - ListToCP | |
| - plus_n_0 | |
| - CPLHdTlOK | |
| - fixLastCP | |
| - CP_ind | |
| - CPLHd | |
| - NoDup_dec | |
| - lengthOfCPToList | |
| - consTailOK | |
| - circPermNCP0 | |
| - circPermNConsOK | |
| - CPLCP | |
| - allButLastCP | |
| - CPToListTl2 | |
| - consHeadCPOK | |
| - consTailCPLast | |
| - PermOKAux | |
| - lastCPTl | |
| - plus_n_1 | |
| - headCP | |
| - consTlHdHdTl | |
| - ListToCPTl | |
| - nthCircPerm2Eq | |
| - app_app_n_1 | |
| - app_n_1_app | |
| - app_1_n_app_eq | |
| - app_1_n_app | |
| - CPCPL | |
| - app_app_1_n | |
| - nthLast | |
| - nthCPTlOK | |
| - NotNoDupDup | |
| - circPermPerm | |
| - nthCircPerm1 | |
| - tailCPbis | |
| - consTailCPTlD | |
| - app_app_2_n_default | |
| - app_2_n_app_default | |
| - consTailCP | |
| - consTailCPTl | |
| - consTailCPAbl | |
| - nthEqOK | |
| - CPToList | |
| - CPToListOK | |
| - consHdTlTlHd | |
| - consTailPerm | |
| - path: theories/Main/Tarski_dev/Ch11_angles.v | |
| theorems: | |
| - l11_50_1 | |
| - lea_distincts | |
| - lea_out4__lea | |
| - lta_not_conga | |
| - lta__lea | |
| - acute_not_per | |
| - orth_symmetry | |
| - bet_conga__bet | |
| - l11_49 | |
| - ex_suppa | |
| - segment_construction_0 | |
| - l11_25 | |
| - cop3_orth_at__orth_at | |
| - conga_inangle2_per__acute | |
| - lta_dec | |
| - col_conga__conga | |
| - orth_at_symmetry | |
| - conga__lea | |
| - l11_17 | |
| - conga_line | |
| - l11_4_2 | |
| - conga_ex_cong3 | |
| - l11_31_1 | |
| - lta_trans | |
| - col2_orth_at__orth_at | |
| - t18_19 | |
| - l11_22 | |
| - in_angle_line | |
| - lea_asym | |
| - l11_22a | |
| - col_conga_col | |
| - conga_pseudo_refl | |
| - orth_at2__eq | |
| - lta__nlea | |
| - le2_per2__le | |
| - acute_per__lta | |
| - l11_4_1 | |
| - t18_18 | |
| - l11_18_2 | |
| - conga_right_comm | |
| - l11_44_2bis | |
| - lta_right_comm | |
| - reflectl__conga | |
| - l11_47 | |
| - inangle_dec | |
| - suppa2__conga456 | |
| - l11_24 | |
| - l11_29_b | |
| - col_conga_cop_reflectl__col | |
| - angle_bisector | |
| - triangle_inequality | |
| - inangle__lta | |
| - conga_distinct | |
| - col_in_angle_out | |
| - acute_conga__acute | |
| - cong3_diff | |
| - acute_cop_perp__one_side | |
| - l11_30 | |
| - cong2_per2__cong_3 | |
| - conga__acute | |
| - in_angle_one_side | |
| - eq_conga_out | |
| - bet_in_angle_bet | |
| - l11_41 | |
| - l11_10 | |
| - inangle_distincts | |
| - acute_one_side_aux | |
| - acute_distincts | |
| - conga_diff45 | |
| - acute_one_side_aux0 | |
| - lt2_per2__lt | |
| - obtuse_distincts | |
| - inangle3123 | |
| - conga2_suppa__suppa | |
| - out341__inangle | |
| - per2__suppa | |
| - acute_obtuse__lta | |
| - lea_total | |
| - acute_trivial | |
| - conga_os__out | |
| - l11_15 | |
| - conga_comm | |
| - ncol_conga_ncol | |
| - lta_os__ts | |
| - conga_inangle_per__acute | |
| - l11_57 | |
| - bet_le__lt | |
| - bet_out__suppa | |
| - ex_conga_ts | |
| - acute_suppa__obtuse | |
| - inangle2__lea | |
| - l11_50_2 | |
| - bet_suppa__out | |
| - conga_refl | |
| - l11_13 | |
| - l11_14 | |
| - lea_in_angle | |
| - cong2_per2__cong_conga2 | |
| - in_angle_asym | |
| - triangle_inequality_2 | |
| - l11_29_a | |
| - nlta__lea | |
| - acute_chara | |
| - conga__lea456123 | |
| - suppa_distincts | |
| - cong4_cop2__eq | |
| - angle_construction_1 | |
| - nlta | |
| - ts2__inangle | |
| - l11_36 | |
| - orth_dec | |
| - tsp_dec | |
| - l11_3_bis | |
| - l11_25_aux | |
| - out_lea__out | |
| - lea_suppa2__lea | |
| - conga_trivial_1 | |
| - inangle1123 | |
| - lea_refl | |
| - l11_43_aux | |
| - angle_construction_2 | |
| - inangle_one_side | |
| - l11_28 | |
| - in_angle_trans | |
| - orth_at__ncop | |
| - suppa_comm | |
| - orth_at__ncop1 | |
| - conga2_cop2__col_1 | |
| - obtuse_out2__obtuse | |
| - col2_orth__orth | |
| - orth_at_distincts | |
| - l11_44_2_b | |
| - l11_44_2 | |
| - conga_sym_equiv | |
| - col_inangle2__out | |
| - mid2_orth_at2__cong | |
| - lea_obtuse_obtuse | |
| - conga_diff2 | |
| - bet_obtuse__acute | |
| - lea121345 | |
| - lea__nlta | |
| - obtuse_suppa__acute | |
| - per_suppa__per | |
| - lta_comm | |
| - l11_44_2_a | |
| - col_in_angle | |
| - conga_diff56 | |
| - lea_dec | |
| - osp_dec | |
| - lea_left_comm | |
| - col_lta__out | |
| - inangle__lea | |
| - two_sides_in_angle | |
| - conga_diff1 | |
| - cong_le_per2__le | |
| - bet2_lta__lta | |
| - conga_cop__or_out_ts | |
| - conga2_cop2__col | |
| - bet__suppa | |
| - col_lta__bet | |
| - lta_distincts | |
| - cong2_conga_cong | |
| - l11_52 | |
| - out__acute | |
| - l11_63_existence | |
| - inangle__lea_1 | |
| - conga_left_comm | |
| - not_conga | |
| - os_ts__inangle | |
| - orth_at_chara | |
| - obtuse_per__lta | |
| - acute_lea_acute | |
| - cop_inangle__ex_col_inangle | |
| - in_angle_two_sides | |
| - cong2__ncol | |
| - lea_right_comm | |
| - triangle_strict_inequality | |
| - col_cop_orth__orth_at | |
| - l11_41_aux | |
| - cong3_preserves_out | |
| - lea_trans | |
| - cong3_conga | |
| - l11_44_1 | |
| - col_cop_orth_at__eq | |
| - in_angle_trans2 | |
| - orth_distincts | |
| - conga_trans | |
| - l11_44_1_b | |
| - l11_51 | |
| - l11_43 | |
| - l11_22b | |
| - l8_21_3 | |
| - lea123456_lta__lta | |
| - out321__inangle | |
| - suppa_left_comm | |
| - lea_comm | |
| - acute_bet__obtuse | |
| - acute_col__out | |
| - lta_left_comm | |
| - triangle_reverse_inequality | |
| - obtuse_sym | |
| - l11_60_aux | |
| - l11_53 | |
| - col_obtuse__bet | |
| - l11_21_b | |
| - orth_at2_tsp__ts | |
| - os2__inangle | |
| - not_conga_sym | |
| - suppa_right_comm | |
| - bet_lea__bet | |
| - l11_62_existence | |
| - l11_22_bet | |
| - l11_60 | |
| - l11_16 | |
| - suppa_dec | |
| - l11_63_aux | |
| - l11_44_1_a | |
| - l11_aux | |
| - acute__not_obtuse | |
| - lea456789_lta__lta | |
| - nlea__lta | |
| - not_lta_and_conga | |
| - os3__lta | |
| - angle_construction_4 | |
| - l11_62_unicity_bis | |
| - or_lta2_conga | |
| - conga_cop_out_reflectl__out | |
| - cong3_conga2 | |
| - acute_out2__acute | |
| - le_lt_per2__lt | |
| - path: theories/Main/Tarski_dev/Ch13_4_cos.v | |
| theorems: | |
| - lcos_morphism | |
| - perp_acute_out | |
| - lcos_per | |
| - lcos_const0 | |
| - lcos_eqa_lcos | |
| - lcos2_eq_lcos3_eq | |
| - lcos_lg_distincts | |
| - acute_not_obtuse | |
| - lcos3_eq_sym | |
| - lcos3_permut2 | |
| - lcos_lg | |
| - lcos2_eq_trans | |
| - out_acute | |
| - lcos_const_o | |
| - lcos_lg_anga | |
| - lcos3_eql_lcos3 | |
| - lcos_eql_lcos | |
| - perp_out__acute | |
| - acute_comp_not_acute | |
| - perp_acute | |
| - eql_lcos_null | |
| - lcos2_lg_not_null | |
| - lcos2_eq_sym | |
| - lcos_eq_lcos3_eq | |
| - lcos_const_cb | |
| - lcos_eq_lcos2_eq | |
| - lcos_const | |
| - lcos3_lcos_1_2 | |
| - lcos2_exists' | |
| - null_lcos_eql | |
| - lcos_lg_not_null | |
| - lcos_const1 | |
| - l13_7 | |
| - lcos_const_ab | |
| - lcos3_lg_not_null | |
| - lcos3_permut3 | |
| - lcos_exists | |
| - lcos2_eq_refl | |
| - flat_not_acute | |
| - perp_obtuse_bet | |
| - lcos2_lg_anga | |
| - lcos3_exists | |
| - lcos_uniqueness | |
| - lcos_const_a | |
| - lcos2_eql_lcos2 | |
| - lcos3_uniqueness | |
| - lcos_eq_trans | |
| - lcos3_eq_refl | |
| - lcos3_lcos_2_1 | |
| - path: theories/Main/Tarski_dev/Ch06_out_lines.v | |
| theorems: | |
| - l6_2 | |
| - l6_7 | |
| - diff_col_ex3 | |
| - colx | |
| - l6_3_2 | |
| - out2_bet_out | |
| - out_diff2 | |
| - not_bet_and_out | |
| - out_cong_cong | |
| - l6_21 | |
| - out2__bet | |
| - l6_13_1 | |
| - segment_reverse | |
| - cong_preserves_bet | |
| - col_transitivity_1 | |
| - l6_16_1 | |
| - Out_cases | |
| - bet2__out | |
| - out_diff1 | |
| - segment_construction_3 | |
| - out_distinct | |
| - bet2_out_out | |
| - out_bet_out_1 | |
| - out_col | |
| - diff_col_ex | |
| - out_bet_out_2 | |
| - l6_3_1 | |
| - bet_out_1 | |
| - out_dec | |
| - out_to_bet | |
| - out_trivial | |
| - not_bet_out | |
| - not_out_bet | |
| - l6_13_2 | |
| - or_bet_out | |
| - l6_11_uniqueness | |
| - bet2_le2__le1245 | |
| - bet_out__bet | |
| - l6_6 | |
| - bet_out_out_bet | |
| - col_transitivity_2 | |
| - col3 | |
| - not_col_exists | |
| - col2__eq | |
| - diff_bet_ex3 | |
| - col_out2_col | |
| - l6_4_1 | |
| - path: theories/Algebraic/POF_to_Tarski.v | |
| theorems: | |
| - dotC' | |
| - markov_betE | |
| - dot2B | |
| - col_2D_aux | |
| - betS_gt0 | |
| - b2D_neq0 | |
| - col_2D_aux_1 | |
| - contraction_betR | |
| - betE_axx | |
| - Rcf_to_GI_PED | |
| - ratio_cp'_aux_1 | |
| - Rcf_to_GI_euclidean | |
| - betSP' | |
| - cong_identity | |
| - ratio_inv | |
| - addrDBB | |
| - inner_pasch | |
| - congP_aux | |
| - upper_dim_dgc1 | |
| - quad_ge0 | |
| - point_equality_decidability | |
| - ratio_eq | |
| - ratiovv | |
| - b_neq0 | |
| - upper_dim_dgc_aux0 | |
| - ratio_cp | |
| - add2r_eq | |
| - bet_betE | |
| - ab_neq | |
| - bet_inner_transitivity | |
| - betS_neq12 | |
| - Rcf_to_T_PED | |
| - betSP | |
| - cong_perp | |
| - bet_neq0 | |
| - extension_col | |
| - add_ratio_1 | |
| - lower_dim | |
| - bet_cong_ratio_eq | |
| - euclid' | |
| - dotE | |
| - ratio_e0_n1 | |
| - ratio_bet'' | |
| - cong_perp_aux2 | |
| - inner_pasch' | |
| - vector2_eq0 | |
| - bet_gt0 | |
| - betEP | |
| - congP | |
| - col_2D | |
| - contraction_bet | |
| - inner_pasch_gen | |
| - tcp | |
| - betR_abc | |
| - thing_gt0 | |
| - upper_dim_dgc1_aux | |
| - contraction_eq | |
| - bet_lt | |
| - segment_construction | |
| - ratioNr | |
| - bet_sym | |
| - row2_eq_nD | |
| - cong_eq' | |
| - add11_neq0 | |
| - bet_opp | |
| - ratio_betS | |
| - all_v_neq0 | |
| - ratio_bet | |
| - a2D_eq0 | |
| - ratioP | |
| - upper_dim_dgc_aux1 | |
| - stuff_gt0 | |
| - betE_sym | |
| - dotvN | |
| - dotNv | |
| - cosine_rule | |
| - congP_aux' | |
| - Rcf_to_T2D | |
| - addrBDD | |
| - add_ratio | |
| - five_segment | |
| - stuff_neq0 | |
| - ca_neq | |
| - addf_divrr | |
| - ratiorN | |
| - bet_lt1 | |
| - dot2N | |
| - betS_neq23 | |
| - betS_neq13 | |
| - thing_neq0 | |
| - ratio0p | |
| - bet_col | |
| - bet_neq0' | |
| - cong_pseudo_reflexivity | |
| - bet_gt0' | |
| - addrBDB | |
| - betS_extension | |
| - upper_dim | |
| - quad_eq0' | |
| - quad_neq0 | |
| - proclus | |
| - euclid | |
| - ratio_lt0_v2_neq0 | |
| - extension_bet | |
| - cosine_rule' | |
| - dot2_eq0 | |
| - upper_dim_dgc2_aux | |
| - betS_id | |
| - dotC | |
| - ratio_lt0_v1_neq0 | |
| - Rcf_to_T_euclidean | |
| - row2_eq0_nD | |
| - col_2D' | |
| - Rcf_to_GI2D | |
| - betSP'_aux | |
| - extensionP | |
| - bet_xxa | |
| - funmxN | |
| - betE_xxa | |
| - bet_xax | |
| - Rcf_to_T | |
| - sub_1_ratio | |
| - eq_pick_neq0 | |
| - eq_inv_scale | |
| - ratio_e1_n0 | |
| - cong_inner_transitivity | |
| - ab_neq_2D | |
| - c_neq0 | |
| - ca_neq_2D | |
| - bc_neq_2D | |
| - vector2_eq | |
| - congC | |
| - addrDBD | |
| - col_2D_aux_2 | |
| - betR_bca | |
| - upper_dim_dgc2 | |
| - col_2D_aux_3 | |
| - bet_ratio | |
| - bc_neq | |
| - inner_pasch'' | |
| - ratio_cp' | |
| - dot2D | |
| - a_eq0 | |
| - ratio_neq0 | |
| - row2_eq | |
| - vector2_neq0 | |
| - bet_trans | |
| - betR_cab | |
| - path: theories/Main/Tarski_dev/Ch14_sum.v | |
| theorems: | |
| - diff_stable | |
| - pj_left_comm | |
| - sum3_col | |
| - sum_uniquenessB | |
| - sum_to_sum3 | |
| - opp_uniqueness | |
| - proj_preserves_sum | |
| - sum_x_axis_unit_change | |
| - proj_col | |
| - sum_cong2 | |
| - project_trivial | |
| - sum_ar2 | |
| - midpoint_opp | |
| - sum_to_sump | |
| - project_col_project | |
| - sum_assoc | |
| - sum22_comm | |
| - change_grid_sum_0 | |
| - sum_A_B_A | |
| - sum_opp | |
| - sum_diff2_diff_sum2_b | |
| - sum_diff_diff_a | |
| - opp_comm | |
| - sum_A_B_B | |
| - sum_O_B_eq | |
| - diff_opp | |
| - sum_plg | |
| - sum_par_strict | |
| - pj_comm | |
| - Pj_exists | |
| - opp_exists | |
| - sum_A_exists | |
| - sum_B_exists | |
| - opp0_uniqueness | |
| - sum_A_null | |
| - diff_uniqueness | |
| - diff_to_plg | |
| - sum_diff_diff | |
| - sum_y_axis_change | |
| - pj_right_comm | |
| - sum_A_O | |
| - sum3_to_sum4 | |
| - sum_uniquenessA | |
| - double_null_null | |
| - sum22_col | |
| - sum3_comm_1_2 | |
| - diff_A_O | |
| - sum22_permut | |
| - sum_O_B | |
| - sum_abcd | |
| - diff_null_eq | |
| - sum_comm | |
| - pj_uniqueness | |
| - pj_col_project | |
| - sum_iff_cong | |
| - sum_A_O_eq | |
| - sum_assoc_2 | |
| - sum_O_O | |
| - sum4_comm | |
| - opp_midpoint | |
| - plg_to_sum | |
| - diff_exists | |
| - diff_null | |
| - not_null_double_not_null | |
| - diff_sum | |
| - sum3_comm_2_3 | |
| - sum4_col | |
| - sum3_exists | |
| - sum_B_null | |
| - sum3_permut | |
| - diff_ar2 | |
| - sum_diff_diff_b | |
| - sum_assoc_1 | |
| - sump_to_sum | |
| - proj_id | |
| - diff_O_A_opp | |
| - sum3_uniqueness | |
| - sum_uniqueness | |
| - diff_uniquenessB | |
| - diff_O_A | |
| - sum4_permut | |
| - sum_diff2_diff_sum2_a | |
| - path: theories/Main/Meta_theory/Continuity/angle_archimedes.v | |
| theorems: | |
| - grada_distincts | |
| - archi__gradaexp_destruction | |
| - grada_out__out | |
| - grada__lea | |
| - gradaexp_destruction_aux | |
| - grada2_sams_suma__grada | |
| - archi__grada_destruction | |
| - acute_archi_aux1 | |
| - acute_archi_aux | |
| - conga2_grada__grada | |
| - acute_archi_aux2 | |
| - angles_archi_aux1 | |
| - archi_in_angles | |
| - gradaexp__grada | |
| - path: theories/Main/Annexes/saccheri.v | |
| theorems: | |
| - per2_os__pars | |
| - acute_sac__aah | |
| - sac__perp3414 | |
| - t22_17__oah | |
| - t22_7__per | |
| - lam6534_mid2__sac | |
| - lam__pars1234 | |
| - sac__perp1214 | |
| - t22_12__rah | |
| - lt_os_per2__lta | |
| - conga_per2_os__cong | |
| - t22_14__rah | |
| - t22_8__lt5612 | |
| - t22_7__lt5612 | |
| - t22_12__aah | |
| - acute_lam__lt | |
| - t22_9_aux | |
| - mid2_sac__perp_lower | |
| - absolute_exterior_angle_theorem | |
| - t22_7__obtuse | |
| - lam_per__rah | |
| - lam_obtuse__lt | |
| - cong_mid__suma | |
| - t22_20 | |
| - cong2_sac2__cong | |
| - t22_8__acute | |
| - t22_14__aah | |
| - t22_14__sams_nbet_aux | |
| - ex_lambert | |
| - cong_sac__per | |
| - lta123234_os_per2__lt | |
| - not_rah | |
| - obtuse_sac__oah | |
| - mid2_sac__lam6534 | |
| - lam_obtuse__oah | |
| - lt4321_os_per2__lta | |
| - t22_14__nsams | |
| - t22_9__acute | |
| - mid2_sac__perp_upper | |
| - t22_11__obtuse | |
| - per2_os__ncol234 | |
| - lt_sac__obtuse | |
| - lam__par1423 | |
| - lam_perm | |
| - per__ex_saccheri | |
| - sac_perm | |
| - t22_14_aux | |
| - t22_17__rah | |
| - t22_12__oah | |
| - t22_8__obtuse | |
| - t22_14__bet | |
| - lta_os_per2__lt | |
| - t22_14__sams_nbet | |
| - lam_acute__aah | |
| - t22_14__nsams_aux | |
| - t22_7__lt1256 | |
| - t22_9__per | |
| - not_aah | |
| - t22_8_aux | |
| - sac__cong | |
| - lam__pars1423 | |
| - sac__par1423 | |
| - not_oah | |
| - t22_11__per | |
| - cong_lam__per | |
| - sac__conga | |
| - t22_11__acute | |
| - per_sac__rah | |
| - lam__os | |
| - t22_8__per | |
| - t22_9__obtuse | |
| - lt_sac__acute | |
| - cong2_lam2__cong_conga | |
| - lam6521_mid2__sac | |
| - sac__pars1423 | |
| - saccheri_s_three_hypotheses | |
| - three_hypotheses_aux | |
| - sac_distincts | |
| - lam_lt__obtuse | |
| - lam_per__cong | |
| - t22_8__lt1256 | |
| - lam__par1234 | |
| - ex_saccheri | |
| - t22_17__aah | |
| - t22_8__cong | |
| - t22_7__acute | |
| - per2_os__ncol123 | |
| - lam_lt__acute | |
| - cop_sac2__sac | |
| - path: theories/Main/Tarski_dev/Ch12_parallel_inter_dec.v | |
| theorems: | |
| - par_trans | |
| - not_par_inter | |
| - cop_npar__inter | |
| - l12_19 | |
| - col_par_par_col | |
| - l12_21_a | |
| - not_par_inter_exists | |
| - par_not_par | |
| - cop_npars__inter_exists | |
| - not_par_strict_inter_exists | |
| - l12_20 | |
| - l12_22_a | |
| - l12_16_2D | |
| - cop2_npar__inter | |
| - cop_par__inter | |
| - cop_npar__inter_exists | |
| - cop_par_perp__perp | |
| - l12_21 | |
| - cop4_par_perp2__par | |
| - par_perp__perp | |
| - par_perp2__par | |
| - par_inter | |
| - l12_16 | |
| - l12_22 | |
| - inter__npar | |
| - path: theories/Main/Tarski_dev/Ch16_coordinates_with_functions.v | |
