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"CompactnessConjecture.symplecticPoint_sup_finrank": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLine_eq_of_points": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplectic_isotropic_finrank_le_two": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplectic_triangle_points_collinear": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplectic_triangle_lines_eq": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_incidence_adj": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_adjacent_to_point": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_adjacent_to_line": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_common_neighbor_unique": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_four_cycle_free": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_six_cycle_free": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticPoint_card": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticIncidence_card_by_points": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticIncidence_card_by_lines": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLine_card": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticIncidence_card": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_vertex_card": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticIncidenceToEdge_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticIncidenceToEdge_surjective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_edge_card": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.quadrangle_density_certificate": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.quadrangle_rpow_density": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.quadrangleVertexCount_mul_le": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.jThetaVertex_mem": [ "propext", "Quot.sound" ], "CompactnessConjecture.not_acyclic_of_eight_cycle_copy": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.encodeFiniteGraph_not_acyclic": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.jTheta_quotient_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.jQuotient_not_acyclic": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.kGamma_quotient_injective": [ "propext", "Quot.sound" ], "CompactnessConjecture.kQuotient_not_acyclic": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.four_cycle_not_acyclic": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.six_cycle_not_acyclic": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamily_isCyclic": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticPointRelated_symm": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticPointRelated_iff_orthogonal": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticPointRelated_of_quadrangle_common_neighbor": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionGraph_base_pair_adj": [ "propext", "Quot.sound" ], "CompactnessConjecture.subdivisionGraph_center_pair_adj": [ "propext", "Quot.sound" ], "CompactnessConjecture.subdivisionPoint_pair_incidence": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionPoint_center_of_point_base": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionPoint_base_of_point_base": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionPoint_base_center_related": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionPoint_bases_unrelated": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticPointSpan_orthogonal_finrank": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticPoint_centers_span_orthogonal": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticPoint_mem_span_of_two_centers": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticPoint_point_class_avoidance": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.jTemplate_adj_color_ne": [ "propext", "Quot.sound" ], "CompactnessConjecture.kTemplate_adj_color_ne": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_no_point_jTemplate": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_no_point_jTemplate_of_bases": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_no_point_jTemplate_of_first_base": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_jTemplate_first_base_is_line": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_kTemplate_has_line_gamma": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionLine_pair_incidence": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionLine_center_of_line_base": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionLine_base_of_line_center": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionLine_base_of_line_base": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionLine_centers_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionLine_bases_disjoint": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.free_map_of_no_isolated": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.extremalNumber_monotone_of_no_isolated": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.cycleGraph_no_isolated": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.quotientGraph_no_isolated": [ "propext", "Quot.sound" ], "CompactnessConjecture.map_equiv_no_isolated": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.encodeFiniteGraph_no_isolated": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.jTemplate_no_isolated": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.kTemplate_no_isolated": [ "propext", "Quot.sound" ], "CompactnessConjecture.encodedJQuotient_no_isolated": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.encodedKQuotient_no_isolated": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamily_member_no_isolated": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.nat_le_pow_of_two_le": [ "propext", "Quot.sound" ], "CompactnessConjecture.quadrangleVertexCount_parameter_lt": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.quadrangle_prime_power_bracketing": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.quadrangle_extremal_lower_of_free": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.quadrangle_extremal_lower_padded_of_free": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.quadrangle_manuscript_scaled_density_of_gap": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.quadrangle_uniform_lower_of_prime_power_avoidance": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.four_cycle_uniform_manuscript_lower": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.six_cycle_uniform_manuscript_lower": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.common_neighbor_unique_of_four_cycle_free": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.bipartite_coloring_eq_of_common_neighbor": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.common_neighbors_triangle_eq_of_cycle_free": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.common_second_neighbors_pairwise_unrelated": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.card_nonbacktrackingNeighbor": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.fintype_card_sigma_lower": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.card_nonbacktrackingFourPath_lower": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.nonbacktrackingFourPath_endpoint_ne": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.nonbacktrackingFourPath_endpoint_unrelated": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.nonbacktrackingFourPathPair_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.nonbacktrackingFourPathWitness_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.four_path_endpoint_witness_count_lower": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.fourPathEndpointWitness_card_eq_sum": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.four_path_common_second_neighbor_sum_lower": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.commonNeighborIndependent_neighborhood_injective": [ "propext", "Quot.sound" ], "CompactnessConjecture.commonNeighborIndependent_sum_degree_le_card": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.commonNeighborIndependent_card_mul_degree_le": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.card_nonbacktrackingThreePath_lower": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.nonbacktrackingThreePathEndpoint_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.girthEight_degree_mul_pred_sq_le_card": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionPairVertex_ne_base": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionPairVertex_ne_center": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionVertexImage_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionVertexImage_map_relation": [ "propext", "Quot.sound" ], "CompactnessConjecture.fiberRepresentative_spec": [ "Classical.choice" ], "CompactnessConjecture.kernelNormalForm_spec": [ "Classical.choice" ], "CompactnessConjecture.kernelNormalForm_eq_iff": [ "Classical.choice" ], "CompactnessConjecture.kernelNormalForm_idempotent": [ "Classical.choice" ], "CompactnessConjecture.kernelNormalForm_fixed": [ "Classical.choice" ], "CompactnessConjecture.kernelNormalForm_jAdmissible": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.kernelNormalForm_kAdmissible": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamilyFree_no_jTemplate": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamilyFree_no_kTemplate": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricQuadratic_eq_bilinear": [ "propext" ], "CompactnessConjecture.symmetricDet_zero_diagonal_sub": [ "propext" ], "CompactnessConjecture.symmetricQuadratic_char_two": [ "propext", "Quot.sound" ], "CompactnessConjecture.symmetricQuadratic_char_two_eq_square": [ "propext", "Quot.sound" ], "CompactnessConjecture.square_surjective_char_two": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricQuadratic_char_two_diagonal_zero_of_two_independent_roots": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricQuadraticEvaluationMatrix_det": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricQuadratic_no_three_independent_roots": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricQuadratic_no_three_roots_of_det_ne_zero": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricDet_zero_diagonal_sub_ne_zero": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.extremalScale_pos": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.extremalScale_nonneg": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.manuscriptLowerConstant_pos": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.not_compact_of_separation": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamily_not_compact_of_bounds": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.not_compactnessConjecture_of_bounds": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.finite_heavy_fiber_mass_half": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.unrelated_four_path_endpoint_card_le": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.common_second_neighbor_pairwise_unrelated": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.four_path_heavy_common_second_neighbor_mass_lower": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.mem_thetaBaseExtensions": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gluedJBase_jBase": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gluedJVertex_jThetaVertex": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.inJCopy_iff_exists_jThetaVertex": [ "propext", "Quot.sound" ], "CompactnessConjecture.theta_base_pair_adj": [ "propext", "Quot.sound" ], "CompactnessConjecture.theta_center_pair_adj": [ "propext", "Quot.sound" ], "CompactnessConjecture.gluedJVertex_map_relation": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gluedJHom_injOn_marked_copy": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gluedJBase_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.thetaCopy_base_center_color_eq": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.thetaCopy_base_color_eq": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gluedThetaBase_color_eq": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gluedJBase_color_eq": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gluedJVertex_color_false_iff": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gluedJHom_color_respecting": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.thetaBaseExtensions_commonNeighborIndependent": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.thetaBaseExtensions_card_mul_degree_le": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.mem_tripleCommonCenters": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.mem_thetaBaseExtensions_of_girthEightCenters": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.mem_thetaBaseExtensions_of_two_common_centers": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.choose_three_factorial_identity": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.choose_three_cubic_lower": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.four_path_common_second_neighbor_triple_mass_lower": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamilyFree_four_path_triple_mass_lower": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.orderedThetaTripleCount_mul_degree_le": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gammaGood_of_three_common_centers": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gamma_base_pair_adj": [ "propext", "Quot.sound" ], "CompactnessConjecture.gamma_center_pair_adj": [ "propext", "Quot.sound" ], "CompactnessConjecture.gammaCopy_vertex_color_false_iff": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.subdivisionRelation_adj": [ "propext", "Quot.sound" ], "CompactnessConjecture.gluedKVertex_map_relation": [ "propext", "Quot.sound" ], "CompactnessConjecture.gluedKHom_injOn_marked_copy": [ "propext", "Quot.sound" ], "CompactnessConjecture.gluedKVertex_color_false_iff": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gluedKHom_color_respecting": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamilyFree_not_adj_gammaGood": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.mem_gammaBadVertices": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamilyFree_edge_has_gammaBad": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.edgeFinset_card_le_sum_degree_of_vertex_cover": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamilyFree_edge_card_le_gammaBad_degree_sum": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.finite_bad_fiber_card_le_two": [ "propext", "Quot.sound" ], "CompactnessConjecture.finite_bad_fiber_mass_le_two": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.mem_commonCenterFinset": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.independentThetaTripleBase_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.independentThetaTripleBase_mem": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.independentThetaTripleBase_surjective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.commonCenterFinset_eq_tripleCommonCenters": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.independentThetaTripleBase_unrelated": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gammaGood_of_independentThetaTriple_fiber": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.independentThetaTripleOrderedWitness_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.orderedThetaWitness_card": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.independentThetaTriple_card_le_orderedThetaTripleCount": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gamma_bad_triple_fiber_mass_le_two_orderedTheta": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.mem_commonSecondNeighborFinset": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.commonSecondNeighborFinset_card": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.fourPathTripleToIndependentThetaTriple_center_mem": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.fourPathTripleToIndependentThetaTriple_endpoint_mem": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.badIndependentThetaTriple_other_center_unique": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.badFourPathTripleToBadIndependentTriple_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.badFourPathTripleWitness_card": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.badIndependentTripleWitness_card": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gammaBad_four_path_triple_mass_le_bad_fiber_mass": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gammaBad_four_path_triple_mass_le_two_orderedTheta": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gammaBad_card_mul_heavyTripleLower_le_two_orderedTheta": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.gammaBad_card_mul_fourpath_power_le": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamilyFree_edge_mul_pred_sq_le_bad_card_mul": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.fourPathHeavyThreshold_low_degree_fourth_le": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.quantitative_minimum_degree_edge_bound": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.quantitative_bad_vertex_edge_bound": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.quantitative_bad_vertex_heavy_triple_bound": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamilyFree_minDegree_polynomial_le": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamilyFree_minDegree_sixteenth_power_le": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricGraphVector_orthogonal": [ "propext", "Quot.sound" ], "CompactnessConjecture.coordinateCenterLinearMap_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricGraphLinearMap_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricGraphVector_mem_center_span_iff": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricGraphLine_coordinateCenter_intersection_iff": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricGraphLine_coordinateCenter_common_point_iff": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.projectiveDirection_nonzero_left": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.projectiveDirection_nonzero_right": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricGraphLine_odd_no_three_actual_centers": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricGraphLines_disjoint_of_difference_det": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricGraphLine_zero_diagonal_disjoint": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricGraphLine_char_two_diagonal_zero_of_actual_centers": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticAutomorphism_form": [ "propext", "Quot.sound" ], "CompactnessConjecture.symplecticAutomorphism_incidence_iff": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticAutomorphism_isotropic_iff": [ "propext", "Quot.sound" ], "CompactnessConjecture.symplecticAutomorphismLineEquiv_apply": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLine_orthogonal_eq": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLine_isCompl_of_disjoint": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLinePairing_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLinePairing_finrank": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLineDualCoordinates_apply": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLinePairing_coordinate_expansion": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLineCoordinateEquiv_apply_add": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLineCoordinateEquiv_form": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLineNormalizer_apply_left": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLineNormalizer_apply_right": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticVerticalLinearMap_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLineNormalizer_map_left": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLineNormalizer_map_right": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticHorizontalProjection_ker": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLineHorizontalProjection_injective": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLineGraphMap_horizontal": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLineGraphMap_symmetric": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLineGraphMap_coordinate_expansion": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLineGraphMap_graphVector": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLine_eq_symmetricGraphLine_of_disjoint_vertical": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symmetricGraphLine_det_ne_zero_of_disjoint_horizontal": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLine_eq_invertible_symmetricGraphLine": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticCanonicalLines_disjoint": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticVertical_mem_coordinateCenter_of_orthogonal": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLine_eq_coordinateCenterLine_of_common_points": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.coordinateCenterLine_direction_det_ne_zero_of_ne": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticAutomorphism_disjoint_iff": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticCanonical_line_no_three_common_centers": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLine_no_three_common_centers": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_no_line_gamma_of_odd": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_no_kQuotient_of_odd": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_encodeFiniteGraph_free_iff": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_no_encoded_kQuotient_of_odd": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.jQuotient_free_of_template_avoidance": [ "propext", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_no_encoded_jQuotient_of_template_avoidance": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_no_jTemplate_of_char_two_line_avoidance": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLine_char_two_canonical_zero_diagonal": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticAutomorphism_commonPoint": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLine_char_two_disjoint_of_two_common_center_pairs": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticLine_char_two_pair_avoidance": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_no_jTemplate_of_char_two": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.symplecticQuadrangle_no_encoded_jQuotient_of_char_two": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.familyExtremal_real_le_of_forall_free": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.familyLittleO_of_eventual_host_bounds": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.familyFree_of_embedded_subgraph": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.eventually_constant_le_positive_nat_rpow": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.extremalScale_sixteenth_power": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.familyLittleO_of_sixteenth_power_host_bound": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamilyFree_minDegree_ambient_sixteenth_power_le": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamilyFree_sixteenth_power_host_bound": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamily_familyLittleO": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.four_cycle_eventual_manuscript_lower": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.six_cycle_eventual_manuscript_lower": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.member_eventual_lower_of_prime_power_avoidance": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.uniformMemberLower_of_characteristic_avoidance": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamily_odd_characteristic_avoidance": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamily_even_characteristic_avoidance": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamily_uniformMemberLower": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamily_not_compact": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.not_erdos_180": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.jTemplate_connected": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.kTemplate_connected": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.quotientGraph_connected_of_colorRespecting": [ "propext", "Quot.sound" ], "CompactnessConjecture.encodeFiniteGraph_connected": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.encodedJQuotient_connected": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.encodedKQuotient_connected": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.finiteCycle_connected": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamily_member_connected": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.colorRespectingQuotient_isBipartite": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.encodeFiniteGraph_isBipartite": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.encodedJQuotient_isBipartite": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.encodedKQuotient_isBipartite": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamily_member_isBipartite": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.finiteNatSup_sixteenth_power_le": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamily_familyExtremal_sixteenth_power_le": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.proposedFamily_familyExtremal_isBigO": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.compactnessSharpHostPowerConstant_pos": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.checkedManuscriptCounterexample": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.quantitativeCompactnessCounterexample": [ "propext", "Classical.choice", "Quot.sound" ], "CompactnessConjecture.compactnessCounterexample_bigO": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.twelve_sevenths_lt_sqrt_three": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.log_two_pos": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryEntropy_nonneg": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryEntropy_le_one": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryEntropy_zero": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryEntropy_one_sub": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryEntropy_continuous": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryEntropy_scale_le": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryEntropy_subadditive": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.abs_binaryEntropy_sub_le_binaryEntropy_abs_sub": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryEntropy_mono_on_half": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryPinskerGap_continuous": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryPinskerGap_hasDerivAt": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryPinskerGapDeriv_hasDerivAt": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryPinskerGapDerivTwo_nonneg": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryPinskerGap_convex": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryPinskerGap_half": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryPinskerGapDeriv_half": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binary_pinsker": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.log_le_tangent": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.log_four_thirds_lt_one_third": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.sqrt_one_add_le": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.normalized_binary_cauchy": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binary_log_sum_bound": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.entropyTangentSigma_pos": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.entropyTangentRho_pos": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.log_entropyTangentSigma": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.log_entropyTangentRho": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.sqrt_three_mul_entropyTangentRho": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.entropyTangentZeroCoefficient_eq": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.entropyTangentOneCoefficient_eq": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.entropyTangentCoefficient_norm": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.entropyTangentLog_constant": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryConditionalLogPotential_tangent_bound": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryConditionalLogPotential_le_kappa": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binaryCoinMass_nonneg": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.independentBinaryPairMass_nonneg": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.independentBinaryPairMass_sum": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.BinaryPairKernel.childMarginal_nonneg": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.BinaryPairKernel.childMarginal_le_one": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.BinaryPairKernel.childMarginal_eq_four_outcomes": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.BinaryPairKernel.conditionalEntropy_mul_log_two": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.BinaryPairKernel.bitDisagreementProbability_mem_Icc": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.BinaryPairKernel.averageDisagreement_eq_four_outcomes": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.BinaryPairKernel.smoothed_childMarginal": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.BinaryPairKernel.smoothed_averageDisagreement": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.BinaryPairKernel.smoothedConditionalEntropy_continuous": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.BinaryPairKernel.smoothed_conditionalEntropy": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.BinaryPairKernel.conditionalEntropy_logsum_reduction": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.BinaryPairKernel.conditionalEntropy_bound_of_marginal_interior": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.BinaryPairKernel.conditionalEntropy_bound": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.withoutReplacementBinaryPairMass_nonneg": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.withoutReplacementBinaryPairMass_sum": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.withoutReplacementBinaryPairExpectation_sub": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.withoutReplacementBinaryPairExpectation_error": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.withoutReplacementBinaryPairExpectation_nonneg": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.withoutReplacementBinaryPairExpectation_le_one": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.empiricalChildMarginal_mem_Icc": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.empiricalChildMarginal_error": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.empiricalConditionalEntropy_error": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.empiricalAverageDisagreement_error": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binomialProbabilityMass_nonneg": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binomialProbabilityMass_succ_mul": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binomialModeRatio_le_of_lt": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binomialModeRatio_le_of_ge": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binomialProbabilityMass_le_succ_of_lt_mode": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.binomialProbabilityMass_succ_le_of_ge_mode": [ "propext", "Classical.choice", "Quot.sound" ], 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"TwoDegenerateGraphs.certificate_ratio_one_lt": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.certifiedWindowWidth_pos": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.tau_pos": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.tau_lt_one_half": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.sqrt_three_pos": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.tau_complement": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.tau_reciprocal_identity": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.log_three_eq_twice_log_sqrt_three": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.entropy_tau_identity": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.certificate_ratio_identity": [ "propext", "Classical.choice", "Quot.sound" ], "TwoDegenerateGraphs.log_certificate_ratio_identity": [ "propext", 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"declarations": [], "fidelity": { "grade": "F1", "reason": "A formal artifact exists, but nothing records that its statement says what the claim says.", "anchor": "", "flags": [] }, "verified": true, "checked_by": [ "qed.bot" ] }, { "id": "erdos:347", "key": "347", "label": "Erdős Problem 347", "collection": "Erdős", "kind": "problem", "title": "Erdős Problem 347", "url": "https://www.erdosproblems.com/347", "page": "s/erdos-347.html", "source_status": "proved (Lean)", "formal_status": "solved", "prize": "no", "prize_value": 0, "aliases": [ "https://www.erdosproblems.com/347", "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/347.lean", "FormalConjectures/ErdosProblems/347.lean" ], "tags": [ "number theory", "complete sequences" ], "formalisations": [ { "system": "lean4", "repository": "google-deepmind/formal-conjectures", "path": "FormalConjectures/ErdosProblems/347.lean", "url": 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No formal statement is attached to the result.", "anchor": "", "flags": [] }, "verified": false, "checked_by": [] }, { "id": "erdos:546", "key": "546", "label": "Erdős Problem 546", "collection": "Erdős", "kind": "problem", "title": "Erdős Problem 546", "url": "https://www.erdosproblems.com/546", "page": "s/erdos-546.html", "source_status": "proved", "formal_status": "solved", "prize": "no", "prize_value": 0, "aliases": [ "https://www.erdosproblems.com/546", "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/546.lean", "FormalConjectures/ErdosProblems/546.lean" ], "tags": [ "graph theory", "ramsey theory" ], "formalisations": [ { "system": "lean4", "repository": "google-deepmind/formal-conjectures", "path": "FormalConjectures/ErdosProblems/546.lean", "url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/546.lean", "labels": [ "research solved" ], "proof_urls": [], "sorry_count": 1, "declarations": 1 } ], "artifacts": [], "claims": [], "checks": [], "mentions": [], "relations": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "verified": false, "checked_by": [] }, { "id": "erdos:547", "key": "547", "label": "Erdős Problem 547", "collection": "Erdős", "kind": "conjecture", "title": "Erdős Problem 547", "url": "https://www.erdosproblems.com/547", "page": "s/erdos-547.html", "source_status": "proved", "formal_status": "mixed", "prize": "no", "prize_value": 0, "aliases": [ "https://www.erdosproblems.com/547", "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/547.lean", "FormalConjectures/ErdosProblems/547.lean" ], "tags": [ "graph theory", "ramsey theory" ], "formalisations": [ { "system": "lean4", "repository": "google-deepmind/formal-conjectures", "path": "FormalConjectures/ErdosProblems/547.lean", "url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/547.lean", "labels": [ "research open", "research solved" ], "proof_urls": [], "sorry_count": 2, "declarations": 2 } ], "artifacts": [], "claims": [], "checks": [], "mentions": [], "relations": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "verified": false, "checked_by": [] }, { "id": "erdos:548", "key": "548", "label": "Erdős Problem 548", "collection": "Erdős", "kind": "problem", "title": "Erdős Problem 548", "url": "https://www.erdosproblems.com/548", "page": "s/erdos-548.html", "source_status": "proved (Lean)", "formal_status": "solved", "prize": "no", "prize_value": 0, "aliases": [ "https://www.erdosproblems.com/548", "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/548.lean", "FormalConjectures/ErdosProblems/548.lean" ], "tags": [ "graph theory" ], "formalisations": [ { "system": "lean4", "repository": "google-deepmind/formal-conjectures", "path": "FormalConjectures/ErdosProblems/548.lean", "url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/548.lean", "labels": [ "research solved" ], "proof_urls": [ "https://github.com/Jayyhk/erdos-lean/blob/4c355efbf8e90c2572aa8d0c8d6f3b4b43f011f4/problems/548/Erdos548.lean#L1182" ], "sorry_count": 1, "declarations": 1 } ], "artifacts": [ { "url": "https://github.com/Jayyhk/erdos-lean/blob/4c355efbf8e90c2572aa8d0c8d6f3b4b43f011f4/problems/548/Erdos548.lean#L1182", "reachable": true, "sorry_count": 0, "axiom_count": 0, "native_decide": false, "admit": false, "theorems": 47, "imports": [ "Mathlib" ] } ], "claims": [], "checks": [], "mentions": [], "relations": [], "declarations": [ { "id": "tadamcz/erdos548/formalization.yaml", "name": "Erdős problem #548 (Erdős–Sós conjecture): proof", "url": "https://github.com/tadamcz/erdos548/blob/HEAD/formalization.yaml", "repo": "tadamcz/erdos548", "directory": "", "container": false, "authors": [], "automation": { "strongest": "autonomous", "models": [ "GPT-6 Astra (pre-release version, OpenAI)" ] }, "review": { "status": "mechanically verified (comparator: lean kernel replay, standard axioms only) in the benchmark harness and again in this repository's ci; preliminary informal reading of the argument by thomas f. bloom; no independent refereeing", "bucket": "other", "reviewers": [ "Thomas F. 