| theorems: | |
| - centroid_theorem | |
| - twice_signed_area_ABB | |
| - sum_f | |
| - characterization_of_perpendicularity_F_bis | |
| - field_prop | |
| - characterization_of_neq_F | |
| - characterization_of_neq_F_bis | |
| - MulF | |
| - characterization_of_right_triangle_F | |
| - characterization_of_midpoint_F | |
| - triangle_area | |
| - leF_Transitive | |
| - coordinates_of_point_F | |
| - sqrt3_square | |
| - characterization_of_equality_F_bis | |
| - axiom_A2b | |
| - DivF | |
| - ringF | |
| - diff_col | |
| - characterization_of_parallelism_F_bis | |
| - eqF_Equivalence | |
| - field_prop_1 | |
| - neq20 | |
| - oppF_neq0 | |
| - characterization_of_parallelism_F_aux | |
| - signed_area_ABB | |
| - triangles_same_base | |
| - Cd_Cd_EqF | |
| - eqF_Transitive | |
| - prod_col | |
| - twice_signed_area_cyclic | |
| - characterization_of_equality_F_aux | |
| - pythrel_null | |
| - sum_col | |
| - inv_f | |
| - Perp_AM_Perp | |
| - div_uniqueness | |
| - neqO_mul_neqO | |
| - InvF | |
| - diff_f | |
| - AM_Perp_AM_Perp_AM_Par | |
| - neg_and_eqF | |
| - fieldF | |
| - co_side | |
| - FRing | |
| - div_f | |
| - chasles_ratios | |
| - Par_AM_Par | |
| - inv_uniqueness | |
| - Fcri | |
| - AddF | |
| - inv_col | |
| - ratio_zero | |
| - prod_f | |
| - pythrel_not_null | |
| - characterization_of_congruence_F | |
| - field_prop_zero | |
| - eqF_Reflexive | |
| - Per_AM_Per | |
| - eq_dec_F | |
| - Py_triv_ABB | |
| - Cong_AM_Cong | |
| - characterization_of_betweenness_F | |
| - AM_Perp_triv1 | |
| - PythF | |
| - signed_area_cyclic | |
| - Pyth_f | |
| - mulF_morphism | |
| - exists_equilateral_triangle | |
| - ncolOEE' | |
| - axiom_A2a | |
| - AM_Perp_triv2 | |
| - OneF | |
| - divF_morphism | |
| - addF_morphism | |
| - invF_morphism | |
| - Fintegral | |
| - AM_Par_ratio_AM_Par | |
| - opp_f | |
| - div_exists | |
| - mulF__eq0 | |
| - div_col | |
| - leF_Antisymmetric | |
| - subF__eq0 | |
| - AM_perp_AM_Par_AM_perp | |
| - subF_morphism | |
| - coordinates_of_point_f | |
| - opp_col | |
| - Ps_One | |
| - eqF_Symmetric | |
| - PythFOk | |
| - pythrelOO | |
| - Fmult_integral | |
| - characterization_of_parallelism_F | |
| - OppF | |
| - perp_triangle_area | |
| - inv_exists_with_notation | |
| - opp_pythrel | |
| - oppF_morphism | |
| - supplement_AM_CongA | |
| - path: theories/Main/Highschool/circumcenter.v | |
| theorems: | |
| - midpoint_thales_reci_circum | |
| - is_circumcenter_uniqueness | |
| - is_circumcenter_coplanar | |
| - is_circumcenter_perm_4 | |
| - is_circumcenter_perm_3 | |
| - circumcenter_per | |
| - is_circumcenter_perm_5 | |
| - is_circumcenter_perm | |
| - circumcenter_cong | |
| - circumcenter_perp_all | |
| - circumcenter_perp | |
| - is_circumcenter_cases | |
| - is_circumcenter_perm_1 | |
| - exists_circumcenter | |
| - circumcenter_intersect | |
| - is_circumcenter_perm_2 | |
| - path: theories/Algebraic/Counter_models/Planar/counter_model_five_segment.v | |
| theorems: | |
| - euclid | |
| - bet_1D | |
| - five_segment | |
| - all_lines_meet | |
| - col_xxa | |
| - bet'P | |
| - bet'_sym | |
| - segment_construction | |
| - bet_1Di | |
| - nbet_cab | |
| - nbet_bca | |
| - bet_symmetry | |
| - nbetS_bca | |
| - cong_inner_transitivity | |
| - col_213 | |
| - neq_mx10 | |
| - col_axx | |
| - inner_pasch | |
| - line_intersection_1 | |
| - nbetS_abc | |
| - permP | |
| - col_xax | |
| - cong_pseudo_reflexivity | |
| - neq_mx | |
| - point_equality_decidability | |
| - dup_meet | |
| - upper_dim | |
| - nbet_abc | |
| - lower_dim | |
| - line_intersection_2 | |
| - line_intersection_3 | |
| - path: theories/Main/Elements_statements/Book_3.v | |
| theorems: | |
| - prop_11_12 | |
| - prop_3_1 | |
| - prop_6 | |
| - prop_18 | |
| - prop_5 | |
| - prop_4 | |
| - prop_2 | |
| - prop_9 | |
| - prop_3_2 | |
| - path: theories/Elements/OriginalProofs/lemma_8_3.v | |
| theorems: | |
| - lemma_8_3 | |
| - path: theories/Algebraic/Counter_models/nD/bet_identity.v | |
| theorems: | |
| - g2_16 | |
| - g2_3 | |
| - bet_identity_aux | |
| - g2_14 | |
| - g2_13 | |
| - g2_7_2 | |
| - g2_4 | |
| - bet_outer_trans | |
| - g2_6_1 | |
| - g2_10 | |
| - g2_9 | |
| - g2_7_1 | |
| - g2_9_aux | |
| - g2_8 | |
| - g2_12 | |
| - g2_15 | |
| - another_point | |
| - path: theories/Main/Tarski_dev/Ch08_orthogonality.v | |
| theorems: | |
| - diff_per_diff | |
| - perp_in_left_comm | |
| - l8_22 | |
| - l8_14_2_1a | |
| - perp_in_comm | |
| - perp_col4 | |
| - Perp_in_cases | |
| - perp_right_comm | |
| - Per_perm | |
| - perp_in_per_1 | |
| - per_cong | |
| - perp_perp_in | |
| - perp_in_right_comm | |
| - perp_not_eq_2 | |
| - perp_per_1 | |
| - l8_10 | |
| - perp_in_per_4 | |
| - per_perp | |
| - l8_24 | |
| - l8_13_2 | |
| - perp_in_perp_bis | |
| - l8_5 | |
| - l8_9 | |
| - l8_18_uniqueness | |
| - perp_in_id | |
| - perp_in_per | |
| - l8_2 | |
| - l8_14_3 | |
| - l8_4 | |
| - perp_not_eq_1 | |
| - l8_20_1 | |
| - l8_21_aux | |
| - perp_col2_bis | |
| - midpoint_distinct | |
| - l8_18_existence | |
| - l8_15_2 | |
| - perp_per_2 | |
| - perp_col1 | |
| - col_col_per_per | |
| - perp_in_sym | |
| - l8_15_1 | |
| - perp_distinct | |
| - midpoint_existence_aux | |
| - perp_in_per_2 | |
| - Perp_in_perm | |
| - perp_not_col2 | |
| - Perp_cases | |
| - per_not_colp | |
| - perp_col0 | |
| - l8_12 | |
| - perp_vector | |
| - perp_left_comm | |
| - col_per2__per | |
| - perp_not_col | |
| - Per_cases | |
| - Perp_perm | |
| - per_distinct | |
| - perp_in_distinct | |
| - per_cong_mid | |
| - l8_14_2_2 | |
| - l8_14_2_1b | |
| - cong_perp_or_mid | |
| - l8_6 | |
| - perp_in_col | |
| - per_col | |
| - l8_7 | |
| - perp_col | |
| - perp_in_dec | |
| - col_per_perp | |
| - l8_22_bis | |
| - per_double_cong | |
| - perp_sym | |
| - perp_in_per_3 | |
| - midpoint_existence | |
| - l8_20_2 | |
| - l8_21 | |
| - l8_14_1 | |
| - perp_comm | |
| - per_not_col | |
| - path: theories/Main/Tarski_dev/Ch13_3_angles.v | |
| theorems: | |
| - anga_out_anga | |
| - acute_not_bet | |
| - eqA_equivalence | |
| - ang_not_null_lg | |
| - ang_const_o | |
| - not_conga_is_ang | |
| - ang_cong_ang | |
| - ang_sym | |
| - ex_eqaa | |
| - ang_const | |
| - is_ang_conga | |
| - is_null_all | |
| - null_anga_null_anga' | |
| - null_anga | |
| - ex_eqa | |
| - anga_col_null | |
| - not_flat_ang_def_equiv | |
| - not_null_ang_ang | |
| - flat_ang | |
| - not_null_ang_def_equiv | |
| - is_anga_to_is_ang | |
| - ex_ang | |
| - anga_distincts | |
| - anga_sym | |
| - ang_not_lg_null | |
| - is_anga_conga | |
| - out_out_anga | |
| - out_is_len_eq | |
| - not_cong_is_ang1 | |
| - is_null_anga_out | |
| - anga_exists | |
| - out_null_anga | |
| - ang_exists | |
| - anga_not_flat | |
| - not_conga_not_ang | |
| - bet_flat_ang | |
| - is_ang_conga_is_ang | |
| - out_null_ang | |
| - ex_points_ang | |
| - all_eqa | |
| - anga_const | |
| - eqA_preserves_anga | |
| - not_cong_is_anga1 | |
| - out_len_eq | |
| - anga_conga | |
| - ang_conga | |
| - ex_anga | |
| - anga_const_o | |
| - ang_distinct | |
| - ex_points_anga | |
| - anga_is_ang | |
| - anga_not_lg_null | |
| - path: theories/Algebraic/Counter_models/nD/counter_model_euclid.v | |
| theorems: | |
| - t'_in_unit_disk | |
| - segment_addition' | |
| - omd_ge0 | |
| - point_equality_decidability | |
| - omd_i1i2 | |
| - cong_inner_transitivity | |
| - euclid | |
| - dist_neq1 | |
| - point_vector_neq | |
| - p_scalar_lt1 | |
| - omd_i2i2 | |
| - cauchy_schwartz | |
| - E_in_disk | |
| - cong_v_bcaa | |
| - five_segment_holds | |
| - segment_construction_holds_aux4 | |
| - cong_aaxx | |
| - omd_ii | |
| - bet_symmetry | |
| - cong_pseudo_reflexivity_vector | |
| - bet_outer_transitivity' | |
| - bet_intersect | |
| - inner_pasch_holds | |
| - segment_construction_holds_aux3 | |
| - five_segment_holds_aux | |
| - dist_eq1' | |
| - addcb_eqt | |
| - segment_construction_holds_aux6 | |
| - b'_in_unit_disk | |
| - nbet_cab | |
| - i'_in_unit_disk | |
| - bet_b1d1c | |
| - b'c'_neq | |
| - cong_id | |
| - bet_in_disk | |
| - b'nD_in_unit_disk | |
| - cong_cdab | |
| - divf_le1 | |
| - omd_oi | |
| - pasch | |
| - five_segment_holds_aux3 | |
| - roots_poly2 | |
| - cong_v_aabc | |
| - betR_abk | |
| - bet_adt | |
| - bet_abx_x01_lt1 | |
| - omd_cb | |
| - one_half_gt0 | |
| - cong_identity | |
| - c'_in_unit_disk | |
| - three_non_collinear_points | |
| - cong_pseudo_reflexivity | |
| - sqrt_invf | |
| - b'nD_neq0 | |
| - a'_in_unit_disk | |
| - inner_pasch_vector | |
| - scaler_eq | |
| - bet_abx_x01_gt | |
| - inner_pasch | |
| - five_segment_holds_aux4 | |
| - omd_eq0 | |
| - one_half_neq0 | |
| - segment_construction | |
| - quad_kNN | |
| - divf_eq1 | |
| - dist_reflexivity | |
| - cong_inner_transitivity' | |
| - one_sub_half | |
| - omd_oi2 | |
| - betR_abc | |
| - one_half_ge0 | |
| - dist_cong' | |
| - euclid_fails | |
| - norm_sqr | |
| - betR_cab | |
| - omd_reflexivity | |
| - Rcf_to_GI2D | |
| - o'_in_unit_disk | |
| - quad_NN | |
| - cong_inner_transitivity_vector | |
| - invf_eq | |
| - bet_dist_bet' | |
| - betR_akb | |
| - invf_lt_invf | |
| - bet_inner_transitivity | |
| - cong_bacd | |
| - Rcf_to_GI_PED' | |
| - one_quarter_lt_one_half | |
| - lower_dim_holds | |
| - omd_ca | |
| - addr_ge_gt0 | |
| - norm_point_lt1 | |
| - c'_norm | |
| - five_segment | |
| - dist'_ge0 | |
| - segment_construction_holds_aux2 | |
| - omd_oo | |
| - one_eighth_lt_one_quarter | |
| - xt_neq | |
| - a'_eq0 | |
| - bet_2D_extension_2D | |
| - omd_cc | |
| - b'_neq0 | |
| - omd_v_reflexivity | |
| - norm_point_le1 | |
| - bet_inner_transitivity_vector | |
| - a'nD_in_unit_disk | |
| - five_segment_holds_aux5 | |
| - omd_oi1 | |
| - upper_dim_aux | |
| - betR_bca | |
| - add_halfhalf | |
| - nbet_abc | |
| - three_non_collinear_points_vector | |
| - upper_dim_aux_2 | |
| - one_half_lt1 | |
| - unit_disk_nD_2D | |
| - dist_gt0 | |
| - euclid_aux | |
| - point_vector_eq | |
| - congA_axt_ab1d1 | |
| - omd_gt0 | |
| - euclid_fails_2D | |
| - dist_v_reflexivity | |
| - betR_neq0 | |
| - cong'_abab | |
| - one_side_cong_eq | |
| - Col_Col | |
| - bet_outer_transitivity | |
| - quarter | |
| - segment_construction_holds_aux5 | |
| - norm_ge0 | |
| - omd_ii2 | |
| - cong_identity_holds_aux | |
| - dist_le1 | |
| - euclid_fails_aux | |
| - segment_construction_holds_aux1 | |
| - divf_lt1 | |
| - dist_aa_eq1 | |
| - x_neq0 | |
| - poly_cong | |
| - bet_bdc | |
| - cauchy_schwartz_eq | |
| - segment_construction_holds | |
| - seg_aux8 | |
| - X_in_disk | |
| - bet_abx_x00_eq0 | |
| - path: theories/Elements/OriginalProofs/lemma_lessthantransitive.v | |
| theorems: | |
| - lemma_lessthantransitive | |
| - path: theories/Main/Tarski_dev/Ch04_cong_bet.v | |
| theorems: | |
| - l4_3 | |
| - cong3_bet_eq | |
| - l4_5 | |
| - l4_2 | |
| - l4_6 | |
| - l4_3_1 | |
| - path: theories/Main/Tarski_dev/Ch15_lengths.v | |
| theorems: | |
| - pythagoras_obtuse | |
| - not_triangular_equality1 | |
| - square_increase_rev | |
| - ltp_to_lep | |
| - inter_circle_per | |
| - pos_neg__prod_neg | |
| - length_uniqueness | |
| - length_sym | |
| - length_id_1 | |
| - ltp__ltps | |
| - length_cong | |
| - inter_circle | |
| - square_increase_strict | |
| - prod_pos__signeq | |
| - l15_7_1 | |
| - lt_to_ltp | |
| - length_existence | |
| - image_preserves_col | |
| - not_signEq_prod_neg | |
| - l15_3 | |
| - length_out | |
| - conga_bet_conga | |
| - ltp__lep_neq | |
| - length_leP_le_2 | |
| - triangular_equality_equiv | |
| - sum_preserves_lep_rev | |
| - inter_circle_obtuse | |
| - length_not_neg | |
| - square_increase | |
| - lta_out_lta | |
| - lep_neq__ltp | |
| - leP_bet | |
| - circle_projp_between | |
| - pythagoras | |
| - triangular_equality_bis | |
| - ltp__diff_pos | |
| - length_id | |
| - sum_pos_null | |
| - prod_col | |
| - length_pos_or_null | |
| - sum_preserves_ltp_rev | |
| - length_eq_cong_1 | |
| - prod_ng___not_signeq | |
| - lea_out_lea | |
| - ltps__ltp | |
| - inter_tangent_circle | |
| - length_Ps | |
| - root_uniqueness | |
| - not_neg_pos | |
| - length_eq_cong_2 | |
| - length_O | |
| - ltp_to_lt | |
| - length_Ar2 | |
| - is_length_exists | |
| - length_pos | |
| - pythagoras_acute | |
| - diff_pos__ltp | |
| - project_preserves_out | |
| - triangular_equality | |
| - length_not_col_null | |
| - image_preserves_bet1 | |
| - sum_preserves_lep | |
| - l15_7_2 | |
| - projp_lt | |
| - sum_preserves_ltp | |
| - length_leP_le_1 | |
| - square_not_neg | |
| - path: theories/Main/Highschool/concyclic.v | |
| theorems: | |
| - concyclic_perm_2 | |
| - concyclic_perm_9 | |
| - concyclic_perm_5 | |
| - concyclic_perm_7 | |
| - concyclic_perm_3 | |
| - concyclic_perm_19 | |
| - concyclic_perm_10 | |
| - concyclic_trans | |
| - concyclic_perm_8 | |
| - concyclic_perm_18 | |
| - concyclic_perm_11 | |
| - concyclic_perm_4 | |
| - concyclic_perm_21 | |
| - concyclic_perm_16 | |
| - concyclic_aux | |
| - concyclic_perm_15 | |
| - concyclic_perm_23 | |
| - concyclic_1123 | |
| - concyclic_perm_20 | |
| - concyclic_perm_6 | |
| - concyclic_perm_1 | |
| - concyclic_perm_13 | |
| - concyclic_perm_14 | |
| - concyclic_perm_12 | |
| - concyclic_perm_22 | |
| - path: theories/Elements/OriginalProofs/lemma_collinear1.v | |
| theorems: | |
| - lemma_collinear1 | |
| - path: theories/Main/Annexes/circles.v | |
| theorems: | |
| - bet_cop_onc2__ex_onc_os_out | |
| - inc112 | |
| - mid_chord__diam_or_ncol | |
| - outcs_exists1 | |
| - circle_cases | |
| - line_circle_two_points | |
| - eqc_chara | |
| - tree_points_onc_cop | |
| - cong2_onc3__eq | |
| - onc2__cong | |
| - concyclic_pseudo_trans | |
| - col_onc2__mid | |
| - onc_not_center | |
| - mid2_onc4__eq | |
| - outcs_exists | |
| - incs_exists | |
| - inc_onc2_out__eq | |
| - tree_points_onc_cop2 | |
| - diam_points | |
| - symmetric_oncircle | |
| - onc212 | |
| - chord_completion | |
| - chord_lt_diam | |
| - bet_onc_le_a | |
| - onc_exists | |
| - bet_onc_lt_b | |
| - col_onc2_perp__mid | |
| - ninc__outcs | |
| - prop_7_8 | |
| - bet_onc_lt_a | |
| - inc__radius | |
| - onc2_mid__incs | |
| - center_onc2_mid__ncol | |
| - mid_onc2__per | |
| - chords_midpoints_col_par | |
| - inc__noutcs | |
| - incs__inc | |
| - inc__outc | |
| - inc_eq | |
| - bet_inc2__incs | |
| - inc2_le_diam | |
| - bet_cong_onc3_cases | |
| - mid_onc2_perp__col | |
| - onc3_mid2__ncol | |
| - eqc_trans | |
| - incs2_lt_diam | |
| - inc_outc__onc | |
| - chord_le_diam | |
| - onc_sym | |
| - diam_cong_incs__outcs | |
| - onc__outc | |
| - bet_onc_le_b | |
| - outc__nincs | |
| - onc2_out__outcs | |
| - chord_intersection | |
| - concyclic_gen_pseudo_trans | |
| - mid_onc__diam | |
| - diam__midpoint | |
| - concyclic_gen_perm_2 | |
| - concyclic_gen_trans_1 | |
| - onc2_mid_cong_col | |
| - concyclic_perm_2 | |
| - cop2_onc6__eqc | |
| - outcs__ninc | |
| - onc4_cong2__eq | |
| - neqc_chara | |
| - concyclic_trans_1 | |
| - cop_mid_onc2_perp__col | |
| - col_inc2_outcs__out | |
| - circle_circle_cop | |
| - eqc_refl | |
| - incs_exists1 | |
| - diam_end_uniqueness | |
| - cong2_cop2_onc3__eq | |
| - diam_sym | |
| - cong_chord_cong_center1 | |
| - incs__outcs | |
| - concyclic_aux | |
| - cong_chord_cong_center | |
| - col_inc_onc2__bet | |
| - concyclic_perm_1 | |
| - diam_exists | |
| - cong_onc3_cases | |
| - onc_col_diam__eq | |
| - mid_onc2__perp | |
| - onc3__ncol | |
| - onc_sym__onc | |
| - incs__noutc | |
| - eqc_sym | |
| - incs_onc_diam__lt | |
| - tree_points_onc | |
| - concyclic_gen_perm_1 | |
| - center_col__diam | |
| - path: theories/Main/Tarski_dev/Ch09_plane.v | |
| theorems: | |
| - sym_preserve_diff | |
| - col2_cop__cop | |
| - bet_cop__tsp | |
| - tsp__nosp | |
| - two_sides_dec | |
| - l9_41_1 | |
| - out_one_side | |
| - tsp_exists | |
| - l9_9 | |
| - l9_10 | |
| - col_one_side_out | |
| - l9_4_2 | |
| - col_preserves_two_sides | |
| - l9_3 | |
| - out_two_sides_two_sides | |
| - tsp__ncop2 | |
| - cop2_os__osp | |
| - l8_21_bis | |
| - l9_19_3 | |
| - cop3_tsp__tsp | |
| - osp__ncop2 | |
| - l9_4_2_aux | |