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The Lean proof's internal counting lemma derives `2|E(G)| ≤ (k-1)n` whenever the tree is absent, which is the sharp classical bound, but only the advertised statement is compared. The vertex type `Fin n` loses no generality; `G.edgeSet.ncard` is the number of edges; `T.IsTree` and `T.IsContained G` are Mathlib's notions (containment as a not necessarily induced subgraph); `k + 1 ≤ n` is `n ≥ k+1`.", "literature_dependencies": 0, "literature": [], "role": "", "substantive": "", "sources": [ { "title": "Proof of Erdős problem #548: Every graph on n ≥ k+1 vertices with at least (k−1)n/2 + 1 edges contains every tree on k+1 vertices.", "relationship": "other", "endorsement": "" }, { "title": "Erdős problem #548 (erdosproblems.com)", "relationship": "background", "endorsement": "" }, { "title": "Extremal problems in graph theory", "relationship": "background", "endorsement": "" }, { "title": "FrontierMath Erdős", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/epoch-research/LeanOpenProblems/blob/77882c437ca1dfefab3b27fa00f1d29788100311/apn/data/erdos_autoformalized/Isolated/Erdos548.erdos_548.lean", "relationship": "builds-on", "note": "The statement file given to the AI system in the FrontierMath Erdős benchmark. 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The compared theorem is this statement, proved." } ], "palomar": [], "checked_by": [], "contradictions": [], "missing": [], "join": "names", "anchor": "", "anchor_moved": false } ], "fidelity": { "grade": "F2", "reason": "The correspondence is declared through an alignment table and written divergences.", "anchor": "", "flags": [ "declares divergences from its source" ] }, "verified": false, "checked_by": [] }, { "id": "erdos:549", "key": "549", "label": "Erdős Problem 549", "collection": "Erdős", "kind": "problem", "title": "Erdős Problem 549", "url": "https://www.erdosproblems.com/549", "page": "s/erdos-549.html", "source_status": "disproved", "formal_status": "solved", "prize": "no", "prize_value": 0, "aliases": [ "https://www.erdosproblems.com/549", "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/549.lean", "FormalConjectures/ErdosProblems/549.lean" ], "tags": [ "graph theory", "ramsey theory" ], "formalisations": [ { "system": "lean4", "repository": "google-deepmind/formal-conjectures", "path": "FormalConjectures/ErdosProblems/549.lean", "url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/549.lean", "labels": [ "research solved" ], "proof_urls": [], "sorry_count": 1, "declarations": 1 } ], "artifacts": [], "claims": [], "checks": [], "mentions": [], "relations": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. 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No formal statement is attached to the result.", "anchor": "", "flags": [] }, "verified": false, "checked_by": [] }, { "id": "erdos:553", "key": "553", "label": "Erdős 553", "collection": "Erdős", "kind": "problem", "title": "", "url": "https://www.erdosproblems.com/553", "page": "s/erdos-553.html", "source_status": "proved", "formal_status": "unclassified", "prize": "no", "prize_value": 0, "aliases": [ "https://www.erdosproblems.com/553" ], "tags": [ "graph theory", "ramsey theory" ], "formalisations": [], "artifacts": [], "claims": [], "checks": [], "mentions": [], "relations": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "verified": false, "checked_by": [] }, { "id": "erdos:554", "key": "554", "label": "Erdős 554", "collection": "Erdős", "kind": "problem", "title": "", "url": "https://www.erdosproblems.com/554", "page": "s/erdos-554.html", "source_status": "open", "formal_status": "unclassified", "prize": "no", "prize_value": 0, "aliases": [ "https://www.erdosproblems.com/554" ], "tags": [ "graph theory", "ramsey theory" ], "formalisations": [], "artifacts": [], "claims": [], "checks": [], "mentions": [], "relations": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "verified": false, "checked_by": [] }, { "id": "erdos:555", "key": "555", "label": "Erdős 555", "collection": "Erdős", "kind": "problem", "title": "", "url": "https://www.erdosproblems.com/555", "page": "s/erdos-555.html", "source_status": "open", "formal_status": "unclassified", "prize": "no", "prize_value": 0, "aliases": [ "https://www.erdosproblems.com/555" ], "tags": [ "graph theory", "ramsey theory" ], "formalisations": [], "artifacts": [], "claims": [], "checks": [], "mentions": [], "relations": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "verified": false, "checked_by": [] }, { "id": "erdos:556", "key": "556", "label": "Erdős 556", "collection": "Erdős", "kind": "problem", "title": "", "url": "https://www.erdosproblems.com/556", "page": "s/erdos-556.html", "source_status": "decidable", "formal_status": "unclassified", "prize": "no", "prize_value": 0, "aliases": [ "https://www.erdosproblems.com/556" ], "tags": [ "graph theory", "ramsey theory" ], "formalisations": [], "artifacts": [], "claims": [], "checks": [], "mentions": [], "relations": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. No formal statement is attached to the result.", "anchor": "", "flags": [] }, "verified": false, "checked_by": [] }, { "id": "erdos:557", "key": "557", "label": "Erdős 557", "collection": "Erdős", "kind": "problem", "title": "", "url": "https://www.erdosproblems.com/557", "page": "s/erdos-557.html", "source_status": "proved (Lean)", "formal_status": "unclassified", "prize": "no", "prize_value": 0, "aliases": [ "https://www.erdosproblems.com/557" ], "tags": [ "graph theory", "ramsey theory" ], "formalisations": [], "artifacts": [], "claims": [], "checks": [], "mentions": [], "relations": [], "declarations": [ { "id": "zhangjun725/erdos557/formalization.yaml", "name": "Erdős problem #557 (multicolour Ramsey numbers of trees): proof", "url": "https://github.com/zhangjun725/erdos557/blob/HEAD/formalization.yaml", "repo": "zhangjun725/erdos557", "directory": "", "container": false, "authors": [], "automation": { "strongest": "other", "models": [ "Claude (Anthropic)" ] }, "review": { "status": "mechanically verified (lean kernel, standard axioms only; see ci); no independent mathematical refereeing", "bucket": "other", "reviewers": [] }, "axioms": [], "nonstandard_axioms": [], "sorry_count": 0, "sorry_in_definitions": 0, "results": 0, "comparator": false, "alignment": true, "divergences": "The informal problem asks whether R_k(T) ≤ kn + O(1) for every tree T on n vertices. `erdos_557` states this with an absolute constant c (c = 3 is exhibited), which is the strongest reading of O(1) (it does not depend on k). A k-colouring of the edges of K_m is modelled by Mathlib's `TopEdgeLabeling (Fin m) (Fin k)`, a function from the edge set of the complete graph to `Fin k`; a monochromatic copy is `T.IsContained (C.labelGraph i)` (not necessarily induced). `multicolourRamsey k G` is the infimum of the set of m for which every such labelling has a monochromatic copy of G; by `Nat.sInf_le` the explicit theorem gives the bound. Trees with n ≥ 1 vertices are covered; for n = 0 there are no trees (`IsTree` requires a vertex). 