| - l9_18 | |
| - l9_9_bis | |
| - tsp_distincts | |
| - sac__coplanar | |
| - one_side_transitivity | |
| - os_ts1324__os | |
| - os__coplanar | |
| - col2_os__os | |
| - cop3_tsp__ts | |
| - l9_30 | |
| - bet_ts__os | |
| - one_or_two_sides_aux | |
| - col_one_side | |
| - col123__nos | |
| - ts__ncol | |
| - l9_18_3 | |
| - bet__ts | |
| - one_side_dec | |
| - bet_ts__ts | |
| - cop_tsp__ex_cop2 | |
| - ex_ncol_cop | |
| - not_two_sides_id | |
| - os_distincts | |
| - cong3_cop2__col | |
| - l9_8_2 | |
| - invert_one_side | |
| - cop_nts__os | |
| - l9_31 | |
| - per_mid_per | |
| - out_out_one_side | |
| - osp__ncop1 | |
| - ts_ts_os | |
| - cop_perp2__col | |
| - osp_bet__osp | |
| - cop3_osp__os | |
| - tsp__ncop1 | |
| - one_side_chara | |
| - col_cop__cop | |
| - coplanar_trans_1 | |
| - l9_2 | |
| - ex_diff_cop | |
| - one_side_not_col124 | |
| - col2_cop2__eq | |
| - mid_two_sides | |
| - col_two_sides | |
| - l9_5 | |
| - ncop_distincts | |
| - out_out_two_sides | |
| - osp_symmetry | |
| - osp_reflexivity | |
| - out_one_side_1 | |
| - ts_distincts | |
| - outer_pasch | |
| - cop_nos__ts | |
| - cop_per2__col | |
| - l9_39 | |
| - cop_dec | |
| - l9_41_2 | |
| - l9_4_1_aux | |
| - l9_8_1 | |
| - one_side_not_col123 | |
| - osp__ntsp | |
| - bet_cop__cop | |
| - col124__nos | |
| - cop2_ts__tsp | |
| - col_two_sides_bet | |
| - l9_4_1 | |
| - osp_distincts | |
| - path: theories/Main/Annexes/inscribed_angle.v | |
| theorems: | |
| - conga_cop2_onc4__os | |
| - cop_onc4_ts__suppa | |
| - high_school_exterior_angle_theorem | |
| - onc3_ts__obtuse | |
| - inscribed_angle_aux1 | |
| - not_obtuse_saccheris | |
| - suma123231__sams | |
| - inscribed_angle_aux | |
| - trisuma__bet | |
| - cop2_onc4__or_conga_suppa | |
| - thales_theorem | |
| - right_saccheris | |
| - onc3_os__acute | |
| - cop2_onc4_suppa__ts | |
| - conga_cop_onc6_os__eqc | |
| - chord_par_diam | |
| - thales_converse_theorem_1 | |
| - bet__trisuma | |
| - diam_onc2_ts__suppa | |
| - cop_onc4_os__conga | |
| - conga_cop_onc3_os__onc | |
| - bet_suma__suma | |
| - acute_cop_onc3__os | |
| - suppa_ts2__suppa | |
| - thales_converse_theorem | |
| - inscribed_angle | |
| - triangle_circumscription | |
| - inscribed_angle_1 | |
| - conga_os__concyclic | |
| - suppa_ts__concyclic | |
| - path: theories/Elements/OriginalProofs/lemma_parallelcollinear1.v | |
| theorems: | |
| - lemma_parallelcollinear1 | |
| - path: theories/Algebraic/Counter_models/nD/dimensional_axioms.v | |
| theorems: | |
| - iP | |
| - nth_new_basis | |
| - eq_nD_2D | |
| - betR_ud_2D | |
| - lower_dim_all1 | |
| - lower_dim | |
| - to_nD0 | |
| - betR_o_i_basis_nth0 | |
| - dist_10_01 | |
| - upper_dimS | |
| - i_neq_basis_nth0 | |
| - betS_nD_2D | |
| - oP | |
| - to_nD_scale | |
| - lower_dim_all2 | |
| - upper_dim | |
| - to_nDB | |
| - pick_to_nD | |
| - bet_nD_2D | |
| - bet_o_i_basis_nth0 | |
| - betR_nD_2D | |
| - basis_nth1 | |
| - basis_nth0 | |
| - path: theories/Main/Annexes/quadrilaterals.v | |
| theorems: | |
| - plg_trivial | |
| - Rectangle_not_triv | |
| - not_col_sym_not_col | |
| - plgf3_mid | |
| - col_not_plgs | |
| - plgs_irreflexive | |
| - plg_existence | |
| - parallelogram_strict_not_col | |
| - plgf_permut | |
| - plgf_trivial1 | |
| - Rectangle_triv | |
| - plg_trivial1 | |
| - plgf_not_comm | |
| - rhombus_cong_square | |
| - midpoint_par_strict | |
| - cong3_id | |
| - plgs_one_side | |
| - plgf_cong | |
| - plg_col_plgf | |
| - plgf_trivial | |
| - bet_double_bet | |
| - plgs_diff | |
| - cong_identity_inv | |
| - midpoint_preserves_bet | |
| - mid_plgf_aux | |
| - plgf_trivial_trans | |
| - plg_bet1 | |
| - bet3_cong3_bet | |
| - symmetry_preserves_two_sides | |
| - symmetry_preserves_one_side | |
| - plg_sym | |
| - symmetry_preseves_bet2 | |
| - plgs__pars | |
| - Square_Parallelogram | |
| - plgf_mid | |
| - Parallelogram_strict_Parallelogram | |
| - mid_par_cong2 | |
| - Rhombus_Plg | |
| - Rectangle_not_triv_2 | |
| - Kite_comm | |
| - Rectangle_Plg | |
| - plgf_irreflexive | |
| - col_cong_mid | |
| - symmetry_preserves_bet | |
| - plgf_not_point | |
| - parallelogram_strict_not_col_3 | |
| - plgs_not_col | |
| - parallelogram_strict_not_col_4 | |
| - plg_cong_rectangle | |
| - symmetry_preseves_bet1 | |
| - plgf_bet | |
| - plgs_sym | |
| - plgf_sym | |
| - mid_par_cong | |
| - plgf_trivial2 | |
| - mid_par_cong1 | |
| - mid_plgf | |
| - plgf_trivial_neq | |
| - midpoint_midpoint_col | |
| - mid_plgs | |
| - plg_irreflexive | |
| - midpoint_cong_uniqueness | |
| - mid_plg | |
| - plgs_existence | |
| - path: theories/Main/Meta_theory/Models/hilbert_to_tarski.v | |
| theorems: | |
| - outH_sym | |
| - Gupta_inspired_variant_neutral_dimensionless_follows_from_Hilbert | |
| - Hilbert_is_a_Col_theory | |
| - between_one | |
| - not_cut3 | |
| - plane_separation | |
| - lower_dim_l | |
| - ncolH_distincts | |
| - ncolH_expand | |
| - cut_morphism | |
| - plane_separation_2D | |
| - cong_distincts | |
| - soustraction_betH | |
| - H_to_T_PED | |
| - col_colh | |
| - betH_congH3_outH_betH | |
| - cut_distinct | |
| - colH_trivial111 | |
| - cut2_not_cut | |
| - TS_upper_dim | |
| - cong_transitivity | |
| - tarski_s_euclid | |
| - morph | |
| - inter_uniquenessH | |
| - cong_preserves_bet | |
| - H_to_T | |
| - coplanar_plane1 | |
| - betH_distincts | |
| - cut_comm | |
| - outH_expand | |
| - ColH_bets | |
| - IncidL_morphism' | |
| - axiom_five_segmentsH | |
| - congH_sym | |
| - congaH_existence_congH | |
| - IncidL_not_IncidL__not_colH | |
| - five_segment | |
| - cong_identity | |
| - line_on_plane' | |
| - inner_pasch_aux | |
| - TS_upper_dim_bis | |
| - betH_trans1 | |
| - segment_construction | |
| - bet_trans | |
| - colH_trivial112 | |
| - IncidP_morphism' | |
| - congH_perm | |
| - same_side_refl | |
| - colH_dec | |
| - colH_permut_312 | |
| - cong_preserves_col_stronger | |
| - ncolH_exists | |
| - decidability_of_intersectionH | |
| - same_side_trans | |
| - out_same_side | |
| - segment_constructionH | |
| - same_side_morphism | |
| - tarski_upper_dim | |
| - betH_congH2__False | |
| - betH_outH__outH | |
| - th18_aux | |
| - th12 | |
| - ColH__Col | |
| - outH_col | |
| - colH_permut_213 | |
| - same_side_not_cut | |
| - cut'_comm | |
| - cong_inner_transitivity | |
| - cut_exists | |
| - not_betH121 | |
| - out_construction | |
| - th15 | |
| - colH_permut_231 | |
| - Col__ColH | |
| - other_point_on_line | |
| - out_distinct | |
| - cong_permT | |
| - inter_incid_uniquenessH | |
| - betH_trans0 | |
| - col_upper_dim | |
| - betH2_out | |
| - lower_dim_e | |
| - H2D_to_T2D | |
| - upper_dim | |
| - colH_trivial121 | |
| - colH_trivial122 | |
| - th17 | |
| - betH_line | |
| - th19 | |
| - IncidLP_morphismr | |
| - coplanar_plane | |
| - par__or_eq_para | |
| - congH_colH_betH | |
| - colH_permut_321 | |
| - out_same_side' | |
| - out2_out | |
| - bet__beth | |
| - Tarski_3D_follows_from_Hilbert_3D | |
| - th14 | |
| - same_side_prime_refl | |
| - betH_dec | |
| - between_only_one' | |
| - congH_perml | |
| - construction_uniqueness | |
| - coplanar_plane0 | |
| - pasch_general_case | |
| - colH_IncidL__IncidL | |
| - EqL_dec | |
| - betH_trans2 | |
| - betH_not_congH | |
| - IncidLP_morphisml | |
| - plane_coplanar | |
| - colH_permut_132 | |
| - congH_permlr | |
| - cong_sym | |
| - same_side_prime_not_colH | |
| - congaH_sym | |
| - bet_cong3_bet | |
| - cut_same_side_cut | |
| - cong_existence' | |
| - IncidLP_morphism | |
| - same_side__plane | |
| - bet_disjoint | |
| - bet_comm | |
| - cong_preserves_col | |
| - path: theories/Main/Tarski_dev/Ch05_bet_le.v | |
| theorems: | |
| - le_comm | |
| - bet3__bet | |
| - nlt__le | |
| - lt_diff | |
| - le_transitivity | |
| - lt1123 | |
| - lt_right_comm | |
| - l5_2 | |
| - bet2_le2__le | |
| - Lt_cases | |
| - l5_6 | |
| - le1221 | |
| - col_dec | |
| - le_trivial | |
| - bet__lt1213 | |
| - l5_3 | |
| - l5_12_b | |
| - cong3_symmetry | |
| - bet_le_eq | |
| - nle__lt | |
| - le_zero | |
| - bet_cong_eq | |
| - lt_transitivity | |
| - segment_construction_2 | |
| - l5_12_a | |
| - le3456_lt__lt | |
| - le_anti_symmetry | |
| - or_lt_cong_gt | |
| - cong2_lt__lt | |
| - fourth_point | |
| - le_bet | |
| - between_cong | |
| - bet__le2313 | |
| - third_point | |
| - le_right_comm | |
| - gt_comm | |
| - between_cong_3 | |
| - lt_left_comm | |
| - le_diff | |
| - ge_left_comm | |
| - l5_5_1 | |
| - le1234_lt__lt | |
| - lt__nle | |
| - bet_dec | |
| - lt__le | |
| - not_and_lt | |
| - cong_dec | |
| - ge_comm | |
| - le_cases | |
| - Le_cases | |
| - l5_5_2 | |
| - bet__le1213 | |
| - cong__le | |
| - l5_1 | |
| - ge_right_comm | |
| - bet__lt2313 | |
| - gt_right_comm | |
| - le_left_comm | |
| - gt_left_comm | |
| - path: theories/Algebraic/Counter_models/Planar/counter_model_bet_symmetry.v | |
| theorems: | |
| - col_xxa | |
| - bet_inner_transitivity | |
| - point_equality_decidability | |
| - bet_col' | |
| - coplanarP | |
| - inner_pasch | |
| - cong_pseudo_reflexivity | |
| - euclid | |
| - five_segment | |
| - col_perm_2 | |
| - A10_segment_construction | |
| - lower_dim | |
| - col_2D_bet' | |
| - bet_on_abscissa | |
| - col_trans | |
| - colP | |
| - upper_dim | |
| - ncolP | |
| - par_strictP | |
| - cong_identity | |
| - col_perm_1 | |
| - bet'_col_2D | |
| - bet_symmetry | |
| - path: theories/Main/Annexes/sums.v | |
| theorems: | |
| - le_lt12_sums2__lt34 | |
| - le_lt56_sums2__lt12 | |
| - sums__eq12 | |
| - le2_sums2__le | |
| - sums112323 | |
| - sums_comm | |
| - sums_assoc | |
| - sums2__cong34 | |
| - sums_right_comm | |
| - eq_sums__eq | |
| - lt2_sums2__lt12 | |
| - le2_sums2__le12 | |
| - sums__cong1245 | |
| - sums__le3456 | |
| - le2_sums2__cong34 | |
| - sums2__cong56 | |
| - le_lt56_sums2__lt34 | |
| - sums123312 | |
| - lt2_sums2__lt34 | |
| - sums__cong2345 | |
| - lt2_sums2__lt | |
| - sums_left_comm | |
| - le_lt34_sums2__lt12 | |
| - le2_sums2__cong12 | |
| - cong3_sums__sums | |
| - sums_middle_comm | |
| - sums2__cong12 | |
| - bet__sums | |
| - sums__le1256 | |
| - ex_sums | |
| - le_lt12_sums2__lt | |
| - le_lt34_sums2__lt | |
| - sums_diff_2 | |
| - sums_assoc_1 | |
| - path: theories/Algebraic/Counter_models/Planar/counter_model_segment_construction.v | |
| theorems: | |
| - point_equality_decidability | |
| - upper_dim | |
| - bet_symmetry | |
| - inner_pasch | |
| - segment_construction | |
| - cong_inner_transitivity | |
| - cong_pseudo_reflexivity | |
| - cong_identity | |
| - lower_dim | |
| - five_segment | |
| - continuity | |
| - euclid | |
| - bet_inner_transitivity | |
| - path: theories/Main/Highschool/gravityCenter.v | |
| theorems: | |
| - intersection_two_medians_exist | |
| - is_gravity_center_perm_4 | |
| - is_gravity_center_perm_1 | |
| - is_gravity_center_exist_unique | |
| - is_gravity_center_coplanar | |
| - is_gravity_center_diff_2 | |
| - three_medians_intersect | |
| - is_gravity_center_third_reci | |
| - is_gravity_center_diff_3 | |
| - is_gravity_center_perm_3 | |
| - is_gravity_center_col | |
| - is_gravity_center_perm | |
| - is_gravity_center_perm_2 | |
| - is_gravity_center_cases | |
| - is_gravity_center_diff_1 | |
| - is_gravity_center_perm_5 | |
| - path: theories/Main/Annexes/coplanar.v | |
| theorems: | |
| - col__coplanar | |
| - coplanar_perm_15 | |
| - coplanar_perm_18 | |
| - coplanar_perm_2 | |
| - ncoplanar_perm_5 | |
| - ncoplanar_perm_7 | |
| - coplanar_perm_4 | |
| - coplanar_trivial | |
| - inangle__coplanar | |
| - ncoplanar_perm_3 | |
| - ncoplanar_perm_19 | |
| - ncop__ncol | |
| - pars__coplanar | |
| - plgf__coplanar | |
| - out__coplanar | |
| - square__coplanar | |
| - coplanar_perm_20 | |
| - coplanar_perm_1 | |
| - ncoplanar_perm_14 | |
| - ncoplanar_perm_1 | |
| - bet__coplanar | |
| - coplanar_perm_7 | |
| - rectangle__coplanar | |
| - ncoplanar_perm_2 | |
| - coplanar_perm_23 | |
| - ncoplanar_perm_23 | |
| - lambert__coplanar | |
| - coplanar_perm_10 | |
| - ncoplanar_perm_9 | |
| - coplanar_perm_14 | |
| - ncoplanar_perm_22 | |
| - ncoplanar_perm_18 | |
| - ncoplanar_perm_21 | |
| - coplanar_perm_5 | |
| - coplanar_perm_16 | |
| - coplanar_perm_12 | |
| - coplanar_perm_13 | |
| - coplanar_perm_6 | |
| - par__coplanar | |
| - ncoplanar_perm_16 | |
| - coplanar_perm_19 | |
| - parallelogram__coplanar | |
| - coplanar_perm_8 | |
| - ncoplanar_perm_15 | |
| - perp__coplanar | |
| - coplanar_perm_21 | |
| - rhombus__coplanar | |
| - ncoplanar_perm_6 | |
| - ncoplanar_perm_20 | |
| - coplanar_perm_22 | |
| - coplanar_perm_17 | |
| - ncoplanar_perm_8 | |
| - plg__coplanar | |
| - plgs__coplanar | |
| - ncoplanar_perm_12 | |
| - path: theories/Main/Tarski_dev/Ch10_line_reflexivity.v | |
| theorems: | |
| - l10_6_uniqueness_spec | |
| - l10_15 | |
| - is_image_rev | |
| - l10_6_uniqueness | |
| - l10_14 | |
| - col_image_spec__eq | |
| - is_image_spec_dec | |
| - image_spec_triv | |
| - image_id | |
| - is_image_col_cong | |
| - l10_2_uniqueness | |
| - l10_6_existence | |
| - col__image_spec | |
| - l10_2_existence | |
| - image_in_col | |
| - col__refl | |
| - image_in_is_image_spec | |
| - l10_2_uniqueness_spec | |
| - l10_7 | |
| - l10_8 | |
| - image__midpoint | |
| - image_image_in | |
| - ex_sym1 | |
| - l10_6_existence_spec | |
| - ex_per_cong | |
| - image_triv | |
| - l10_5 | |
| - exists_cong_per | |
| - image_spec__eq | |
| - cong_midpoint__image | |
| - is_image_spec_rev | |
| - path: theories/Main/Annexes/project.v | |
| theorems: | |
| - project_par_project | |
| - projp_idem | |
| - project_not_id | |
| - project_par | |
| - project_col_eq | |
| - project_par_dir | |
| - project_not_col | |
| - project_preserves_bet | |
| - eqv_eq_project | |
| - projp_id | |
| - triangle_par | |
| - project_id | |
| - proj_distinct | |
| - project_idem | |
| - par_col_project | |
| - ker_col | |
| - perp_projp2_eq | |
| - project_par_eqv | |
| - projp_preserves_eqv | |
| - projp_is_project_perp | |
| - project_project_par | |
| - col_2_par_projp2_cong | |
| - col_par_projp2_eq | |
| - eqv_project_eq_eq | |
| - project_uniqueness | |
| - eqv_cong | |
| - perp_projp | |
| - par3_conga3 | |
| - projp_projp_perp | |
| - ker_par | |
| - col_projp_eq | |
| - project_col | |
| - project_existence | |
| - projp_to_project | |
| - path: theories/Elements/OriginalProofs/lemma_TTflip.v | |
| theorems: | |
| - lemma_TTflip | |
| - path: theories/Main/Meta_theory/Dimension_axioms/upper_dim_2.v | |
| theorems: | |
| - upper_dim_implies_not_one_side_two_sides | |
| - upper_dim_implies_all_coplanar | |
| - upper_dim_stab | |
| - all_coplanar_implies_upper_dim | |
| - all_coplanar_upper_dim | |
| - upper_dim_implies_one_or_two_sides | |
| - upper_dim_implies_perp2__col | |
| - upper_dim_implies_not_two_sides_one_side | |
| - upper_dim_implies_per2__col | |
| - upper_dim_implies_col_perp2__col | |
| - upper_dim_implies_not_two_sides_one_side_aux | |
| - path: theories/Coinc/CoincR.v | |
| theorems: | |
| - CoappDupPerm | |
| - CPToSSHdTl | |
| - pick_varieties_ok_2 | |
| - memCPProper | |
| - memCPAuxProperOK | |
| - proper_0 | |
| - consHdInterpOK | |
| - ss_ok_inter_ok1 | |
| - memMemCPAuxOK | |
| - pick_variety_aux_memCPAux2 | |
| - identify_varieties | |
| - memCPAux | |
| - proper_2 | |
| - CPToSS | |
| - CPToSSOK | |
| - interp_CPTlOK | |
| - collect_wds | |
| - pick_variety_aux_memCPAux1 | |
| - memCPToSSOK | |
| - pick_variety_auxCP | |
| - memCPAuxTlOK | |
| - pick_variety_auxCP_existsF | |
| - CoappDupAux | |
| - ss_ok_empty | |
| - proper_00 | |
| - positive_dec | |
| - pick_varieties_ok_1 | |
| - interp_CPOK | |
| - pick_variety_auxCP_forallT | |
| - proper_3 | |
| - test_coinc_ok | |
| - PropToTagged | |
| - identify_varieties_ok | |
| - st_ok_empty | |
| - mca_pick_variety_aux_pca | |
| - ss_ok_inter_ok2 | |
| - CPToSSOKAux | |
| - exists_witness_ok | |
| - memMemCPOK | |
| - proper_1 | |
| - memCPAuxHdTl | |
| - collect_coincs | |
| - memCPConsHd | |
| - path: theories/Main/Utils/all_equiv.v | |
| theorems: | |
| - stronger2__stronger_right | |
| - stronger2__stronger_left | |
| - stronger_transitivity | |
| - all_equiv'_aux | |
| - all_equiv2_impl2__all_equiv | |
| - incl_preserves_stronger | |
| - all_equiv2_impl__stronger | |
| - all_equiv2_stronger2__all_equiv | |
| - all_equiv_under_chara | |
| - all_equiv' | |
| - all_equiv3_stronger3__all_equiv | |
| - all_equiv_chara | |
| - all_equiv'_auxP | |
| - all_equiv_trivial | |
| - all_equiv__equiv | |
| - incl_preserves_all_equiv | |
| - all_equiv3_impl3__all_equiv | |
| - path: theories/Main/Meta_theory/Models/tarski_to_beeson.v | |
| theorems: | |
| - between_identity_B | |
| - T_dec | |
| - NColB_NDiffCol | |
| - cong_stability | |
| - BetH_Bet | |
| - segment_construction_B | |
| - NT_NBet | |
| - T_Bet | |
| - Bet_T | |
| - between_symmetry_B | |
| - Diff_Col_ColB | |
| - NColB_NColOrEq | |
| - bet_stability | |
| - between_inner_transitivity_B | |
| - lower_dim_B | |
| - Beeson_follows_from_Tarski | |
| - path: theories/Main/Highschool/orientation.v | |
| theorems: | |
| - out_preserves_eq_o | |
| - eqo_refl | |
| - proj_comm | |
| - proj_not_col | |
| - per_preserves_bet_aux1 | |
| - proj_preserves_bet1 | |
| - proj_diff_not_col_inv | |
| - proj_diff_not_col | |
| - proj_eq_col | |
| - proj_preserves_bet | |
| - bet_le_le | |
| - proj_per | |
| - proj_inv_exists | |
| - per_id | |
| - col_proj_proj | |
| - proj_perp_id | |
| - eq_o_one_side | |
| - proj_id | |
| - symmetry_preseves_bet2 | |
| - proj_exists | |
| - bet_half_bet | |
| - bet_double_bet | |
| - one_side_eqo | |
| - perp_not_eq_3 | |
| - eq_o_eqo | |
| - proj3_id | |
| - proj_col_proj | |
| - eq_o_refl | |
| - per_preserves_bet_aux2 | |
| - one_side_eq_o | |
| - eqo_one_side | |
| - midpoint_preserves_bet | |
| - out_preserves_eqo | |
| - le_left_comm | |
| - cong_identity_inv | |
| - proj_uniqueness | |
| - midpoint_par | |
| - proj_col | |
| - le_right_comm | |
| - symmetry_preseves_bet1 | |
| - out_preserves_eqo1 | |
| - col_proj_col | |
| - eqo_eq_o | |
| - midpoint_par_strict | |
| - per_diff | |
| - proj_one_side | |
| - proj_par_strict | |
| - proj_diff | |
| - proj_not_eq_not_col | |
| - per_one_side | |
| - par_cong_mid | |
| - le_cong_le | |
| - per_proj | |
| - le_comm | |
| - proj_not_eq | |
| - path: theories/Main/Annexes/quadrilaterals_inter_dec.v | |
| theorems: | |
| - par_preserves_conga_os | |
| - plgs_permut | |
| - par_cong_plg | |
| - plg_not_comm | |
| - per_rmb | |
| - plg_per_rect2 | |
| - plgf_rect_id | |
| - plgf_plgs_trans | |
| - cop_perp3__perp | |
| - ncol123_plg__pars1234 | |
| - pars_par_plg | |
| - ncol124_plg__pars1234 | |
| - ts_cong_par_cong_par | |
| - par_cong_mid | |
| - rect_per | |
| - plg_not_comm_2 | |
| - plg_conga1 | |
| - par_cong3_rect | |
| - col_cong_cong | |
| - plg_uniqueness | |
| - rect_comm2 | |
| - Square_Rhombus | |
| - rmb_per | |
| - par_2_plg | |
| - conga_to_par_os | |
| - pars_par_pars | |
| - plgs_two_sides | |
| - ncol234_plg__pars1423 | |
| - par_cong_mid_ts | |
| - plgs_comm2 | |
| - rect_permut | |
| - plg_mid_2 | |
| - os_cong_par_cong_par | |
| - not_par_pars_not_cong | |
| - ncol123_plg__plgs | |
| - plgs_pseudo_trans | |
| - rect_per4 | |
| - plgs_cong_1 | |
| - plg_per_rect3 | |
| - plgs_pars_2 | |
| - plgs_trans_trivial | |
| - parallelogram_equiv_plg | |
| - plg_cong | |
| - parallel_2_plg | |
| - square_perp_rectangle | |
| - ncol124_plg__pars1423 | |
| - rect_per2 | |
| - par_cong_cop | |
| - par_cong_mid_os | |
| - plg_par_2 | |
| - sac_plg | |
| - plgs_pars_1 | |
| - parallelogram_strict_midpoint | |
| - plg_permut | |
| - plg_per_rect | |
| - rect_2_rect | |
| - perp3__rect | |
| - plg_cong_1 | |
| - plgf_plgf_plgf | |
| - rmb_cong | |
| - plg_conga | |
| - plg_comm2 | |
| - cong3_par2_par | |
| - ncol134_plg__pars1423 | |
| - par_par_cong_cong_parallelogram | |
| - ncol134_plg__pars1234 | |
| - plgs_par_strict | |
| - exists_square | |
| - sac_rectangle | |
| - plgs_half_plgs | |
| - plgs_in_angle | |
| - rect_per3 | |
| - rect_per1 | |
| - plg_par_1 | |
| - plg_pseudo_trans | |
| - plg_per_rect1 | |
| - par_cong_plg_2 | |
| - plgs_not_comm | |
| - plgs_half_plgs_aux | |
| - plgs_cong | |
| - plgs_cong_2 | |
| - ncol234_plg__pars1234 | |
| - ncol134_plg__plgs | |
| - ncol123_plg__pars1423 | |
| - ncol234_plg__plgs | |
| - plgf_comm2 | |
| - ncol124_plg__plgs | |
| - par_strict_cong_mid1 | |
| - par_cong_cong | |
| - par_strict_trans | |
| - path: theories/Main/Tarski_dev/Ch14_prod.v | |
| theorems: | |
| - prod_to_prodp | |
| - distr_r | |
| - prodp_to_prod | |
| - eq_squares_eq_or_opp | |
| - prod_sym | |
| - change_grid_prod1 | |
| - proj_preserves_prod | |
| - prod_O_l_eq | |
| - prod_O_r_eq | |
| - prod_comm | |
| - change_grid_prod | |
| - prod_0_l | |
| - l14_31_2 | |
| - diff_of_squares | |
| - prod_1_r | |
| - prod_1_l | |
| - distr_l_diff | |
| - project_pj | |
| - prod_exists | |
| - prod_uniqueness | |
| - inv_exists | |
| - change_grid_prod_l_O | |
| - distr_l | |
| - opp_prod | |
| - prod_1_l_eq | |
| - prod_assoc | |
| - prod_assoc2 | |
| - prod_null | |
| - prod_uniquenessB | |
| - diff_2_prod | |
| - prod_uniquenessA | |
| - l14_31_1 | |
| - prod_1_r_eq | |
| - path: theories/Elements/OriginalProofs/proposition_10.v | |
| theorems: | |
| - proposition_10 | |
| - path: theories/Main/Elements_statements/Book_1.v | |
| theorems: | |
| - prop_29_2 | |
| - prop_19 | |
| - prop_9 | |
| - prop_33 | |
| - prop_16 | |
| - prop_12 | |
| - prop_28_1 | |
| - prop_21_1 | |
| - prop_17 | |
| - prop_22 | |
| - prop_25 | |
| - prop_7 | |
| - prop_14 | |
| - prop_26_2 | |
| - prop_1_circle_circle | |
| - prop_5_1 | |
| - prop_24 | |
| - prop_34_1 | |
| - prop_47 | |
| - prop_32_1 | |
| - prop_13 | |
| - prop_1_euclidean | |
| - prop_4 | |
| - prop_6 | |
| - prop_15 | |
| - prop_18 | |
| - prop_23 | |
| - prop_29_3 | |
| - prop_11 | |
| - prop_10 | |
| - prop_26_1 | |
| - path: theories/Main/Meta_theory/Models/tarski_to_hilbert.v | |
| theorems: | |
| - axiom_cong_5' | |
| - Hilbert_3D_follows_from_Tarski_3D | |
| - EqP_Equiv | |
| - cop_plane | |
| - axiom_hcong_1_existence | |
| - axiom_two_points_on_line | |
| - lower_dim_3' | |
| - axiom_plane_uniqueness | |
| - out_outH | |
| - axiom_between_one | |
| - lower_dim' | |
| - axiom_hcong_scott | |
| - axiom_between_only_one | |
| - axiom_between_out | |
| - same_side_sym | |
| - axiom_Incidp_morphism | |
| - Bet_Between_H | |
| - same_side_OS | |
| - axiom_conga_comm | |
| - same_side_one_side | |
| - EqL_Equiv | |
| - plane_cop | |
| - axiom_line_existence | |
| - eqp_symmetry | |
| - axiom_line_on_plane | |
| - cut_two_sides | |
| - incident_Proper | |
| - axiom_conga_permlr | |
| - Para_Par | |
| - outH_out | |
| - eq_reflexivity | |
| - ncols_coincide | |
| - eqp_transitivity | |
| - eqp_reflexivity | |
| - axiom_hcong_4_uniqueness | |
| - incident_eq | |
| - axiom_hcong_1_uniqueness | |
| - axiom_hcong_3 | |
| - cols_coincide_1 | |
| - axiom_line_uniqueness | |
| - axiom_Incid_morphism | |
| - col_disjoint_bet | |
| - axiom_between_diff | |
| - axiom_one_point_on_plane | |
| - exists_not_incident | |
| - axiom_euclid_uniqueness | |
| - axiom_plane_existence | |
| - OS_distinct | |
| - Hilbert_neutral_follows_from_Tarski_neutral | |
| - axiom_Incid_dec | |
| - same_side_trans | |
| - axiom_Incidp_dec | |
| - axiom_conga_refl | |
| - Hilbert_euclidean_follows_from_Tarski_euclidean | |
| - eq_incident | |
| - axiom_congaH_outH_congaH | |
| - Hilbert_euclidean_ID_follows_from_Tarski_euclidean | |
| - eq_symmetry | |
| - axiom_hcong_4_existence | |
| - axiom_between_comm | |
| - cols_coincide_2 | |
| - axiom_pasch | |
| - axiom_between_col | |
| - eq_transitivity | |
| - one_side_same_side | |
| - Incid_line | |
| - cols_coincide | |
| - path: theories/Main/Annexes/vectors.v | |
| theorems: | |
| - vector_construction_uniqueness | |
| - null_sum | |
| - plg_out_plg | |
| - same_dir_to_null | |
| - eqv_permut | |
| - same_dir_null | |
| - bet_same_dir1 | |
| - eqv_trans | |
| - opp_dir_id | |
| - eqv_sym | |
| - same_dir_refl | |
| - same_dir_dec | |
| - opp_dir_to_null | |
| - mid_eqv | |
| - bet_same_dir2 | |
| - par_ts_same_dir | |
| - ise_to_is | |
| - opp_not_same_dir | |
| - eqv_comm | |
| - plgf_out_plgf | |
| - same_dir_ts | |
| - same_or_opp_dir | |
| - same_not_opp_dir | |
| - null_sum_eq | |
| - sum_exists | |
| - sum_uniqueness | |
| - eqv_mid | |
| - same_dir_out | |
| - same_dir_trans | |
| - vector_construction | |
| - same_dir_comm | |
| - plg_opp_dir | |
| - vector_uniqueness | |
| - vector_same_dir_cong | |
| - chasles | |
| - eqv_opp_null | |
| - plgf_plgf_bet | |
| - plgs_out_plgs | |
| - eqv_par | |
| - eqv_trivial | |
| - same_dir_out1 | |
| - null_vector | |
| - eqv_sum | |
| - plgs_plgs_bet | |
| - same_dir_sym | |
| - opposite_sum | |
| - path: theories/Main/Tarski_dev/Ch14_order.v | |
| theorems: | |
| - diff_2_le_le | |
| - prod_pos_pos | |
| - pos_null_neg | |
| - ltP_sum_pos | |
| - bet_lt21_le32 | |
| - ltP_neg | |
| - l14_36_b | |
| - pos_not_neg | |
| - leP_trans | |
| - not_pos_and_neg | |
| - l14_36_a | |
| - square_pos | |
| - bet_lt12_le13 | |
| - bet_lt21_le31 | |
| - leP_asym | |
| - compatibility_of_sum_with_order | |
| - opp_pos_neg | |
| - leP_refl | |
| - ltP_ar2 | |
| - le_pos_prod_le | |
| - pos_opp_neg | |
| - diff_pos_diff_neg | |
| - ltP_neq | |
| - bet_lt12_le23 | |
| - ps_le | |
| - opp_neg_pos | |
| - lt_diff_ps | |
| - opp_2_le_le | |
| - O_not_positive | |
| - col_pos_or_neg | |
| - sum_pos_pos | |
| - compatibility_of_prod_with_order | |
| - leP_sum_leP | |
| - path: theories/Algebraic/Counter_models/Planar/counter_model_euclid.v | |
| theorems: | |
| - five_segment | |
| - cong_inner_transitivity | |
| - euclid | |
| - nbet_abc | |
| - c'_in_unit_disk | |
| - upper_dim_aux | |
| - inner_paschP | |
| - upper_dim | |
| - b'_in_unit_disk | |
| - a'_eq0 | |
| - segment_construction | |
| - cong_pseudo_reflexivity | |
| - point_equality_decidability | |
| - nbet_cab | |
| - bet_symmetry | |
| - bet_inner_transitivity | |
| - betR_bca | |
| - upper_dim_aux_2 | |
| - a'_in_unit_disk | |
| - Rcf_to_GI_PED' | |
| - betR_cab | |
| - path: theories/Main/Meta_theory/Continuity/dedekind_cantor.v | |
| theorems: | |
| - nested__ex_right | |
| - nested__ex_left | |
| - nested_aux1 | |
| - nested_sym | |
| - dedekind__cantor | |
| - nested__diff0 | |
| - nested__bet | |
| - nested_aux2 | |
| - path: theories/Main/Meta_theory/Models/gupta_inspired_to_tarski.v | |
| theorems: | |
| - g2_15 | |
| - inner_paschT | |
| - g2_9 | |
| - between_identityT | |
| - col_decG | |
| - g2_7 | |
| - g2_4 | |
| - g2_2 | |
| - g2_10 | |
| - bet_decG | |
| - g2_1 | |
| - g2_16 | |
| - between_trivialT | |
| - g2_3 | |
| - GI_to_T_PED | |
| - l2_11T | |
| - GI2D_to_T2D | |
| - construction_uniquenessT | |
| - upper_dimT | |
| - cong_trivial_identityT | |
| - g2_8 | |
| - g2_6 | |
| - cong_inner_transitivityT | |
| - g2_14 | |
| - g2_13 | |
| - g2_11 | |
| - euclidT | |
| - g2_12 | |
| - GI_to_T | |
| - GI_euclidean_to_T_euclidean | |
| - path: theories/Main/Annexes/half_angles.v | |
| theorems: | |
| - conga2_ghalfa__ghalfa | |
| - halfa_not_null | |
| - halfa__ts | |
| - halfa_chara1 | |
| - halfa_uniqueness | |
| - ghalfa_out4__ghalfa | |
| - halfa_sym | |
| - halfa1123__out | |
| - halfa__nbet2 | |
| - halfa__suma | |
| - ghalfa__suma | |
| - ghalfa2__conga_2 | |
| - ghalfa_left_comm | |
| - inangle_halfa2__inangle | |
| - halfa_exists | |
| - halfa2_lta__lta2 | |
| - ghalfa_preserves_conga_2 | |
| - suma_preserves_ghalfa | |
| - halfa_distincts | |
| - halfa__lea | |
| - halfa3123__out | |
| - halfa__nbet | |
| - halfa2_lea__lea1 | |
| - ghalfa2__conga_1 | |
| - halfa__coplanar | |
| - acute_ghalfa2_sams_suma2__ghalfa123 | |
| - ghalfa_distincts | |
| - ghalfa_comm | |
| - null_halfa__null | |
| - ghalfa__out | |
| - ghalfa_chara | |
| - conga_halfa__conga2 | |
| - halfa__ghalfa | |
| - conga_halfa__conga1 | |
| - halfa2_lta__lta1 | |
| - halfa__acute | |
| - cop_halfa_perp__os2 | |
| - ghalfa_preserves_lta | |
| - halfa__sams | |
| - halfa2_lea__lea2 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/parallel_postulates.v | |
| theorems: | |
| - aristotle | |
| - equivalent_postulates_assuming_greenberg_s_axiom | |
| - stronger_postulates_ter | |
| - greenberg | |
| - stronger_postulates_bis | |
| - equivalent_postulates_without_any_continuity_bis | |
| - equivalent_postulates_without_decidability_of_intersection_of_lines | |
| - equivalent_postulates_assuming_archimedes_axiom | |
| - postulates_in_euclidean_context | |
| - stronger_postulates | |
| - equivalent_postulates_without_any_continuity | |
| - inter_dec | |
| - path: theories/Elements/OriginalProofs/lemma_rectangleparallelogram.v | |
| theorems: | |
| - lemma_rectangleparallelogram | |
| - path: theories/Main/Highschool/orthocenter.v | |
| theorems: | |
| - is_orthocenter_perm_3 | |
| - altitude_intersect | |
| - is_orthocenter_perm_4 | |
| - construct_intersection | |
| - is_orthocenter_perm | |
| - orthocenter_per | |
| - diff_not_col_col_par4_mid | |
| - is_orthocenter_cases | |
| - not_col_par_col_diff | |
| - orthocenter_col | |
| - is_orthocenter_perm_5 | |
| - is_orthocenter_perm_2 | |
| - is_orthocenter_coplanar | |
| - is_orthocenter_perm_1 | |
| - altitude_is_perp_bisect | |
| - construct_triangle | |
| - path: theories/Main/Meta_theory/Models/tarski_to_euclid.v | |
| theorems: | |
| - InCircCenter | |
| - InCirc_InCirc | |
| - BetS_BetS | |
| - outside | |
| - bet_cases | |
| - Euclid_neutral_follows_from_Tarski_neutral | |
| - circle_circle' | |
| - 'on' | |
| - Euclid5 | |
| - circle_line | |
| - OnCirc_OnCirc | |
| - Col_Col | |
| - eOutCirc_OutCirc | |
| - eOnCirc_OnCirc | |
| - path: theories/Algebraic/Counter_models/counter_model_bet_identity.v | |
| theorems: | |
| - upper_dim | |
| - point_eq_dec | |
| - inner_pasch | |
| - cong_identity | |
| - not_bet_diff | |
| - cong_aux_3 | |
| - lower_dim | |
| - five_segment | |
| - cong_pseudo_reflexivity | |
| - cong_aux | |
| - cong_inner_transitivity | |
| - cong_aux_6 | |
| - cong_aux_5 | |
| - cong_sym | |
| - path: theories/Main/Tarski_dev/Ch13_1.v | |
| theorems: | |
| - is_image_perp_in | |
| - per13_preserves_bet | |
| - perp2_par | |
| - per3_preserves_bet2_aux | |
| - l13_2_1 | |
| - per3_preserves_bet2 | |
| - cop4_perp_in2__col | |
| - per2_col_eq | |
| - perp2_preserves_bet13 | |
| - cong_perp_conga | |
| - perp_in2__col | |
| - perp2_pseudo_trans | |
| - perp2_trans | |
| - perp2_comm | |
| - perp_in_rewrite | |
| - perp2_perp_in | |
| - l13_1_aux | |
| - l13_8 | |
| - col_cop_perp2__pars_bis | |
| - perp_out_acute | |
| - perp2_sym | |
| - perp2_left_comm | |
| - perp2_refl | |
| - per_lt | |
| - l13_2 | |
| - per23_preserves_bet_inv | |
| - perp_per_bet | |
| - per3_preserves_bet1 | |
| - perp_inter_perp_in_n | |
| - perp_bet_obtuse | |
| - per13_preserves_bet_inv | |
| - l13_1 | |
| - perp2_right_comm | |
| - ts_per_per_ts | |
| - per2_preserves_diff | |
| - per23_preserves_bet | |
| - path: theories/Main/Tarski_dev/Ch12_parallel.v | |
| theorems: | |
| - par_not_col_strict | |
| - par_col_par | |
| - par_distincts | |
| - l12_22_b | |
| - conga_cop_inangle_per2__inangle | |
| - par_strict_distinct | |
| - par_strict_irreflexivity | |
| - par_col2_par_bis | |
| - l12_9 | |
| - par_strict_col2_par_strict | |
| - par_comm | |
| - col_cop2_perp2__col | |
| - par_distinct | |
| - inter_distincts | |
| - l12_18_b | |
| - perp_inter_exists | |
| - l12_6 | |
| - par_neq1 | |
| - col_par | |
| - par_one_or_two_sides | |
| - l12_9_2D | |
| - not_par_not_col | |
| - inter_uniqueness_not_par | |
| - par_strict_not_col_1 | |
| - par_strict_col_par_strict | |
| - perp_inter_perp_in | |
| - par_id | |
| - col_cop_perp2__pars | |
| - par_reflexivity | |
| - par_left_comm | |
| - perp_not_par | |
| - inter_right_comm | |
| - l12_18_d | |
| - parallel_existence1 | |
| - Par_strict_perm | |
| - l12_18_a | |
| - cong_conga_perp | |
| - par_col_par_2 | |
| - par_strict_comm | |
| - par_right_comm | |