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🟢 Chaffee and Noble (2016)" }, "date": "2026-01-19", "date_text": "19 Jan, 2026", "outcome": "full", "outcome_text": "New proof found (Lean)", "outcome_scope": "statement", "lean_claimed": true, "machine_checked": true, "autonomy": "A2", "autonomy_reason": "AI building on literature supplied to it", "verification": "V3", "contribution": "primary", "section": "1(c)", "section_title": "AI building on literature", "sources": [ { "label": "Erdős problems AI wiki", "url": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems" } ], "checked_by": [ "Palomar" ], "fidelity": "F3" }, { "id": "53d6135a9ab5", "statement": "erdos:1007", "systems": [ "Aristotle" ], "humans": [], "details": { "proof to formalize": "House (2013)" }, "date": "2026-01-19", "date_text": "19 Jan, 2026", "outcome": "full", "outcome_text": "House (2013)", "outcome_scope": "task", "lean_claimed": false, "machine_checked": true, "autonomy": "A1", "autonomy_reason": "Secondary contribution: formalization", "verification": "V3", "contribution": "secondary", "section": "2(b)", "section_title": "Formalization", "sources": [ { "label": "Erdős problems AI wiki", "url": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems" } ], "checked_by": [ "Palomar" ], "fidelity": "F3" } ], "checks": [ { "method": "palomar", "by": "Palomar", "id": "PALOMAR-2026-09-23-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-23-000003", "repo": "Dishah3241/Erdos1007", "commit": "43f89415a6663848a1445effbda8b3abe77b052b", "project_path": "", "artifact": "https://github.com/Dishah3241/Erdos1007/tree/43f89415a6663848a1445effbda8b3abe77b052b", "theorems": [ "Erdos1007.Palomar.target" ], "trust": "high", "attached_by": "anchor", "verdict": "verified", "detail": "Registered by Palomar at 43f89415: Comparator confirmed 1 theorems prove the recorded statement within Palomar's axiom policy, replayed through Lean's kernel and the independent nanoda kernel", "checked": "2026-09-23", "nonstandard_axioms": [], "uses_sorry": false }, { "method": "palomar", "by": "Palomar", "id": "PALOMAR-2026-09-25-000003", "url": "https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-25-000003", "repo": "Dishah3241/Erdos1007Dim5", "commit": "85032a43ca6db7d92b492efd05deeea0739c4570", "project_path": "", "artifact": "https://github.com/Dishah3241/Erdos1007Dim5/tree/85032a43ca6db7d92b492efd05deeea0739c4570", "theorems": [ "Erdos1007Dim5.Palomar.target" ], "trust": "high", "attached_by": "anchor", "verdict": "verified", "detail": "Registered by Palomar at 85032a43: Comparator confirmed 1 theorems prove the recorded statement within Palomar's axiom policy, replayed through Lean's kernel and the independent nanoda kernel", "checked": "2026-09-25", "nonstandard_axioms": [], "uses_sorry": false } ], "mentions": [], "relations": [], "declarations": [ { "id": "Dishah3241/Erdos1007/formalization.yaml", "name": "Unit-distance graphs of dimension four with nine edges", "url": "https://github.com/Dishah3241/Erdos1007/blob/HEAD/formalization.yaml", "repo": "Dishah3241/Erdos1007", "directory": "", "container": false, "authors": [ "Dishant Shah" ], "automation": { "strongest": "agent", "models": [ "claude-opus-5", "claude-opus-5-5", "glm-5.3-flash", "gpt-6-astra", "grok-4.7" ] }, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "gpt-6-astra (Codex CLI): adversarial review of the statement", "claude-opus-5-5 (Claude Code): independent review of the proof", "Dishant Shah: sign-off on the meaning of the statement-level declarations" ] }, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "sorry_count": 0, "sorry_in_definitions": 0, "results": 3, "comparator": true, "alignment": true, "divergences": "None in meaning. The statement is formal-conjectures' erdos_1007.variants.dimension_four_extremal with its supporting definitions inlined and its ℝ^n notation written out as EuclideanSpace ℝ (Fin n). With both builds loaded together, the inlined statement is equal to the upstream declaration's type by rfl, and the proof elaborates against that type using only propext, Classical.choice and Quot.sound (docs/upstream-check-2026-09-22.md). Two earlier reviews checked the same equality against copies of the upstream definitions (docs/red-team-2026-09-20.md, docs/review-2026-09-22.md). The statement carries the upstream hypothesis that no vertex is isolated. 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That statement requires a nine-edge graph of dimension four and the representability in ℝ³ of every graph with at most eight edges, both of which are also prove" } ], "palomar": [ "PALOMAR-2026-09-23-000003" ], "checked_by": [ "Palomar" ], "contradictions": [], "missing": [], "join": "anchor", "anchor": "https://github.com/google-deepmind/formal-conjectures/blob/2a46c7bd74505b85f4967475bb733ded0ef8d348/FormalConjectures/ErdosProblems/1007.lean", "anchor_moved": false }, { "id": "Dishah3241/Erdos1007Dim5/formalization.yaml", "name": "Erdős 1007, dimension five: fifteen edges, attained by K6 and K1,3,3", "url": "https://github.com/Dishah3241/Erdos1007Dim5/blob/HEAD/formalization.yaml", "repo": "Dishah3241/Erdos1007Dim5", "directory": "", "container": false, "authors": [ "Dishant Shah" ], "automation": { "strongest": "agent", "models": [ "Grok 4.7 500K Extra High Fast", "claude-opus-5-5", "glm-5.3-flash", "grok-4.7-build" ] }, "review": { "status": "agent-reviewed", "bucket": "agent-reviewed", "reviewers": [ "grok-4.7-build (Grok CLI): independent check of the statements against upstream, by rfl", "claude-opus-5-5 (Claude Code): independent review of the proof", "Dishant Shah: sign-off on the meaning of the statement-level declarations" ] }, "axioms": [ "Classical.choice", "Quot.sound", "propext" ], "nonstandard_axioms": [], "sorry_count": 0, "sorry_in_definitions": 0, "results": 1, "comparator": true, "alignment": true, "divergences": "None in meaning. The statements are formal-conjectures' erdos_1007.variants.dimension_five and erdos_1007.variants.dimension_five_extremal, with their supporting definitions inlined (UnitDistanceEmbeddable, HasDimension and K133) and the ℝ^n notation written out as EuclideanSpace ℝ (Fin n). With both builds loaded together, each inlined statement equals the upstream declaration's type by rfl. A model of a different lineage from the one that wrote the statements ran that check (docs/upstream-check-stage1.md). Both proofs also elaborate against upstream's declaration types in one environment (docs/upstream-check-2026-09-23.md). The proof works with the library's copies of the two definitions, which have the same bodies, and rfl identifies them (docs/review-2026-09-23.md, check 2). Neither statement has a hypothesis, so there is none to drop. 