| - par_strict_symmetry | |
| - not_par_inter_uniqueness | |
| - par_strict_not_col_3 | |
| - inter_left_comm | |
| - pars__os3412 | |
| - par_strict_not_cols | |
| - not_strict_par2 | |
| - col2_par__col4 | |
| - par_strict_left_comm | |
| - parallel_existence | |
| - par_neq2 | |
| - perp_dec | |
| - not_strict_par | |
| - inter_comm | |
| - col_not_col_not_par | |
| - Par_cases | |
| - l12_21_b | |
| - par_strict_one_side | |
| - l12_18_c | |
| - all_one_side_par_strict | |
| - acute_col_perp__out | |
| - par_two_sides_two_sides | |
| - par_strict_not_col_4 | |
| - inter_trivial | |
| - par_symmetry | |
| - l12_22_aux | |
| - inter_sym | |
| - l12_18 | |
| - not_strict_par1 | |
| - acute_col_perp__out_1 | |
| - par_strict_col4__par_strict | |
| - path: theories/Algebraic/Counter_models/Planar/counter_model_upper_dim.v | |
| theorems: | |
| - ab_neq | |
| - congP_aux' | |
| - oner_eqm1 | |
| - neq | |
| - congP_aux | |
| - betR_bca | |
| - eq3 | |
| - de_neq | |
| - upper_dim | |
| - betR_cab | |
| - a_eq0 | |
| - row3_eq | |
| - oner_neqm1 | |
| - ca_neq | |
| - ac_neq | |
| - cong_bdbe | |
| - cong_adae | |
| - betR_abc | |
| - cong_cdce | |
| - negb_and3 | |
| - path: theories/Coinc/ColR.v | |
| theorems: | |
| - CTcol_permutation_4 | |
| - collect_diffs | |
| - proper_1 | |
| - proper_9 | |
| - test_col_ok | |
| - pick_lines_ok_2 | |
| - CTcol_trivial_1 | |
| - CTcol_permutation_3 | |
| - subst_sp_ok | |
| - strong_induction | |
| - proper_8 | |
| - exists_witness_ok | |
| - collect_cols | |
| - ss_ok_empty | |
| - proper_4 | |
| - identify_lines_ok | |
| - CTcol_permutation_2 | |
| - proper_2 | |
| - proper_00 | |
| - pick_lines_ok_1 | |
| - proper_0 | |
| - proper_5 | |
| - sp_ok_empty | |
| - proper_6 | |
| - subst_ss_ok | |
| - eq_eq_tagged | |
| - CTcol_trivial_2 | |
| - path: theories/Algebraic/Counter_models/nD/counter_model_upper_dim.v | |
| theorems: | |
| - my_enum_ordSr | |
| - upper_dim | |
| - nth_basis | |
| - betS_nD'_2D | |
| - b2DE | |
| - betR_nD'_2D | |
| - to_nD'B | |
| - lower_dim | |
| - last_new_basis | |
| - to_nD'_scale | |
| - pick_to_nD' | |
| - upper_dim_all2 | |
| - lower_dim_all2 | |
| - lower_dim_all1 | |
| - upper_dim_all1 | |
| - eq_nD'_2D | |
| - a2DE | |
| - to_nD'E | |
| - c2DE | |
| - iP | |
| - nth_new_basis | |
| - path: theories/Algebraic/coplanarity.v | |
| theorems: | |
| - InPlane'__Coplanar | |
| - Cop_4321 | |
| - Cop_2341 | |
| - Cop_2134 | |
| - Cop__Coplanar | |
| - Copl_2431 | |
| - Cop_3421 | |
| - Copl_3412 | |
| - Copl_2341 | |
| - Copl_4123 | |
| - Copl_3421 | |
| - Cop_1324 | |
| - Copl_1423 | |
| - Cop_4132 | |
| - Copl_1243 | |
| - Copl_3214 | |
| - Cop_1342 | |
| - Cop_aabc | |
| - Copl_4231 | |
| - Cop_2143 | |
| - Cop_3241 | |
| - Cop_2431 | |
| - Copl_2413 | |
| - Cop_4231 | |
| - InPlane__Coplanar | |
| - Cop_2314 | |
| - Copl_2314 | |
| - Copl_4321 | |
| - Cop_3142 | |
| - Copl_3241 | |
| - Copl_2143 | |
| - Cop_4312 | |
| - Cop_1423 | |
| - Copl_4213 | |
| - Cop_1432 | |
| - Copl_4132 | |
| - Copl_3124 | |
| - Cop_3124 | |
| - path: theories/Main/Tarski_dev/Ch07_midpoint.v | |
| theorems: | |
| - col_cong2_bet2 | |
| - symmetry_preserves_midpoint | |
| - l7_20_bis | |
| - swap_diff | |
| - is_midpoint_id | |
| - midpoint_not_midpoint | |
| - l7_3_2 | |
| - l7_20 | |
| - midpoint_distinct_2 | |
| - le_mid2__le12 | |
| - l7_15 | |
| - lt_mid2__lt12 | |
| - l7_9_bis | |
| - midpoint_distinct_3 | |
| - l7_9 | |
| - l7_17_bis | |
| - bet_col1 | |
| - col_cong2_bet3 | |
| - cong_mid2__cong | |
| - bet2_lt_le__lt | |
| - l7_22 | |
| - midpoint_out | |
| - is_midpoint_id_2 | |
| - symmetric_point_construction | |
| - l7_2 | |
| - midpoint_bet | |
| - midpoint_col | |
| - l7_17 | |
| - mid__lt | |
| - col_cong_bet | |
| - col_cong2_bet1 | |
| - cong_cong_half_1 | |
| - symmetric_point_uniqueness | |
| - Mid_perm | |
| - midpoint_cong | |
| - cong_cong_half_2 | |
| - lt_mid2__lt13 | |
| - l7_25 | |
| - le_mid2__le13 | |
| - l7_21 | |
| - cong_col_mid | |
| - l7_3 | |
| - l7_22_aux | |
| - col_cong2_bet4 | |
| - l7_16 | |
| - col_bet2_cong1 | |
| - midpoint_preserves_out | |
| - midpoint_dec | |
| - l7_13 | |
| - path: theories/Elements/OriginalProofs/lemma_9_5.v | |
| theorems: | |
| - lemma_9_5 | |
| - path: theories/Algebraic/Counter_models/Planar/counter_model_cong_identity.v | |
| theorems: | |
| - cong_identity | |
| - cong_pseudo_reflexivity | |
| - bet_symmetry | |
| - col_alt_def | |
| - segment_construction | |
| - euclid | |
| - continuity | |
| - bet_inner_transitivity | |
| - point_equality_decidability | |
| - inner_pasch | |
| - cong_inner_transitivity | |
| - lower_dim | |
| - upper_dim | |
| - path: theories/Algebraic/Counter_models/Planar/counter_model_cong_inner_transitivity.v | |
| theorems: | |
| - col_alt_def | |
| - bet_inner_transitivity | |
| - point_equality_decidability | |
| - upper_dim | |
| - continuity | |
| - segment_construction | |
| - lower_dim | |
| - cong_inner_transitivity | |
| - cong_identity | |
| - five_segment | |
| - bet_symmetry | |
| - inner_pasch | |
| - path: theories/Algebraic/tcp_ndc.v | |
| theorems: | |
| - tcp_bet_sa_b_inter_mab_mac_b_sa | |
| - tcp_aligned_inplane | |
| - tcp_ts_mab_mac_a_ssa | |
| - tcp_ts_mmab_mac_a_sa | |
| - tcp_npars_mab_mmac_b_sa | |
| - tcp_os_b_sa_mab_mac | |
| - tcp_pars | |
| - tcp_ncol_inplane_3_3 | |
| - tcp_ts_a_ssa_mab_mac | |
| - tcp_ncol_inplane_1_4 | |
| - tcp_npars_mab_mac_b_ssa | |
| - tcp_ncol_inplane_3_1 | |
| - tcp_ts_mmab_mac_a_mbc | |
| - tcp_ts_a_sa_mab_mac | |
| - tcp_ncol_inplane_3_6 | |
| - tcp_os_mmab_mac_b_sa | |
| - tcp_npars_mab_mmac_b_ssa | |
| - tcp_os_mab_mac_b_sa | |
| - tcp_ts_mab_mac_a_sa | |
| - tcp_ts_a_mbc_mab_mac | |
| - tcp_ncol_inplane_3_5 | |
| - tcp_ncol_inplane_2_5 | |
| - tcp_ncol_midpoints | |
| - tcp_ncol_inplane_1_1 | |
| - tcp_os_mmab_mac_b_ssa | |
| - tcp_os_b_ssa_mmab_mac | |
| - tcp_os_b_ssa_mab_mmac | |
| - tcp_npars_mab_mac_b_sa | |
| - tcp_npars_mmab_mac_b_sa | |
| - tcp_ncol_inplane_1_3 | |
| - tcp_ncol_inplane | |
| - tcp_ncol_inplane_3_4 | |
| - tcp_ncol_inplane_2_6 | |
| - tcp_ncol_inplane_1_6 | |
| - tcp_ncol_inplane_2_1 | |
| - tcp_ncol_inplane_1_2 | |
| - tcp_ncol_ndc_choice_col | |
| - tcp_os_mab_mmac_b_sa | |
| - tcp_aligned_plane_exists | |
| - tcp_ncol_ndc_choice | |
| - tcp_os_mab_mac_b_ssa | |
| - tcp_ts_mab_mac_a_mbc | |
| - tcp_ncol_inplane_2_2 | |
| - tcp_ncol_inplane_aux | |
| - tcp_ncol_inplane_1_5 | |
| - tcp_ncol_inplane_2_3 | |
| - tcp_ncols_ndc | |
| - tcp_ts_mab_mmac_a_sa | |
| - tcp_ncol_inplane_4 | |
| - tcp_ncol_inplane_3_2 | |
| - tcp_os_b_sa_mab_mmac | |
| - tcp_ncol_ndc_ncol | |
| - path: theories/Main/Highschool/triangles.v | |
| theorems: | |
| - equilateral_rot_2 | |
| - equilateral_isosceles_3 | |
| - equilateral_isosceles_2 | |
| - equilateral_strict_swap_4 | |
| - conga_isosceles | |
| - equilateral_swap | |
| - equilateral_strict_equilateral | |
| - equilateral_strict_conga_3 | |
| - equilateral_strict_conga_2 | |
| - equilateral_strict_conga_1 | |
| - equilateral_swap_rot | |
| - equilateral_strict_swap_1 | |
| - equilateral_rot | |
| - isosceles_conga | |
| - isosceles_foot__midpoint_conga | |
| - equilateral_strict_neq | |
| - equilateral_cong | |
| - conga3_equilateral | |
| - path: theories/Main/Highschool/SegmTrisect.v | |
| theorems: | |
| - SegmTrisectUniqueness | |
| - SegmTrisectExistence | |
| - SegmTrisectExistenceUniqueness | |
| - SegmTrisectFirstThirdUniqueness | |
| - SegmTrisectFirstThirdExistence | |
| - path: theories/Main/Annexes/rhombus.v | |
| theorems: | |
| - PlgLeft | |
| - PlgExABC2 | |
| - RhombusExABC1 | |
| - PlgEx | |
| - PlgAABB | |
| - RhombusUnicity | |
| - RhombusEx | |
| - PlgEquivDef | |
| - RhombusExABC2 | |
| - PlgExABC1 | |
| - ColCongMid | |
| - path: theories/Main/Annexes/suma.v | |
| theorems: | |
| - nsams__obtuse | |
| - trisuma_perm_213 | |
| - sams_lea_lta789_suma2__lta123 | |
| - sams_lta2_suma2__lta456 | |
| - sams_suma__lea456789 | |
| - bet_per2__suma | |
| - sams_lea_lta123_suma2__lta | |
| - sams_lea123_suma2__lea | |
| - sams_suma__lea123789 | |
| - nbet_sams_suma__acute | |
| - sams_lea2_suma2__conga123 | |
| - out213__sams | |
| - ex_trisuma | |
| - sams_lea_lta456_suma2__lta123 | |
| - sams_lea_lta456_suma2__lta | |
| - sams_lea2_suma2__lea123 | |
| - out546_suma__conga | |
| - sams_sym | |
| - sams_lea_lta123_suma2__lta456 | |
| - bet_suma2__or_conga | |
| - suma_middle_comm | |
| - sams_assoc_1 | |
| - suppa__sams | |
| - trisuma_dec | |
| - trisuma_perm_321 | |
| - sams_lta2_suma2__lta | |
| - suma_left_comm | |
| - out546__sams | |
| - acute_suma__nbet | |
| - suma_dec | |
| - trisuma_perm_132 | |
| - sams_right_comm | |
| - acute2_suma2__conga | |
| - per2__sams | |
| - suma_right_comm | |
| - sams_assoc | |
| - os_ts__sams | |
| - trisuma2__conga | |
| - conga3_trisuma__trisuma | |
| - sams_assoc_2 | |
| - bet_suma__per | |
| - bet_per_suma__per123 | |
| - col_suma__col | |
| - sams2_suma2__conga456 | |
| - out546__suma | |
| - sams_suma__out213 | |
| - bet_suma__suppa | |
| - sams2_suma2__conga | |
| - bet2_suma__suma | |
| - trisuma_perm_312 | |
| - ex_suma | |
| - sams_lea2_suma2__conga456 | |
| - sams_lea456_suma2__lea | |
| - suma_sym | |
| - inangle__suma | |
| - conga_trisuma__trisuma | |
| - suma2__conga | |
| - sams_lta2_suma2__lta123 | |
| - bet_per_suma__per456 | |
| - suma2_obtuse2__conga | |
| - bet_suma__sams | |
| - acute_per__sams | |
| - acute2__sams | |
| - obtuse__nsams | |
| - ts__suma | |
| - out6_suma__suma | |
| - sams123231 | |
| - sams_left_comm | |
| - sams_comm | |
| - sams_dec | |
| - suma_assoc_1 | |
| - out213_suma__conga | |
| - acute2_suma__nbet | |
| - conga_sams_nos__nts | |
| - suma2__or_conga_suppa | |
| - suma_suppa__bet | |
| - ncol_suma__ncol | |
| - per2_suma__bet | |
| - sams_distincts | |
| - sams2_suma2__conga123 | |
| - suma_distincts | |
| - suma_comm | |
| - sams_lea2__sams | |
| - bet__suma | |
| - acute__sams | |
| - path: theories/Main/Tarski_dev/Ch02_cong.v | |
| theorems: | |
| - not_cong_4312 | |
| - cong_trivial_identity | |
| - cong_diff | |
| - cong_transitivity | |
| - cong_3_sym | |
| - cong_reverse_identity | |
| - construction_uniqueness | |
| - cong_diff_2 | |
| - cong3_transitivity | |
| - eq_dec_points | |
| - cong_right_commutativity | |
| - five_segment_with_def | |
| - l2_11 | |
| - not_cong_1243 | |
| - cong_diff_3 | |
| - Cong_cases | |
| - cong_diff_4 | |
| - bet_cong3 | |
| - cong_commutativity | |
| - cong_left_commutativity | |
| - not_cong_2134 | |
| - cong_3_swap | |
| - not_cong_3412 | |
| - cong_3421 | |
| - Cong_perm | |
| - not_cong_2143 | |
| - cong_3_swap_2 | |
| - cong_4321 | |
| - not_cong_4321 | |
| - path: theories/Main/Meta_theory/Models/beeson_to_tarski.v | |
| theorems: | |
| - BetT_symmetry | |
| - IT_chara | |
| - pasch | |
| - another_point | |
| - bet_id | |
| - cong_stability_eq_stability | |
| - pasch_col_case | |
| - IT_to_T | |
| - five_segment | |
| - segment_construction | |
| - path: theories/Elements/OriginalProofs/proposition_09.v | |
| theorems: | |
| - proposition_09 | |
| - path: theories/Algebraic/Counter_models/nD/counter_model_five_segment.v | |
| theorems: | |
| - cong_inner_transitivity | |
| - five_segment | |
| - inner_pasch | |
| - col__colI | |
| - bet_bet' | |
| - row2_cong_nD | |
| - cong_pseudo_reflexivity | |
| - row_mx_behead | |
| - to_nD_behead | |
| - Col_Col | |
| - euclid | |
| - cong_identity | |
| - to_nD_head | |
| - bet_inner_transitivity | |
| - bet'_abc | |
| - segment_construction | |
| - eq_head_behead | |
| - path: theories/Algebraic/Counter_models/nD/counter_model_bet_inner_transitivity.v | |
| theorems: | |
| - coplanar_midpoint | |
| - col__Col | |
| - Midpoint_midpoint | |
| - midpointBR | |
| - midpointxx | |
| - cong_midpoint | |
| - onemx_neq0 | |
| - bet'_midpoint | |
| - midpointLR | |
| - colI__Col | |
| - ColD | |
| - col__colI | |
| - euclid | |
| - midpointBL | |
| - colI__Col_13 | |
| - betR_midpoint | |
| - ncol__ncolI | |
| - bet_midpoint | |
| - inner_pasch | |
| - bet_inner_transitivity | |
| - bet_symmetry | |
| - Col__col | |
| - Col_23 | |
| - five_segment | |
| - cong_identity | |
| - cong_pseudo_reflexivity | |
| - segment_construction | |
| - path: theories/Algebraic/Counter_models/nD/independent_version_to_beeson.v | |
| theorems: | |
| - ITbetween_identity | |
| - bet_stability | |
| - ITsegment_construction | |
| - altIT_to_IT | |
| - ITinner_pasch | |
| - ITfive_segment | |
| - weak_Bstability | |
| - cong_stability | |
| - ITbetween_symmetry | |
| - ITlower_dim | |
| - path: theories/Coinc/CongR.v | |
| theorems: | |
| - eq_pseudo_refl | |
| - identify_lengths_ok | |
| - proper_3 | |
| - collect_congs | |
| - proper_2_aux | |
| - proper_1 | |
| - exists_witness_ok | |
| - pick_lengths_ok_1 | |
| - CTcong_right_comm | |
| - ss_ok_empty | |
| - pick_lengths_ok_2 | |
| - positive_dec | |
| - identify_lengths | |
| - test_cong_ok | |
| - proper_0 | |
| - proper_2 | |
| - path: theories/Elements/OriginalProofs/lemma_TTorder.v | |
| theorems: | |
| - lemma_TTorder | |
| - path: theories/Main/Meta_theory/Models/tarski_to_gupta_inspired.v | |
| theorems: | |
| - T_to_GI | |
| - cong_inner_transitivity' | |
| - T2D_to_TG2D | |
| - path: theories/Algebraic/Counter_models/nD/counter_model_cong_identity.v | |
| theorems: | |
| - bet_inner_transitivityP | |
| - inner_pasch | |
| - cong_identity | |
| - segment_construction | |
| - bet_symmetry | |
| - cong_inner_transitivity | |
| - upper_dim | |
| - lower_dim | |
| - path: theories/Elements/OriginalProofs/lemma_crossbar2.v | |
| theorems: | |
| - lemma_crossbar2 | |
| - path: theories/Elements/OriginalProofs/lemma_9_5b.v | |
| theorems: | |
| - lemma_9_5b | |
| - path: theories/Main/Meta_theory/Parallel_postulates/TCP_tarski.v | |
| theorems: | |
| - impossible_case_3 | |
| - impossible_case_4_1 | |
| - impossible_case_4 | |
| - triangle_circumscription_implies_tarski_s_euclid | |
| - impossible_case_2 | |
| - impossible_case_1 | |
| - triangle_circumscription_implies_tarski_s_euclid_aux1 | |
| - triangle_circumscription_implies_tarski_s_euclid_aux | |
| - path: theories/Elements/OriginalProofs/euclidean_tactics.v | |
| theorems: | |
| - nCol_not_Col | |
| - not_nCol_Col | |
| - Col_or_nCol | |
| - Col_nCol_False | |
| - eq_or_neq | |
| - nCol_or_Col | |
| - path: theories/Main/Meta_theory/Models/makarios_to_tarski.v | |
| theorems: | |
| - Tarski_follows_from_Makarios_Variant | |
| - LmCoghGrab | |
| - Tarski_follows_from_Makarios_Variant_with_decidable_point_equality' | |
| - Mcong_reflexivity | |
| - five_segment | |
| - Mcong_trivial_identity | |
| - Mcong_symmetry | |
| - cong_pre_pseudo_reflexivity | |
| - cong_pseudo_reflexivity | |
| - between_trivial | |
| - path: theories/Main/Annexes/midpoint_theorems.v | |
| theorems: | |
| - triangle_mid_par_flat | |
| - triangle_mid_par_cong | |
| - col123_124__col234 | |
| - col_permut231 | |
| - col_permut321 | |
| - triangle_mid_par_flat_cong_2 | |
| - triangle_mid_par | |
| - col_permut213 | |
| - triangle_mid_par_flat_cong_aux | |
| - triangle_mid_par_strict_cong_aux | |
| - triangle_mid_par_strict_cong_simp | |
| - triangle_mid_par_strict_cong_1 | |
| - triangle_mid_par_flat_cong | |
| - triangle_mid_par_strict | |
| - triangle_mid_par_strict_cong | |
| - col_permut312 | |
| - triangle_mid_par_cong_1 | |
| - triangle_mid_par_flat_cong_1 | |
| - triangle_par_mid | |
| - triangle_mid_par_strict_cong_2 | |
| - path: theories/Elements/OriginalProofs/lemma_3_7a.v | |
| theorems: | |
| - lemma_3_7a | |
| - path: theories/Algebraic/Counter_models/nD/counter_model_cong_inner_transitivity.v | |
| theorems: | |
| - bet_symmetry | |
| - cong_inner_transitivity | |
| - bet_inner_transitivity | |
| - segment_construction | |
| - cong_axay | |
| - continuity | |
| - inner_pasch | |
| - five_segment | |
| - cong_pseudo_reflexivity | |
| - path: theories/Main/Tarski_dev/Ch10_line_reflexivity_2.v | |
| theorems: | |
| - image_preserves_col | |
| - image_spec_preserves_per | |