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On the erratum, the displayed C2 form, the pairing consequence of the paper's hypothesis (4.1), and the emptiness of the class at every positive block order not congruent to 1 mod 4 are formalized, which closes both printed readings of the corollary's output order; what remains informal is the transcription of the printed text, human-audited in Hadamard-M, and the bibliographic propagation from the unsupported order-103 entry to the order-515 list entry and its order-2060 claim.", "literature_dependencies": 0, "literature": [], "role": "", "substantive": "", "sources": [ { "title": "Hadamard-T: T-matrix witnesses and the Hadamard orders they close", "relationship": "formalizes", "endorsement": "participated" }, { "title": "Hadamard-M: an erratum at order 515, and an explicit Hadamard matrix of order 7796", "relationship": "formalizes", "endorsement": "participated" }, { "title": "A construction of Hadamard matrices", "relationship": "background", "endorsement": "" }, { "title": "A database of constructions of Hadamard matrices", "relationship": "background", "endorsement": "" } ], "related": [ { "id": "https://github.com/leanprover-community/mathlib4/blob/v4.33.0/Mathlib/LinearAlgebra/Matrix/HadamardMatrix.lean", "relationship": "builds-on", "note": "Mathlib's Hadamard-matrix module (added 2026-06-16, mathlib4 PR #38582) supplies the Matrix.IsHadamard predicate the compared statements use, closure lemmas including the Kronecker product, and the 4-divides-n obstruction. It exhibits no Hadamard matrix of any order and contains no existence theorem, and Mathlib at this revision has no content on Williamson quadruples, T-matrices, or the Goethals-" }, { "id": "https://github.com/google-deepmind/formal-conjectures/blob/abe6ef73129989fb8cf94edb72b1ac21d30a411b/FormalConjectures/Wikipedia/Hadamard.lean", "relationship": "other", "note": "States the Hadamard conjecture and its known-for-4k-with-k-at-most-166 variant in Lean as unproved benchmark targets; as of that commit, every Hadamard statement there, including an explicit order-12 matrix, is left as sorry. 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It inlines the plane notation and instances, names the rigid-motion topology instance so that Comparator sees the same name in Challenge and Solution, and leaves out the tests, metadata and open shape-uniqueness conjecture found there. Compared with the paper: an AI review of the statement found the following differences, none of which weakens the theorem. Sofas satisfy m 0 = id and start inside the horizontal arm, where the paper allows any initial translation; canonical_paper_motion_bridge proves that a sofa of either kind is a translate of one of the other kind. Sofas are closed and connected, as in the paper's Definition 1.2, so the theorem says nothing about disconnected sets. Motions lie in the full isometry group E(2), where the paper uses SE(2); orientation preservation is proved. Boundedness and measurability are not assumed; every moving sofa is proved compact. The supremum is taken in the extended nonnegative reals and is proved finite, at least 11/5 and attained. ABφθSpec uses non-strict inequalities, whi", "literature_dependencies": 0, "literature": [], "role": "", "substantive": "", "sources": [ { "title": "Optimality of Gerver's Sofa", "relationship": "formalizes", "endorsement": "not-contacted" }, { "title": "On moving a sofa around a corner", "relationship": "adapts", "endorsement": "not-contacted" }, { "title": "Differential equations and exact solutions in the moving sofa problem", "relationship": "background", "endorsement": "not-contacted" } ], "related": [ { "id": "https://github.com/google-deepmind/formal-conjectures/tree/ddfbaf90f4482030d88aae5233fe933874296a23", "relationship": "adapts", "note": "The source of the statement; it contains no proof. MovingSofaSubmission/Challenge.lean copies the moving sofa definitions and the three compared statements, unchanged, from FormalConjectures/Wikipedia/MovingSofa.lean at this commit. Four things differ: the plane notation and instances are inlined, the rigid-motion topology instance is given a name, docstrings are added to the definitions that had " }, { "id": "https://github.com/dawidmtrela-dotcom/GerverSofaLean/releases/tag/v1.1.0", "relationship": "builds-on", "note": "Dawid Trela's Lean development (MIT). Used here: its kernel-checked proof that Gerver's constants exist and are unique, its proof that Gerver's sofa moves around the corner, and its certified enclosures of the constants and of the transcendental functions. MovingSofa/Canonical/GerverDefinitions.lean and MovingSofa/Gerver/Motion.lean adapt these to the Challenge definitions. The v1.1.0 release arch" }, { "id": "https://github.com/rkirov/jordan_pick/tree/b3c9b7cf7358bf81a077d78ad67e6e8247869ddd", "relationship": "builds-on", "note": "The Jordan curve theorem, used here through MovingSofa.jordan_separation. Registered with Palomar as PALOMAR-2026-08-19-000001." }, { "id": "https://github.com/Vilin97/lean-pool/tree/bb74ee07fc23bc81358d75a9c40303e5e27fced8", "relationship": "builds-on", "note": "Jonathan Ho's Brunn-Minkowski inequality with its Prekopa-Leindler prerequisites. It keeps the area above the threshold along convex combinations of convex bodies. The three files used are copied unchanged into vendor/lean-pool (Apache-2.0) and built as a library of this package, because Lake records the package name lean-pool in the manifest as «lean-pool», which Palomar's verifier rejects." } ], "palomar": [], "checked_by": [], "contradictions": [], "missing": [], "join": "artifact", "anchor": "", "anchor_moved": false } ], "fidelity": { "grade": "F2", "reason": "The correspondence is declared through an alignment table and written divergences.", "anchor": "", "flags": [ "declares divergences from its source", "reviewed by an agent", "source authors not contacted" ] }, "verified": false, "checked_by": [] }, { "id": "wikipedia:NoThreeInLineProblem", "key": "NoThreeInLineProblem", "label": "No-three-in-line problem", "collection": "Wikipedia", "kind": "problem", "title": "No-three-in-line problem", "url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/NoThreeInLineProblem.lean", "page": "s/wikipedia-nothreeinlineproblem.html", "source_status": null, "formal_status": "unclassified", "prize": null, "prize_value": 0, "aliases": [ "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/NoThreeInLineProblem.lean", "FormalConjectures/Wikipedia/NoThreeInLineProblem.lean" ], "tags": [], "formalisations": [ { "system": "lean4", "repository": "google-deepmind/formal-conjectures", "path": "FormalConjectures/Wikipedia/NoThreeInLineProblem.lean", "url": "https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/NoThreeInLineProblem.lean", "labels": [], "proof_urls": [], "sorry_count": 0, "declarations": 0 } ], "artifacts": [], "claims": [], "checks": [], "mentions": [], "relations": [], "declarations": [], "fidelity": { "grade": "F0", "reason": "None. 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