| - cong_cop_per2_1 | |
| - l10_16 | |
| - cop_not_par_other_side | |
| - cop__cong_on_bissect | |
| - cong_cop_per2 | |
| - image_preserves_bet | |
| - cop_image_in2__col | |
| - all_coplanar | |
| - hilbert_s_version_of_pasch_aux | |
| - intersection_with_image_gen | |
| - image_preserves_per | |
| - l10_10 | |
| - perp2__col | |
| - image_gen_preserves_ncol | |
| - cong_cop_per2_gen | |
| - l10_10_spec | |
| - cong_cop_image__col | |
| - cop_not_par_same_side | |
| - l10_12 | |
| - two_sides_cases | |
| - per2__col | |
| - not_par_two_sides | |
| - hilbert_s_version_of_pasch | |
| - image_gen_preserves_bet | |
| - image_gen_preserves_col | |
| - path: theories/Main/Tarski_dev/Ch13_2_length.v | |
| theorems: | |
| - ex_points_lg | |
| - ex_eqL | |
| - all_eql | |
| - null_len | |
| - is_len_cong_is_len | |
| - ex_eql | |
| - lg_cong | |
| - lg_null_trivial | |
| - not_cong_is_len1 | |
| - lg_cong_lg | |
| - lg_exists | |
| - ex_point_lg | |
| - not_cong_is_len | |
| - ex_point_lg_bet | |
| - eqL_equivalence | |
| - lg_null_dec | |
| - lg_sym | |
| - ex_point_lg_out | |
| - ex_lg | |
| - path: theories/Coinc/Utils/sets.v | |
| theorems: | |
| - compare_spec | |
| - eqListRefl | |
| - eqST_dec | |
| - OCPTlOK | |
| - OCP | |
| - lt_antiref | |
| - lengthOne | |
| - eqb_eq | |
| - OCPALengthOK | |
| - STadd_iff | |
| - lt | |
| - ltListIrrefl | |
| - eq_equiv | |
| - PermSorted | |
| - lt_irrefl | |
| - OCPPerm | |
| - OCPSortedAux | |
| - eqbListEqList | |
| - leb_total | |
| - lt | |
| - eqListSortOCP | |
| - compare_spec | |
| - eqListOK | |
| - ltTrans | |
| - lt_trans | |
| - eqbST_eqST | |
| - CPLOCPTlOK | |
| - lengthAtLeastOne | |
| - eqListSym | |
| - OCPSortedTl | |
| - lt_trans | |
| - ltListTrans | |
| - eqListTrans | |
| - STempty_b | |
| - compareListSpec | |
| - path: theories/Main/Tarski_dev/Ch16_coordinates.v | |
| theorems: | |
| - characterization_of_betweenness | |
| - exists_coord | |
| - point_of_coordinates_on_an_axis | |
| - grid_exchange_axes | |
| - characterization_of_collinearity | |
| - coord_exchange_axes | |
| - l16_9_2 | |
| - eq_points_coordinates | |
| - Cs_not_Col | |
| - characterization_of_betweenness_aux | |
| - length_eq_or_opp | |
| - cong_3_3_cong_5 | |
| - point_of_coordinates | |
| - point_of_coordinates_origin | |
| - square_distance_formula_aux | |
| - exists_grid_spec | |
| - bet_betCood | |
| - Cd_Col | |
| - exists_projp | |
| - same_abscissa_col | |
| - path: theories/Main/Annexes/Tagged_predicates.v | |
| theorems: | |
| - Par_tagged_Par | |
| - Cong_Cong_tagged | |
| - Perp_in_Perp_in_tagged | |
| - Bet_tagged_Bet | |
| - Diff_perm | |
| - Per_Per_tagged | |
| - Par_strict_Par_strict_tagged | |
| - Plg_Plg_tagged | |
| - Perp_tagged_Perp | |
| - Cong_tagged_Cong | |
| - Diff_Diff_tagged | |
| - Bet_Bet_tagged | |
| - Mid_Mid_tagged | |
| - Perp_in_tagged_Perp_in | |
| - Par_strict_tagged_Par_strict | |
| - Per_tagged_Per | |
| - NCol_NCol_tagged | |
| - Col_Col_tagged | |
| - Col_tagged_Col | |
| - Perp_Perp_tagged | |
| - Plg_tagged_Plg | |
| - path: theories/Elements/OriginalProofs/lemma_supplements2.v | |
| theorems: | |
| - lemma_supplements2 | |
| - path: theories/Algebraic/Counter_models/nD/independent_version_to_tarski.v | |
| theorems: | |
| - bet_axx | |
| - Coplanar_dec | |
| - Col_dec | |
| - Col_TCol | |
| - Cop_TCop | |
| - altIT_to_T_PED | |
| - altIT_euclidean_to_T_euclidean | |
| - altIT_to_T | |
| - NCol_TNCol | |
| - path: theories/Algebraic/Counter_models/nD/counter_model_bet_symmetry.v | |
| theorems: | |
| - ColP | |
| - bet_inner_transitivity | |
| - five_segment | |
| - CopP | |
| - cong_pseudo_reflexivity | |
| - cong_identity | |
| - col__colI | |
| - bet_symmetry | |
| - inner_pasch | |
| - euclid | |
| - cong_inner_transitivity | |
| - path: theories/Main/Tarski_dev/Ch13_6_Desargues_Hessenberg.v | |
| theorems: | |
| - l13_15 | |
| - l13_19_par_aux | |
| - l13_18 | |
| - l13_15_par | |
| - l13_19_par | |
| - l13_19_aux | |
| - l13_15_2 | |
| - l13_18_2 | |
| - l13_15_1 | |
| - l13_15_2_aux | |
| - l13_18_3 | |
| - path: theories/Main/Tactics/CoincR_for_concy.v | |
| theorems: | |
| - collect_coincs_for_concy | |
| - collect_wds_for_concy | |
| - test_coinc_ok_for_concy | |
| - path: theories/Elements/OriginalProofs/lemma_parallelflip.v | |
| theorems: | |
| - lemma_parallelflip | |
| - path: theories/Main/Meta_theory/Models/tarski_to_coinc_theory_for_col.v | |
| theorems: | |
| - Tarski_is_a_Coinc_predicates_for_col | |
| - diff_perm_1 | |
| - Tarski_is_a_Arity_for_col | |
| - col_3 | |
| - col_perm_2 | |
| - Tarski_is_a_Coinc_theory_for_col | |
| - diff_perm_2 | |
| - col_bd | |
| - path: theories/Elements/OriginalProofs/proposition_44A.v | |
| theorems: | |
| - proposition_44A | |
| - path: theories/Main/Tarski_dev/Ch13_5_Pappus_Pascal.v | |
| theorems: | |
| - cop_par__perp2 | |
| - l13_10_aux3 | |
| - l13_6_bis | |
| - per13_preserves_bet_inv | |
| - lcos3_lcos2 | |
| - cop_per2__perp | |
| - lcos_lcos_cop__col | |
| - l13_10_aux1 | |
| - l13_11 | |
| - l13_14 | |
| - l13_10_aux5 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/legendre.v | |
| theorems: | |
| - legendre_s_third_theorem_aux | |
| - legendre_s_fourth_theorem | |
| - legendre_aux | |
| - legendre_s_fourth_theorem_aux | |
| - legendre_s_second_theorem | |
| - stronger_legendre_s_first_theorem | |
| - legendre_aux1 | |
| - legendre_s_first_theorem | |
| - path: theories/Algebraic/Counter_models/Planar/counter_model_cong_pseudo_reflexivity.v | |
| theorems: | |
| - bet_inner_transitivity | |
| - lower_dim | |
| - bet_symmetry | |
| - inner_pasch | |
| - euclid | |
| - cong_identity | |
| - upper_dim | |
| - five_segment | |
| - cong_pseudo_reflexivity | |
| - path: theories/Main/Meta_theory/Parallel_postulates/par_trans_NID.v | |
| theorems: | |
| - col2_dec | |
| - playfair__playfair_ter | |
| - not_ex_forall_not_7 | |
| - playfair_quater__playfair | |
| - par_trans__par_dec | |
| - par_dec_NID | |
| - playfair_ter__playfair | |
| - path: theories/Algebraic/Counter_models/nD/counter_model_segment_construction.v | |
| theorems: | |
| - Midpoint_midpoint | |
| - bet'__bet | |
| - coplanar_1324 | |
| - nCol__ncol | |
| - cong_midpoint | |
| - euclid | |
| - col__ColT | |
| - midpointBR | |
| - five_segment | |
| - Col_aaa | |
| - bet_midpoint | |
| - bet_symmetry | |
| - midpointLR | |
| - midpointBL | |
| - col__colI | |
| - nCol__nCol' | |
| - cong_identity | |
| - segment_construction | |
| - cong_pseudo_reflexivity | |
| - onemx_neq0 | |
| - betR_midpoint | |
| - bet_inner_transitivity | |
| - ncol__nCol | |
| - ColT__Col | |
| - coplanar_midpoint | |
| - midpointxx | |
| - cong_inner_transitivity | |
| - coplanar_3214 | |
| - Col'__Col | |
| - cong_abab | |
| - inner_pasch | |
| - path: theories/Elements/OriginalProofs/proposition_39.v | |
| theorems: | |
| - proposition_39 | |
| - path: theories/Algebraic/Counter_models/nD/independence.v | |
| theorems: | |
| - decidability__stability | |
| - big_andb_and | |
| - tnth_nseq | |
| - nth_nseq | |
| - path: theories/Main/Meta_theory/Continuity/elementary_continuity_props.v | |
| theorems: | |
| - equivalent_variants_of_line_circle | |
| - equivalent_variants_of_circle_circle | |
| - circle_circle_bis__euclid_22 | |
| - circle_circle__circle_circle_two | |
| - circle_circle_bis__circle_circle_axiom | |
| - euclid_22__circle_circle | |
| - circle_circle_bis__one_point_line_circle | |
| - circle_circle__circle_circle_bis | |
| - path: theories/Main/Highschool/midpoint_thales.v | |
| theorems: | |
| - midpoint_thales_reci | |
| - midpoint_thales | |
| - path: theories/Elements/OriginalProofs/proposition_14.v | |
| theorems: | |
| - proposition_14 | |
| - path: theories/Main/Tarski_dev/Ch03_bet.v | |
| theorems: | |
| - l3_9_4 | |
| - bet_neq21__neq | |
| - outer_transitivity_between | |
| - l3_17 | |
| - bet_dec_eq_dec_b | |
| - cong_dec_eq_dec_b | |
| - bet_neq12__neq | |
| - between_equality | |
| - not_bet_distincts | |
| - between_exchange2 | |
| - between_exchange4 | |
| - l2_11_b | |
| - Bet_perm | |
| - between_symmetry | |
| - between_trivial2 | |
| - between_exchange3 | |
| - outer_transitivity_between2 | |
| - between_inner_transitivity | |
| - bet_col | |
| - between_trivial | |
| - between_equality_2 | |
| - bet_neq32__neq | |
| - another_point | |
| - BetSEq | |
| - Bet_cases | |
| - point_construction_different | |
| - two_distinct_points | |
| - path: theories/Elements/OriginalProofs/proposition_40.v | |
| theorems: | |
| - proposition_40 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/rah_thales_postulate.v | |
| theorems: | |
| - rah__thales_postulate | |
| - path: theories/Main/Meta_theory/Dimension_axioms/upper_dim_3.v | |
| theorems: | |
| - median_planes_implies_upper_dim | |
| - upper_dim_3_equivalent_axioms | |
| - upper_dim_3_stab | |
| - median_planes_aux | |
| - orthonormal_family_axiom_implies_space_separation | |
| - upper_dim_implies_orthonormal_family_axiom | |
| - orthonormal_family_axiom_implies_orth_at2__col | |
| - orthonormal_family_aux | |
| - space_separation_implies_median_planes | |
| - orthonormal_family_axiom_implies_not_two_sides_one_side | |
| - plane_intersection_implies_space_separation | |
| - space_separation_implies_plane_intersection | |
| - path: theories/Main/Annexes/tangency.v | |
| theorems: | |
| - tangentat_perp | |
| - tangency_chara2 | |
| - tangent_neq | |
| - intercc__neq | |
| - tangency_chara | |
| - interccat__ncol | |
| - tangent_out | |
| - tangentcc__neq | |
| - onc2__oreq | |
| - tangency_chara3 | |
| - cop_onc2__oreq | |
| - diam_not_tangent | |
| - path: theories/Elements/OriginalProofs/proposition_26A.v | |
| theorems: | |
| - proposition_26A | |
| - path: theories/Main/Meta_theory/Parallel_postulates/triangle_playfair_bis.v | |
| theorems: | |
| - legendre_aux1 | |
| - triangle__playfair_bis | |
| - legendre_aux2 | |
| - legendre_aux | |
| - path: theories/Elements/OriginalProofs/lemma_angledistinct.v | |
| theorems: | |
| - lemma_angledistinct | |
| - path: theories/Elements/OriginalProofs/lemma_squaresequal.v | |
| theorems: | |
| - lemma_squaresequal | |
| - path: theories/Main/Annexes/perp_bisect.v | |
| theorems: | |
| - cong_mid_perp_bisect | |
| - perp_bisect_existence_cop | |
| - perp_bisect_cong | |
| - perp_bisect_equiv_def | |
| - perp_bisect_sym_1 | |
| - perp_bisect_cong_1 | |
| - perp_bisect_cop2_existence | |
| - cong_cop_perp_bisect | |
| - perp_bisect_sym_2 | |
| - cong_cop2_perp_bisect_col | |
| - perp_bisect_cong2 | |
| - perp_bisect_sym_3 | |
| - perp_mid_perp_bisect | |
| - perp_bisect_cong_2 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/rectangle_existence_rah.v | |
| theorems: | |
| - rectangle_existence__rah | |
| - path: theories/Main/Meta_theory/Models/tarski_to_cong_theory.v | |
| theorems: | |
| - Tarski_is_a_Cong_theory | |
| - path: theories/Main/Meta_theory/Continuity/completeness.v | |
| theorems: | |
| - line_completeness_aux | |
| - ncompleteness_for_planes__upper_dim | |
| - nupper_dim_3__completeness_for_3d_spaces | |
| - extension_to_3d__upper_dim_3 | |
| - line_completeness__completeness_for_planes | |
| - line_extension_reverse_bet | |
| - line_completeness__completeness_for_3d_spaces | |
| - not_archimedes__line_completeness | |
| - line_extension_stability | |
| - extension_reverse_col | |
| - pres_bet_line__col | |
| - ncompleteness_for_3d_spaces__upper_dim_3 | |
| - extension__line_extension | |
| - nupper_dim__completeness_for_planes | |
| - archimedes_aux | |
| - line_extension_symmetry | |
| - path: theories/Elements/OriginalProofs/lemma_parallelsymmetric.v | |
| theorems: | |
| - lemma_parallelsymmetric | |
| - path: theories/Elements/OriginalProofs/proposition_45.v | |
| theorems: | |
| - proposition_45 | |
| - path: theories/Main/Highschool/bisector.v | |
| theorems: | |
| - bisector_perp_equality | |
| - not_col_bfoot_not_equality | |
| - perp_equality_bisector | |
| - not_col_efoot_not_equality | |
| - bisector_existence | |
| - equality_foot_out_out | |
| - path: theories/Elements/OriginalProofs/lemma_Euclid4.v | |
| theorems: | |
| - lemma_Euclid4 | |
| - path: theories/Main/Annexes/defect.v | |
| theorems: | |
| - defect_distincts | |
| - t22_16_1bis | |
| - ex_defect | |
| - defect_perm_231 | |
| - defect_perm_321 | |
| - t22_16_1 | |
| - defect2__conga | |
| - defect_perm_213 | |
| - rah_defect__out | |
| - t22_16_2aux1 | |
| - conga_defect__defect | |
| - col_defect__out | |
| - conga3_defect__defect | |
| - defect_ncol_out__rah | |
| - defect_perm_132 | |
| - t22_16_2aux | |
| - t22_16_2 | |
| - path: theories/Main/Meta_theory/Continuity/grad.v | |
| theorems: | |
| - grad__col | |
| - gradexp_clos_trans | |
| - gradexp2__gradexp123 | |
| - gradexp2__gradexp456 | |
| - grad112__eq | |
| - grad121__eq | |
| - grad__ex_gradexp_le | |
| - bet_cong2_grad__grad2 | |
| - grad_neq__neq13 | |
| - gradexp__grad | |
| - grad2__grad123 | |
| - gradexpinv_clos_trans | |
| - grad2__grad456 | |
| - reach__ex_gradexp_le | |
| - gradexp__gradexpinv | |
| - reach__ex_gradexp_lt | |
| - grad__le | |
| - grad__bet | |
| - reach__grad_min | |
| - path: theories/Elements/OriginalProofs/lemma_lessthancongruence.v | |
| theorems: | |
| - lemma_lessthancongruence | |
| - path: theories/Main/Meta_theory/Parallel_postulates/triangle_existential_triangle.v | |
| theorems: | |
| - triangle__existential_triangle | |
| - path: theories/Main/Tarski_dev/Ch04_col.v | |
| theorems: | |
| - col_permutation_4 | |
| - l4_19 | |
| - col_permutation_2 | |
| - l4_16 | |
| - l4_18 | |
| - not_col_permutation_1 | |
| - l4_14 | |
| - not_col_permutation_4 | |
| - not_col_permutation_2 | |
| - col_permutation_1 | |
| - l4_13 | |
| - Col_cases | |
| - NCol_cases | |
| - NCol_perm | |
| - col_trivial_1 | |
| - not_col_distincts | |
| - not_col_permutation_3 | |
| - col_trivial_2 | |
| - l4_17 | |
| - Col_perm | |
| - not_col_permutation_5 | |
| - col_cong_3_cong_3_eq | |
| - col_trivial_3 | |
| - col_permutation_3 | |
| - path: theories/Elements/euclid_to_tarski.v | |
| theorems: | |
| - nullsegment2 | |
| - Tarski_follows_Euclid | |
| - cong_sym | |
| - Tcong_symmetric | |
| - Bet_sym | |
| - nullsegment1 | |
| - lemma_congruenceflip | |
| - cong_eq | |
| - Tcong_reflexive | |
| - TCong_neq_Cong | |
| - path: theories/Algebraic/Counter_models/nD/stronger_pasch.v | |
| theorems: | |
| - addition_segments | |
| - stronger_inner_pasch | |
| - bet_outer_connectivity | |
| - g2_24 | |
| - path: theories/Elements/OriginalProofs/lemma_10_12.v | |
| theorems: | |
| - lemma_10_12 | |
| - path: theories/Elements/OriginalProofs/lemma_angleorderrespectscongruence.v | |
| theorems: | |
| - lemma_angleorderrespectscongruence | |
| - path: theories/Main/Meta_theory/Parallel_postulates/thales_converse_postulate_weak_triangle_circumscription_principle.v | |
| theorems: | |
| - thales_converse_postulate__weak_triangle_circumscription_principle | |
| - path: theories/Elements/OriginalProofs/lemma_supplementsymmetric.v | |
| theorems: | |
| - lemma_supplementsymmetric | |
| - path: theories/Elements/OriginalProofs/lemma_squareparallelogram.v | |
| theorems: | |
| - lemma_squareparallelogram | |
| - path: theories/Main/Tactics/CoincR_for_cop.v | |
| theorems: | |
| - collect_coincs_for_cop | |
| - st_ok_empty_for_cop | |
| - test_coinc_ok_for_cop | |
| - collect_wds_for_cop | |
| - path: theories/Elements/OriginalProofs/lemma_collinear4.v | |
| theorems: | |
| - lemma_collinear4 | |
| - path: theories/Elements/OriginalProofs/lemma_droppedperpendicularunique.v | |
| theorems: | |
| - lemma_droppedperpendicularunique | |
| - path: theories/Elements/OriginalProofs/lemma_together.v | |
| theorems: | |
| - lemma_together | |
| - path: theories/Main/Meta_theory/Continuity/aristotle.v | |
| theorems: | |
| - aristotle__obtuse_case_elimination | |
| - greenberg__aristotle | |
| - aristotle__greenberg | |
| - aristotle__acute_or_right | |
| - path: theories/Algebraic/Counter_models/nD/gupta_inspired_to_independent_version.v | |
| theorems: | |
| - GI_to_IT | |
| - GI_euclidean_to_IT_euclidean | |
| - exists_cong_per_f | |
| - bet__col | |
| - basisG | |
| - GI2D_to_T2D | |
| - path: theories/Elements/OriginalProofs/proposition_32.v | |
| theorems: | |
| - proposition_32 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/playfair_midpoint.v | |
| theorems: | |
| - playfair_s_postulate_implies_midpoint_converse_postulate | |
| - path: theories/Elements/OriginalProofs/lemma_paralleldef2A.v | |
| theorems: | |
| - lemma_paralleldef2A | |
| - path: theories/Main/Meta_theory/Continuity/dedekind_variant.v | |
| theorems: | |
| - dedekind_equiv | |
| - path: theories/Elements/OriginalProofs/lemma_fiveline.v | |
| theorems: | |
| - lemma_fiveline | |
| - path: theories/Elements/OriginalProofs/lemma_diagonalsbisect.v | |
| theorems: | |
| - lemma_diagonalsbisect | |
| - path: theories/Algebraic/Counter_models/Planar/counter_model_bet_inner_transitivity.v | |
| theorems: | |
| - inner_pasch | |
| - lower_dim | |
| - cong_pseudo_reflexivity | |
| - bet_symmetry | |
| - five_segment | |
| - cong_inner_transitivity | |
| - col_alt_def | |
| - cong_identity | |
| - upper_dim | |
| - bet_inner_transitivity | |
| - path: theories/Elements/OriginalProofs/lemma_congruencesymmetric.v | |
| theorems: | |
| - lemma_congruencesymmetric | |
| - path: theories/Elements/OriginalProofs/lemma_collinearorder.v | |
| theorems: | |
| - lemma_collinearorder | |
| - path: theories/Elements/OriginalProofs/proposition_22.v | |
| theorems: | |
| - lemma_togethera | |
| - proposition_22 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/original_spp_inverse_projection_postulate.v | |
| theorems: | |
| - original_spp__inverse_projection_postulate | |
| - path: theories/Elements/OriginalProofs/lemma_paralleldef2B.v | |
| theorems: | |
| - lemma_paralleldef2B | |
| - path: theories/Main/Meta_theory/Parallel_postulates/alternate_interior_angles_proclus.v | |
| theorems: | |
| - alternate_interior__proclus_aux | |
| - alternate_interior__proclus | |
| - path: theories/Elements/OriginalProofs/lemma_parallelNC.v | |
| theorems: | |
| - lemma_parallelNC | |
| - path: theories/Main/Meta_theory/Models/tarski_to_coinc_theory_for_concyclic.v | |
| theorems: | |
| - concy_perm_1 | |
| - concy_perm_2 | |
| - Tarski_is_a_Arity_for_concy | |
| - concyclic_gen_1123 | |
| - concy_bd | |
| - Tarski_is_a_Coinc_theory_for_concy | |
| - not_col_perm_2 | |
| - Tarski_is_a_Coinc_predicates_for_concy | |
| - path: theories/Main/Tactics/CoincR_for_col.v | |
| theorems: | |
| - st_ok_empty_for_col | |
| - collect_coincs_for_col | |
| - collect_wds_for_col | |
| - test_coinc_ok_for_col | |
| - path: theories/Elements/OriginalProofs/lemma_angleordertransitive.v | |
| theorems: | |
| - lemma_angleordertransitive | |
| - path: theories/Main/Highschool/Euler_line.v | |
| theorems: | |
| - Euler_line_special_case | |
| - gravity_center_change_triangle | |
| - concyclic_not_col_or_eq_aux | |
| - Euler_line | |
| - concyclic_not_col_or_eq | |
| - path: theories/Main/Meta_theory/Parallel_postulates/par_perp_2_par_par_perp_perp.v | |
| theorems: | |
| - par_perp_2_par_implies_par_perp_perp | |
| - path: theories/Elements/OriginalProofs/lemma_rectanglerotate.v | |
| theorems: | |
| - lemma_rectanglerotate | |
| - path: theories/Main/Meta_theory/Parallel_postulates/SPP_ID.v | |
| theorems: | |
| - strong_parallel_postulate_implies_inter_dec | |
| - path: theories/Elements/OriginalProofs/lemma_26helper.v | |
| theorems: | |
| - lemma_26helper | |
| - path: theories/Elements/OriginalProofs/lemma_supplementofright.v | |
| theorems: | |
| - lemma_supplementofright | |
| - path: theories/Elements/OriginalProofs/lemma_samesidetransitive.v | |
| theorems: | |
| - lemma_samesidetransitive | |
| - path: theories/Elements/OriginalProofs/lemma_lessthanadditive.v | |
| theorems: | |
| - lemma_lessthanadditive | |
| - path: theories/Elements/OriginalProofs/proposition_39A.v | |
| theorems: | |
| - proposition_39A | |
| - path: theories/Main/Meta_theory/Continuity/dedekind_completeness.v | |
| theorems: | |
| - dedekind_variant__completeness | |
| - extension_grad | |
| - extension_gradexp | |
| - path: theories/Elements/OriginalProofs/lemma_partnotequalwhole.v | |
| theorems: | |
| - lemma_partnotequalwhole | |
| - path: theories/Main/Meta_theory/Continuity/archimedes.v | |
| theorems: | |
| - t22_24 | |
| - archi__obtuse_case_elimination | |
| - t22_24_aux1 | |
| - t22_19 | |
| - t22_23 | |
| - t22_18 | |
| - t22_24_aux | |
| - path: theories/Main/Meta_theory/Parallel_postulates/weak_triangle_circumscription_principle_bachmann_s_lotschnittaxiom.v | |
| theorems: | |
| - weak_triangle_circumscription_principle__bachmann_s_lotschnittaxiom | |
| - path: theories/Elements/OriginalProofs/proposition_36.v | |
| theorems: | |
| - proposition_36 | |
| - path: theories/Elements/OriginalProofs/lemma_3_7b.v | |
| theorems: | |
| - lemma_3_7b | |
| - path: theories/Main/Meta_theory/Decidability/equivalence_between_decidability_properties_of_basic_relations.v | |
| theorems: | |
| - equivalence_between_decidability_properties_of_basic_relations | |
| - eq_dec_bet_dec | |
| - cong_dec_eq_dec | |
| - eq_dec_cong_dec | |
| - path: theories/Elements/OriginalProofs/lemma_oppositesidesymmetric.v | |
| theorems: | |
| - lemma_oppositesidesymmetric | |
| - path: theories/Main/Highschool/sesamath_exercises.v | |
| theorems: | |
| - sesamath_4ieme_G2_ex39 | |
| - sesamath_4ieme_G2_ex36 | |
| - sesamath_4ieme_G2_ex38 | |
| - sesamath_4ieme_G2_ex47 | |
| - sesamath_4ieme_G2_ex41 | |
| - sesamath_4ieme_G2_ex36_aux | |
| - sesamath_4ieme_G2_ex35 | |
| - path: theories/Elements/OriginalProofs/lemma_collinearright.v | |
| theorems: | |
| - lemma_collinearright | |
| - path: theories/Elements/OriginalProofs/lemma_equalanglestransitive.v | |
| theorems: | |
| - lemma_equalanglestransitive | |
| - path: theories/Algebraic/Counter_models/nD/counter_model_cong_pseudo_reflexivity.v | |
| theorems: | |
| - bet_symmetry | |
| - inner_pasch | |
| - euclid | |
| - five_segment | |
| - cong_pseudo_reflexivity | |
| - cong_identity | |
| - cong_inner_transitivity | |
| - onemx_neq0 | |
| - ColP | |
| - path: theories/Elements/OriginalProofs/lemma_Pasch_outer2.v | |
| theorems: | |
| - lemma_Pasch_outer2 | |
| - path: theories/Elements/OriginalProofs/lemma_differenceofparts.v | |
| theorems: | |
| - lemma_differenceofparts | |
| - path: theories/Elements/OriginalProofs/lemma_Playfairhelper2.v | |
| theorems: | |
| - lemma_Playfairhelper2 | |
| - path: theories/Elements/OriginalProofs/lemma_ABCequalsCBA.v | |
| theorems: | |
| - lemma_ABCequalsCBA | |
| - path: theories/Algebraic/Counter_models/nD/counter_model_lower_dim.v | |
| theorems: | |
| - upper_dim | |
| - nth_basis | |
| - mulmx11_eq0 | |
| - bet_cong__mid | |
| - zeromxE | |
| - path: theories/Elements/OriginalProofs/lemma_altitudebisectsbase.v | |
| theorems: | |
| - lemma_altitudebisectsbase | |
| - path: theories/Main/Meta_theory/Parallel_postulates/midpoint_playfair.v | |
| theorems: | |
| - midpoint_converse_postulate_implies_playfair | |
| - path: theories/Main/Meta_theory/Parallel_postulates/SPP_tarski.v | |
| theorems: | |
| - impossible_case_8 | |
| - strong_parallel_postulate_implies_tarski_s_euclid_aux | |
| - impossible_case_6 | |
| - impossible_case_5 | |
| - path: theories/Elements/OriginalProofs/lemma_angleaddition.v | |
| theorems: | |
| - lemma_angleaddition | |
| - path: theories/Elements/OriginalProofs/proposition_28A.v | |
| theorems: | |
| - proposition_28A | |
| - path: theories/Algebraic/Counter_models/Planar/counter_model_lower_dim.v | |
| theorems: | |
| - segment_construction | |
| - point_equality_decidability | |
| - cong_identity | |
| - five_segment | |
| - cong_pseudo_reflexivity | |
| - inner_pasch | |
| - cong_inner_transitivity | |
| - upper_dim | |
| - lower_dim | |
| - path: theories/Elements/OriginalProofs/lemma_equaltorightisright.v | |
| theorems: | |
| - lemma_equaltorightisright | |
| - path: theories/Elements/OriginalProofs/proposition_01.v | |
| theorems: | |
| - proposition_01 | |
| - path: theories/Elements/OriginalProofs/lemma_tworays.v | |
| theorems: | |
| - lemma_tworays | |
| - path: theories/Main/Meta_theory/Parallel_postulates/original_euclid_original_spp.v | |
| theorems: | |
| - original_euclid__original_spp | |
| - path: theories/Main/Meta_theory/Continuity/first_order.v | |
| theorems: | |
| - dedekind__fod | |
| - path: theories/Algebraic/Counter_models/Planar/gupta_inspired_to_independent_tarski.v | |
| theorems: | |
| - GI_euclidean_to_IT_euclidean | |
| - GI2D_to_IT2D | |
| - GI_to_IT | |
| - path: theories/Elements/OriginalProofs/lemma_supplements.v | |
| theorems: | |
| - lemma_supplements | |
| - path: theories/Main/Meta_theory/Parallel_postulates/par_trans_playfair.v | |
| theorems: | |
| - par_trans_implies_playfair | |
| - path: theories/Elements/OriginalProofs/lemma_parallelcollinear2.v | |
| theorems: | |
| - lemma_parallelcollinear2 | |
| - path: theories/Elements/OriginalProofs/proposition_03.v | |
| theorems: | |
| - proposition_03 | |
| - path: theories/Elements/OriginalProofs/lemma_parallelbetween.v | |
| theorems: | |
| - lemma_parallelbetween | |
| - path: theories/Main/Highschool/incenter.v | |
| theorems: | |
| - incenter_permut132 | |
| - incenter_dec | |
| - incenter_permut231 | |
| - incenter_permut321 | |
| - incenter_exists | |
| - path: theories/Elements/OriginalProofs/lemma_raystrict.v | |
| theorems: | |
| - lemma_raystrict | |
| - path: theories/Elements/OriginalProofs/lemma_NCdistinct.v | |
| theorems: | |
| - lemma_NCdistinct | |
| - path: theories/Elements/OriginalProofs/lemma_tarskiparallelflip.v | |
| theorems: | |
| - lemma_tarskiparallelflip | |
| - path: theories/Main/Meta_theory/Parallel_postulates/szmielew.v | |
| theorems: | |
| - szmielew_s_theorem | |
| - aah__hpp | |
| - path: theories/Coinc/Utils/general_tactics.v | |
| theorems: | |
| - ltac_something_eq | |
| - ltac_something_hide | |
| - path: theories/Main/Meta_theory/Continuity/first_order_dedekind_circle_circle.v | |
| theorems: | |
| - circle_circle_aux | |
| - path: theories/Main/Highschool/exercises.v | |
| theorems: | |
| - Per_mid_rectangle | |
| - path: theories/Elements/OriginalProofs/lemma_doublereverse.v | |
| theorems: | |
| - lemma_doublereverse | |
| - path: theories/Main/Meta_theory/Parallel_postulates/bachmann_s_lotschnittaxiom_legendre_s_parallel_postulate.v | |
| theorems: | |
| - bachmann_s_lotschnittaxiom__legendre_s_parallel_postulate | |
| - path: theories/Main/Meta_theory/Parallel_postulates/weak_tarski_s_parallel_postulate_weak_inverse_projection_postulate.v | |
| theorems: | |
| - weak_tarski_s_parallel_postulate__weak_inverse_projection_postulate_aux | |
| - path: theories/Main/Meta_theory/Parallel_postulates/posidonius_postulate_rah.v | |
| theorems: | |
| - posidonius_postulate__rah | |
| - path: theories/Main/Meta_theory/Models/tarski_to_coinc_theory_for_cop.v | |
| theorems: | |
| - cop_3 | |
| - cop_bd | |
| - cop_perm_2 | |
| - Tarski_is_a_Arity_for_cop | |
| - cop_perm_1 | |
| - not_col_perm_2 | |
| - Tarski_is_a_Coinc_theory_for_cop | |
| - path: theories/Elements/OriginalProofs/lemma_subtractequals.v | |
| theorems: | |
| - lemma_subtractequals | |
| - path: theories/Elements/OriginalProofs/proposition_35.v | |
| theorems: | |
| - proposition_35 | |
| - path: theories/Elements/OriginalProofs/proposition_33.v | |
| theorems: | |
| - proposition_33 | |
| - path: theories/Elements/OriginalProofs/lemma_RTcongruence.v | |
| theorems: | |
| - lemma_RTcongruence | |
| - path: theories/Elements/OriginalProofs/proposition_30A.v | |
| theorems: | |
| - proposition_30A | |
| - path: theories/Main/Meta_theory/Parallel_postulates/proclus_SPP.v | |
| theorems: | |
| - proclus_s_postulate_implies_strong_parallel_postulate | |
| - path: theories/Main/Meta_theory/Parallel_postulates/rectangle_principle_rectangle_existence.v | |
| theorems: | |
| - rectangle_principle__rectangle_existence | |
| - path: theories/Elements/OriginalProofs/proposition_06.v | |
| theorems: | |
| - proposition_06 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/proclus_bis_proclus.v | |
| theorems: | |
| - proclus_bis__proclus | |
| - path: theories/Coinc/Permutations.v | |
| theorems: | |
| - PermCoincOK | |
| - PermWdOK | |
| - path: theories/Elements/OriginalProofs/lemma_PGsymmetric.v | |
| theorems: | |
| - lemma_PGsymmetric | |
| - path: theories/Elements/OriginalProofs/lemma_layoffunique.v | |
| theorems: | |
| - lemma_layoffunique | |
| - path: theories/Elements/OriginalProofs/proposition_04.v | |
| theorems: | |
| - proposition_04 | |
| - path: theories/Elements/OriginalProofs/proposition_28D.v | |
| theorems: | |
| - proposition_28D | |
| - path: theories/Main/Meta_theory/Models/tarski_continuous_to_trc.v | |
| theorems: | |
| - TC_to_TRC | |
| - path: theories/Elements/OriginalProofs/lemma_TGsymmetric.v | |
| theorems: | |
| - lemma_TGsymmetric | |
| - path: theories/Main/Meta_theory/Parallel_postulates/consecutive_interior_angles_alternate_interior_angles.v | |
| theorems: | |
| - consecutive_interior__alternate_interior | |
| - path: theories/Elements/OriginalProofs/lemma_diagonalsmeet.v | |
| theorems: | |
| - lemma_diagonalsmeet | |
| - path: theories/Elements/OriginalProofs/lemma_samesidesymmetric.v | |
| theorems: | |
| - lemma_samesidesymmetric | |
| - path: theories/Elements/OriginalProofs/lemma_betweennesspreserved.v | |
| theorems: | |
| - lemma_betweennesspreserved | |
| - path: theories/Elements/OriginalProofs/lemma_equalitysymmetric.v | |
| theorems: | |
| - lemma_equalitysymmetric | |
| - path: theories/Elements/OriginalProofs/lemma_ray2.v | |
| theorems: | |
| - lemma_ray2 | |
| - path: theories/Elements/OriginalProofs/lemma_oppositesideflip.v | |
| theorems: | |
| - lemma_oppositesideflip | |
| - path: theories/Elements/OriginalProofs/lemma_samesideflip.v | |
| theorems: | |
| - lemma_samesideflip | |
| - path: theories/Elements/OriginalProofs/lemma_8_2.v | |
| theorems: | |
| - lemma_8_2 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/playfair_par_trans.v | |
| theorems: | |
| - playfair_implies_par_trans | |
| - path: theories/Main/Meta_theory/Parallel_postulates/alternate_interior_angles_playfair_bis.v | |
| theorems: | |
| - alternate_interior__playfair_aux | |
| - alternate_interior__playfair_bis | |
| - path: theories/Main/Meta_theory/Parallel_postulates/par_perp_perp_par_perp_2_par.v | |
| theorems: | |
| - par_perp_perp_implies_par_perp_2_par | |
| - path: theories/Main/Tarski_dev/Ch15_pyth_rel.v | |
| theorems: | |
| - opp_same_square | |
| - PythRel_uniqueness | |
| - PythOK | |
| - PythRel_exists | |
| - path: theories/Elements/OriginalProofs/lemma_30helper.v | |
| theorems: | |
| - lemma_30helper | |
| - path: theories/Algebraic/Counter_models/Planar/independent_tarski_to_gupta_inspired.v | |
| theorems: | |
| - IT_euclidean_to_GI_euclidean | |
| - IT_to_GI | |
| - IT2D_to_GI2D | |
| - path: theories/Main/Meta_theory/Parallel_postulates/par_perp_perp_TCP.v | |
| theorems: | |
| - inter_dec_plus_par_perp_perp_imply_triangle_circumscription | |
| - path: theories/Main/Meta_theory/Parallel_postulates/weak_inverse_projection_postulate_bachmann_s_lotschnittaxiom.v | |
| theorems: | |
| - weak_inverse_projection_postulate__bachmann_s_lotschnittaxiom_aux | |
| - weak_inverse_projection_postulate__bachmann_s_lotschnittaxiom | |
| - path: theories/Main/Meta_theory/Parallel_postulates/alternate_interior_angles_triangle.v | |
| theorems: | |
| - alternate_interior__triangle | |
| - path: theories/Elements/OriginalProofs/proposition_36A.v | |
| theorems: | |
| - proposition_36A | |
| - path: theories/Elements/OriginalProofs/lemma_equalanglessymmetric.v | |
| theorems: | |
| - lemma_equalanglessymmetric | |
| - path: theories/Main/Meta_theory/Parallel_postulates/thales_postulate_thales_converse_postulate.v | |
| theorems: | |
| - thales_postulate__thales_converse_postulate | |
| - path: theories/Main/Highschool/varignon.v | |
| theorems: | |
| - varignon_aux | |
| - varignon' | |
| - varignon | |
| - path: theories/Main/Meta_theory/Parallel_postulates/tarski_playfair.v | |
| theorems: | |
| - tarski_s_euclid_implies_playfair | |
| - path: theories/Main/Meta_theory/Parallel_postulates/playfair_bis_playfair.v | |
| theorems: | |
| - playfair_bis__playfair | |
| - path: theories/Main/Meta_theory/Models/tarski_to_col_theory.v | |
| theorems: | |
| - Tarski_is_a_Col_theory | |
| - path: theories/Elements/OriginalProofs/proposition_25.v | |
| theorems: | |
| - proposition_25 | |
| - path: theories/Main/Meta_theory/Continuity/dedekind_archimedes.v | |
| theorems: | |
| - archimedes_aux | |
| - dedekind_variant__archimedes | |
| - path: theories/Elements/OriginalProofs/lemma_crossbar.v | |
| theorems: | |
| - lemma_crossbar | |
| - path: theories/Elements/OriginalProofs/proposition_11.v | |
| theorems: | |
| - proposition_11 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/existential_playfair_rah.v | |
| theorems: | |
| - existential_playfair__rah | |
| - path: theories/Algebraic/Counter_models/nD/stability_properties.v | |
| theorems: | |
| - two_points | |
| - point_equality_stability__betS_stability | |
| - cong_stability__point_equality_stability | |
| - point_equality_stability__cong_stability | |
| - path: theories/Main/Meta_theory/Models/tarski_to_makarios.v | |
| theorems: | |
| - Makarios_Variant_follows_from_Tarski | |
| - lower_dim_ex | |
| - path: theories/Elements/OriginalProofs/lemma_ondiameter.v | |
| theorems: | |
| - lemma_ondiameter | |
| - path: theories/Main/Meta_theory/Parallel_postulates/alternate_interior_angles_consecutive_interior_angles.v | |
| theorems: | |
| - alternate_interior__consecutive_interior | |
| - path: theories/Elements/OriginalProofs/proposition_38.v | |
| theorems: | |
| - proposition_38 | |
| - path: theories/Elements/OriginalProofs/proposition_26B.v | |
| theorems: | |
| - proposition_26B | |
| - path: theories/Elements/OriginalProofs/proposition_19.v | |
| theorems: | |
| - proposition_19 | |
| - path: theories/Elements/OriginalProofs/lemma_trichotomy1.v | |
| theorems: | |
| - lemma_trichotomy1 | |
| - path: theories/Elements/OriginalProofs/lemma_rayimpliescollinear.v | |
| theorems: | |
| - lemma_rayimpliescollinear | |
| - path: theories/Elements/OriginalProofs/lemma_collinearparallel.v | |
| theorems: | |
| - lemma_collinearparallel | |
| - path: theories/Elements/OriginalProofs/lemma_twoperpsparallel.v | |
| theorems: | |
| - lemma_twoperpsparallel | |
| - path: theories/Elements/OriginalProofs/lemma_35helper.v | |
| theorems: | |
| - lemma_35helper | |
| - path: theories/Elements/OriginalProofs/lemma_extensionunique.v | |
| theorems: | |
| - lemma_extensionunique | |
| - path: theories/Elements/OriginalProofs/lemma_9_5a.v | |
| theorems: | |
| - lemma_9_5a | |
| - path: theories/Elements/OriginalProofs/lemma_trichotomy2.v | |
| theorems: | |
| - lemma_trichotomy2 | |
| - path: theories/Elements/OriginalProofs/lemma_righttogether.v | |
| theorems: | |
| - lemma_righttogether | |
| - path: theories/Main/Meta_theory/Parallel_postulates/euclid_5_original_euclid.v | |
| theorems: | |
| - euclid_5__original_euclid | |
| - path: theories/Main/Meta_theory/Parallel_postulates/playfair_existential_playfair.v | |
| theorems: | |
| - playfair__existential_playfair | |
| - path: theories/Elements/OriginalProofs/proposition_48A.v | |
| theorems: | |
| - proposition_48A | |
| - path: theories/Elements/OriginalProofs/proposition_28B.v | |
| theorems: | |
| - proposition_28B | |
| - path: theories/Elements/OriginalProofs/proposition_15.v | |
| theorems: | |
| - proposition_15b | |
| - proposition_15a | |
| - proposition_15 | |
| - path: theories/Elements/OriginalProofs/proposition_41.v | |
| theorems: | |
| - proposition_41 | |
| - path: theories/Elements/OriginalProofs/lemma_EFreflexive.v | |
| theorems: | |
| - lemma_EFreflexive | |
| - path: theories/Elements/OriginalProofs/lemma_equalanglesNC.v | |
| theorems: | |
| - lemma_equalanglesNC | |
| - path: theories/Elements/OriginalProofs/lemma_rectanglereverse.v | |
| theorems: | |
| - lemma_rectanglereverse | |
| - path: theories/Elements/OriginalProofs/lemma_3_6a.v | |
| theorems: | |
| - lemma_3_6a | |
| - path: theories/Elements/OriginalProofs/proposition_16.v | |
| theorems: | |
| - proposition_16 | |
| - path: theories/Elements/OriginalProofs/proposition_35A.v | |
| theorems: | |
| - proposition_35A | |
| - path: theories/Elements/OriginalProofs/proposition_43B.v | |
| theorems: | |
| - proposition_43B | |
| - path: theories/Elements/OriginalProofs/lemma_equalanglesflip.v | |
| theorems: | |
| - lemma_equalanglesflip | |
| - path: theories/Elements/OriginalProofs/lemma_rightreverse.v | |
| theorems: | |
| - lemma_rightreverse | |
| - path: theories/Elements/OriginalProofs/lemma_notperp.v | |
| theorems: | |
| - lemma_notperp | |
| - path: theories/Main/Meta_theory/Parallel_postulates/weak_inverse_projection_postulate_weak_tarski_s_parallel_postulate.v | |
| theorems: | |
| - weak_inverse_projection_postulate__weak_tarski_s_parallel_postulate | |
| - path: theories/Elements/OriginalProofs/lemma_squarerectangle.v | |
| theorems: | |
| - lemma_squarerectangle | |
| - path: theories/Elements/OriginalProofs/lemma_paste5.v | |
| theorems: | |
| - lemma_paste5 | |
| - path: theories/Elements/OriginalProofs/lemma_angletrichotomy2.v | |
| theorems: | |
| - lemma_angletrichotomy2 | |
| - path: theories/Main/Meta_theory/Continuity/cantor_variant.v | |
| theorems: | |
| - nested_bis__nested | |
| - cantor__cantor_variant | |
| - path: theories/Elements/OriginalProofs/lemma_NCorder.v | |
| theorems: | |
| - lemma_NCorder | |
| - path: theories/Elements/OriginalProofs/lemma_PGrotate.v | |
| theorems: | |
| - lemma_PGrotate | |
| - path: theories/Elements/OriginalProofs/lemma_equalangleshelper.v | |
| theorems: | |
| - lemma_equalangleshelper | |
| - path: theories/Elements/OriginalProofs/lemma_triangletoparallelogram.v | |
| theorems: | |
| - lemma_triangletoparallelogram | |
| - path: theories/Elements/OriginalProofs/proposition_06a.v | |
| theorems: | |
| - proposition_06a | |
| - path: theories/Main/Meta_theory/Parallel_postulates/rah_posidonius_postulate.v | |
| theorems: | |
| - rah__posidonius | |
| - path: theories/Elements/OriginalProofs/lemma_outerconnectivity.v | |
| theorems: | |
| - lemma_outerconnectivity | |
| - path: theories/Elements/OriginalProofs/proposition_42.v | |
| theorems: | |
| - proposition_42 | |
| - path: theories/Elements/OriginalProofs/lemma_congruenceflip.v | |
| theorems: | |
| - lemma_congruenceflip | |
| - path: theories/Main/Meta_theory/Parallel_postulates/par_perp_perp_playfair.v | |
| theorems: | |
| - par_perp_perp_implies_playfair | |
| - path: theories/Elements/OriginalProofs/lemma_localextension.v | |
| theorems: | |
| - lemma_localextension | |
| - path: theories/Elements/OriginalProofs/lemma_rightangleNC.v | |
| theorems: | |
| - lemma_rightangleNC | |
| - path: theories/Elements/OriginalProofs/lemma_layoff.v | |
| theorems: | |
| - lemma_layoff | |
| - path: theories/Elements/OriginalProofs/lemma_ETreflexive.v | |
| theorems: | |
| - lemma_ETreflexive | |
| - path: theories/Main/Meta_theory/Parallel_postulates/tarski_s_euclid_remove_degenerated_cases.v | |
| theorems: | |
| - tarski_s_euclid_remove_degenerated_cases | |
| - path: theories/Elements/OriginalProofs/lemma_8_7.v | |
| theorems: | |
| - lemma_8_7 | |
| - path: theories/Elements/OriginalProofs/lemma_twolines2.v | |
| theorems: | |
| - lemma_twolines2 | |
| - path: theories/Elements/OriginalProofs/lemma_TTflip2.v | |
| theorems: | |
| - lemma_TTflip2 | |
| - path: theories/Elements/OriginalProofs/lemma_PGflip.v | |
| theorems: | |
| - lemma_PGflip | |
| - path: theories/Elements/OriginalProofs/proposition_05.v | |
| theorems: | |
| - proposition_05 | |
| - path: theories/Elements/OriginalProofs/lemma_crossimpliesopposite.v | |
| theorems: | |
| - lemma_crossimpliesopposite | |
| - path: theories/Elements/OriginalProofs/lemma_NChelper.v | |
| theorems: | |
| - lemma_NChelper | |
| - path: theories/Main/Meta_theory/Parallel_postulates/existential_saccheri_rah.v | |
| theorems: | |
| - existential_saccheri__rah | |
| - path: theories/Elements/OriginalProofs/lemma_ray4.v | |
| theorems: | |
| - lemma_ray4 | |
| - path: theories/Elements/OriginalProofs/lemma_trapezoiddiagonals.v | |
| theorems: | |
| - lemma_trapezoiddiagonals | |
| - path: theories/Elements/OriginalProofs/lemma_supplementinequality.v | |
| theorems: | |
| - lemma_supplementinequality | |
| - path: theories/Elements/OriginalProofs/proposition_21.v | |
| theorems: | |
| - proposition_21 | |
| - path: theories/Elements/OriginalProofs/lemma_collinear2.v | |
| theorems: | |
| - lemma_collinear2 | |
| - path: theories/Elements/OriginalProofs/lemma_Playfairhelper.v | |
| theorems: | |
| - lemma_Playfairhelper | |
| - path: theories/Elements/OriginalProofs/proposition_33B.v | |
| theorems: | |
| - proposition_33B | |
| - path: theories/Elements/OriginalProofs/lemma_lessthanbetween.v | |
| theorems: | |
| - lemma_lessthanbetween | |
| - path: theories/Elements/OriginalProofs/lemma_angletrichotomy.v | |
| theorems: | |
| - lemma_angletrichotomy | |
| - path: theories/Elements/OriginalProofs/lemma_collinearparallel2.v | |
| theorems: | |
| - lemma_collinearparallel2 | |
| - path: theories/Elements/OriginalProofs/proposition_20.v | |
| theorems: | |
| - proposition_20 | |
| - path: theories/Elements/OriginalProofs/lemma_TTtransitive.v | |
| theorems: | |
| - lemma_TTtransitive | |
| - path: theories/Elements/OriginalProofs/proposition_34.v | |
| theorems: | |
| - proposition_34 | |
| - path: theories/Elements/OriginalProofs/lemma_equalanglesreflexive.v | |
| theorems: | |
| - lemma_equalanglesreflexive | |
| - path: theories/Elements/OriginalProofs/lemma_midpointunique.v | |
| theorems: | |
| - lemma_midpointunique | |
| - path: theories/Main/Meta_theory/Parallel_postulates/playfair_alternate_interior_angles.v | |
| theorems: | |
| - playfair__alternate_interior | |
| - path: theories/Elements/OriginalProofs/lemma_squareflip.v | |
| theorems: | |
| - lemma_squareflip | |
| - path: theories/Elements/OriginalProofs/lemma_interior5.v | |
| theorems: | |
| - lemma_interior5 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/universal_posidonius_postulate_par_perp_perp.v | |
| theorems: | |
| - universal_posidonius_postulate__perpendicular_transversal_postulate_aux | |
| - universal_posidonius_postulate__perpendicular_transversal_postulate | |
| - path: theories/Elements/OriginalProofs/proposition_31.v | |
| theorems: | |
| - proposition_31 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/similar_rah.v | |
| theorems: | |
| - similar__rah | |
| - similar__rah_aux | |
| - path: theories/Elements/OriginalProofs/proposition_23B.v | |
| theorems: | |
| - proposition_23B | |
| - path: theories/Elements/OriginalProofs/proposition_27B.v | |
| theorems: | |
| - proposition_27B | |
| - path: theories/Elements/OriginalProofs/lemma_sameside2.v | |
| theorems: | |
| - lemma_sameside2 | |
| - path: theories/Elements/OriginalProofs/lemma_parallelcollinear.v | |
| theorems: | |
| - lemma_parallelcollinear | |
| - path: theories/Elements/OriginalProofs/proposition_23.v | |
| theorems: | |
| - proposition_23 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/playfair_universal_posidonius_postulate.v | |
| theorems: | |
| - playfair__universal_posidonius_postulate | |
| - path: theories/Elements/OriginalProofs/proposition_29C.v | |
| theorems: | |
| - proposition_29C | |
| - path: theories/Elements/OriginalProofs/lemma_inequalitysymmetric.v | |
| theorems: | |
| - lemma_inequalitysymmetric | |
| - path: theories/Main/Meta_theory/Parallel_postulates/bachmann_s_lotschnittaxiom_weak_triangle_circumscription_principle.v | |
| theorems: | |
| - bachmann_s_lotschnittaxiom__weak_triangle_circumscription_principle | |
| - path: theories/Main/Meta_theory/Parallel_postulates/bachmann_s_lotschnittaxiom_variant.v | |
| theorems: | |
| - bachmann_s_lotschnittaxiom_aux | |
| - path: theories/Elements/OriginalProofs/lemma_betweennotequal.v | |
| theorems: | |
| - lemma_betweennotequal | |
| - path: theories/Elements/OriginalProofs/lemma_parallelPasch.v | |
| theorems: | |
| - lemma_parallelPasch | |
| - path: theories/Elements/OriginalProofs/lemma_Playfair.v | |
| theorems: | |
| - lemma_Playfair | |
| - path: theories/Elements/OriginalProofs/lemma_ray1.v | |
| theorems: | |
| - lemma_ray1 | |
| - path: theories/Elements/OriginalProofs/lemma_crisscross.v | |
| theorems: | |
| - lemma_crisscross | |
| - path: theories/Elements/OriginalProofs/proposition_18.v | |
| theorems: | |
| - proposition_18 | |
| - path: theories/Elements/OriginalProofs/proposition_17.v | |
| theorems: | |
| - proposition_17 | |
| - path: theories/Elements/OriginalProofs/proposition_46.v | |
| theorems: | |
| - proposition_46 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/proclus_aristotle.v | |
| theorems: | |
| - proclus__aristotle | |
| - path: theories/Elements/OriginalProofs/proposition_27.v | |
| theorems: | |
| - proposition_27 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/existential_triangle_rah.v | |
| theorems: | |
| - existential_triangle__rah | |
| - path: theories/Main/Meta_theory/Continuity/cantor_completeness.v | |
| theorems: | |
| - cantor__completeness | |
| - path: theories/Elements/OriginalProofs/lemma_ray3.v | |
| theorems: | |
| - lemma_ray3 | |
| - path: theories/Elements/OriginalProofs/proposition_13.v | |
| theorems: | |
| - proposition_13 | |
| - path: theories/Main/Meta_theory/Parallel_postulates/inverse_projection_postulate_proclus_bis.v | |
| theorems: | |
| - inverse_projection_postulate__proclus_bis | |
| - path: theories/Elements/OriginalProofs/lemma_squareunique.v | |
| theorems: | |
| - lemma_squareunique | |
| - path: theories/Elements/OriginalProofs/lemma_3_5b.v | |
| theorems: | |
| - lemma_3_5b | |
| - path: theories/Elements/OriginalProofs/lemma_angleorderrespectscongruence2.v | |
| theorems: | |
| - lemma_angleorderrespectscongruence2 | |
| - path: theories/Elements/OriginalProofs/lemma_altitudeofrighttriangle.v | |
| theorems: | |
| - lemma_altitudeofrighttriangle | |
| - path: theories/Elements/OriginalProofs/proposition_42B.v | |
| theorems: | |
| - proposition_42B | |
| - path: theories/Elements/OriginalProofs/proposition_29.v | |
| theorems: | |
| - proposition_29 | |
| - path: theories/Elements/OriginalProofs/proposition_47.v | |
| theorems: | |
| - proposition_47 | |
| - path: theories/Elements/OriginalProofs/lemma_PGrectangle.v | |
| theorems: | |
| - lemma_PGrectangle | |
| - path: theories/Elements/OriginalProofs/proposition_48.v | |
| theorems: | |
| - proposition_48 | |
| - path: theories/Main/Meta_theory/Continuity/archimedes_cantor_dedekind.v | |
| theorems: | |
| - archimedes_cantor__dedekind_variant | |
| - path: theories/Elements/OriginalProofs/proposition_47B.v | |
| theorems: | |
| - proposition_47B | |
| - path: theories/Elements/OriginalProofs/lemma_extension.v | |
| theorems: | |
| - lemma_extension | |
| - path: theories/Main/Meta_theory/Parallel_postulates/rah_triangle.v | |
| theorems: | |
| - rah__triangle | |
| - path: theories/Main/Meta_theory/Parallel_postulates/rah_similar.v | |
| theorems: | |
| - rah__similar | |
| - path: theories/Elements/OriginalProofs/proposition_11B.v | |
| theorems: | |
| - proposition_11B | |
| - path: theories/Elements/OriginalProofs/lemma_lessthannotequal.v | |
| theorems: | |
| - lemma_lessthannotequal | |
| - path: theories/Main/Meta_theory/Parallel_postulates/bachmann_s_lotschnittaxiom_weak_inverse_projection_postulate.v | |
| theorems: | |
| - bachmann_s_lotschnittaxiom__weak_inverse_projection_postulate | |
| - path: theories/Elements/OriginalProofs/lemma_together2.v | |
| theorems: | |
| - lemma_together2 | |