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"OAI.Combinatorics.Crossing.DrawingExteriorReplacement": [ "drawing_exterior_polygonal_replacement" ], "OAI.Combinatorics.Crossing.DrawingMarks": [ "drawingMarks", "drawingMarks_finite", "crossing_not_vertex", "marked_parameters_finite", "marked_parameter_zero", "marked_parameter_one", "unmarked_edge_point_interior", "unmarked_intersection_same_edge" ], "OAI.Combinatorics.Crossing.DrawingPassages": [ "DrawingPassages", "nonempty_drawingPassages" ], "OAI.Combinatorics.Crossing.DrawingPorts": [ "Port", "portPoint", "portMark", "portPoint_mem_arc", "portPoint_injective", "portPoint_on_circle", "PortsAt", "portsAt_on_circle" ], "OAI.Combinatorics.Crossing.DrawingRangeCrossings": [ "edgeInterior_subset_range", "edgeInterior_of_range_not_vertices", "edgeInterior_not_vertex", "crossing_of_ranges_not_vertices" ], "OAI.Combinatorics.Crossing.DrawingRestriction": [ "restrictEdge", "restrictContinuousDrawing", "restrict_edgeInterior", "restrict_crossingPoints_subset", "restrictDrawing", "crossingCount_restrict_le", "restrict_noAdjacentCrossings" ], "OAI.Combinatorics.Crossing.DrawingSegmentIncidence": [ "drawing_curve_meeting_mark", "DrawingPiece", "drawingPieceStart", "drawingPieceEnd", "drawingPiece_endpoints_ne", "drawing_pieces_meeting_endpoints" ], "OAI.Combinatorics.Crossing.DrawingSpokePorts": [ "openSegmentTrace_reverse_mem", "squaredLength_sub_comm", "PolygonalChain.startPort", "PolygonalChain.endPort", "drawing_chain_startPort_interior", "drawing_chain_endPort_interior", "drawing_chain_startPort_circle", "drawing_chain_endPort_circle" ], "OAI.Combinatorics.Crossing.DrawingSubdivisionGraph": [ "drawingSubdivisionGraph", "drawingSubdivisionGraph_piece_adj", "common_endpoint_alternatives", "drawingSubdivisionGraph_embedding", "drawingSubdivisionGraph_finite_support" ], "OAI.Combinatorics.Crossing.DrawingVertexPlacement": [ "vertexMark", "vertexMark_injective", "exists_drawing_vertex_placement", "vertex_spoke_injective", "vertex_spoke_interior" ], "OAI.Combinatorics.Crossing.DrawingWalkCutTraces": [ "graphWalkPath_range_cases", "drawingEdgeWalk_take_trace", "drawingEdgeWalk_drop_trace" ], "OAI.Combinatorics.Crossing.DrawnEdgeChains": [ "trackChain", "chordChain", "shiftedTrackChain", "connectorChain", "shiftedEdgeChain", "shiftedChordChain", "shiftedEdgeChain_pieces", "shiftedEdgeChain_intersection_right" ], "OAI.Combinatorics.Crossing.EmbeddedGraphPaths": [ "StraightEdgeEmbedding", "graphWalkPath", "graphWalkPath_hasFiniteAffineCover", "graphWalkPath_injective", "exists_simple_graphWalkPath" ], "OAI.Combinatorics.Crossing.EndpointFlagPairing": [ "IncidentFlag", "dartSourceFlag", "dartTargetFlag", "incidentCast_refl", "incidentFlag_cast", "dartPassageNext_flag", "flagPairing", "vertexChordPairing_flags", "passagePairing_flags", "edgeEndpointPairing", "flagPairing_source_target", "flagPairing_target_source", "dart_endpoint_pairing" ], "OAI.Combinatorics.Crossing.EndpointVisits": [ "last_sublevel_visit", "first_sublevel_visit" ], "OAI.Combinatorics.Crossing.EvaluationAmbient": [ "constrainedInclusion", "constrainedInclusion_eval", "ambientEval", "ambientEvalRange", "finrank_ambientEvalRange", "finrank_ambientEvalRange_off_node", "ambientEvalRange_mono_coeff", "ambientEvalRange_le_coeff", "ambientEvalRange_at_node_le" ], "OAI.Combinatorics.Crossing.EvenCrossingLowerBound": [ "even_successor_crossing_lower_bound", "crossing_lower_bound" ], "OAI.Combinatorics.Crossing.ExpectedTrackGeometry": [ "actual_normalized_tracks" ], "OAI.Combinatorics.Crossing.ExteriorArcReplacement": [ "exterior_arc_polygonal_replacement" ], "OAI.Combinatorics.Crossing.ExteriorChordMeeting": [ "chord_parameter_endpoint", "exterior_chord_meeting_parameters" ], "OAI.Combinatorics.Crossing.ExteriorCrossingCut": [ "ExteriorCrossingCut", "PolygonalExterior.nonempty_exteriorCrossingCut", "prefixInSurgery", "suffixInSurgery", "prefixInSurgery_trace", "suffixInSurgery_trace" ], "OAI.Combinatorics.Crossing.ExteriorSegmentGraph": [ "exteriorSegmentGraph", "exteriorSegmentGraph_le", "graphWalkPath_edge_subset_range", "exteriorGraphWalk", "exteriorGraphWalk_getVert", "exteriorGraphWalk_length", "exteriorGraphWalk_trace" ], "OAI.Combinatorics.Crossing.ExteriorSpokeMeeting": [ "exterior_spoke_meeting_parameters" ], "OAI.Combinatorics.Crossing.ExtraDrawingSubdivision": [ "exists_drawing_subdivision_at", "exists_assembled_port_subdivision" ], "OAI.Combinatorics.Crossing.FiniteAffineCover": [ "HasFiniteAffineCover", "hasFiniteAffineCover_refl", "hasFiniteAffineCover_segment", "HasFiniteAffineCover.trans", "HasFiniteAffineCover.cast", "polygonalArcPath_hasFiniteAffineCover", "PolygonalChain.hasFiniteAffineCover" ], "OAI.Combinatorics.Crossing.FiniteCircleCoordinates": [ "exists_circle_cut_vector", "exists_finite_circle_coordinates" ], "OAI.Combinatorics.Crossing.FiniteConePoint": [ "exists_polygon_cone_point" ], "OAI.Combinatorics.Crossing.FiniteDiskCollar": [ "outsideDisks", "isOpen_disk_exterior", "isOpen_outsideDisks", "exists_finite_disk_collar" ], "OAI.Combinatorics.Crossing.FiniteEventualRadius": [ "finite_eventual_radius" ], "OAI.Combinatorics.Crossing.FiniteIntersectionDrawing": [ "FiniteIntersectionDrawing", "AdmissibleDrawing.toFiniteIntersectionDrawing", "finiteCrossingCount", "finiteCrossingCount_of_admissible", "finiteCrossingCount_eq_ncard" ], "OAI.Combinatorics.Crossing.FiniteLineAvoidance": [ "lineAvoidancePolynomial", "lineAvoidancePolynomial_eval", "lineAvoidancePolynomial_ne_zero", "exists_near_avoiding_lines" ], "OAI.Combinatorics.Crossing.FiniteMovingPoints": [ "finite_moving_points_separation" ], "OAI.Combinatorics.Crossing.FinitePathChain": [ "finitePathChain", "finitePathChain_range_iff", "finitePathChain_injective" ], "OAI.Combinatorics.Crossing.FinitePolygonalComposition": [ "finitePolygonalChain", "finitePolygonalChain_intersection_left", "finitePolygonalChain_intersection_right", "finitePolygonalChain_transverse_left", "finitePolygonalChain_transverse_right", "finitePolygonalChain_intersection_double" ], "OAI.Combinatorics.Crossing.FiniteScaledRadius": [ "finite_scaled_radius" ], "OAI.Combinatorics.Crossing.FiniteSeparation": [ "finite_positive_lower_bound", "finite_node_separation" ], "OAI.Combinatorics.Crossing.FirstLastVisits": [ "first_last_sublevel_visits" ], "OAI.Combinatorics.Crossing.FloorArithmetic": [ "axisPairs", "square_quarter_even", "square_quarter_odd", "axisPairs_even", "axisPairs_odd", "diagonal_floor_identity" ], "OAI.Combinatorics.Crossing.FourCyclePairing": [ "edgeBlock", "blockRectangle", "fourCycle_pairing_rectangle" ], "OAI.Combinatorics.Crossing.FourCycles": [ "completeBipartite", "orientedEdgeStep", "walkEdgeChain", "fourCycleWalk", "fourCycleWalk_isCycle", "fourCycleWalk_chain" ], "OAI.Combinatorics.Crossing.GeneralOrderAveraging": [ "orderSign_of_close", "orderSign_straddling_average" ], "OAI.Combinatorics.Crossing.GeneralSpokeAveraging": [ "small_perturbation_injective", "fixed_node_ne_small_perturbation", "local_spoke_straddling_average" ], "OAI.Combinatorics.Crossing.GenericPointExtension": [ "exists_generic_point_extension" ], "OAI.Combinatorics.Crossing.GenericPointFamily": [ "exists_generic_point_family" ], "OAI.Combinatorics.Crossing.GenericPolygonIntersection": [ "PolygonLinePosition", "polygon_intersection_zero_in_line_position" ], "OAI.Combinatorics.Crossing.GenericPolynomialBox": [ "exists_box_avoiding_polynomials" ], "OAI.Combinatorics.Crossing.GlobalCycleAveraging": [ "closed_cycle_averaged_sum", "graph_cycles_averaged_sum" ], "OAI.Combinatorics.Crossing.GlobalCycleIntersection": [ "exists_closed_cycle_intersection_radius" ], "OAI.Combinatorics.Crossing.GraphSequenceWalk": [ "graphSequenceWalk", "graphSequenceWalk_length", "graphSequenceWalk_getVert", "graphSequenceWalk_isPath" ], "OAI.Combinatorics.Crossing.GraphWalkRealizationLaws": [ "graphWalkPath_range_iff", "graphWalkPath_copy_range", "graphWalkPath_mapLe_range", "graphWalkPath_append_range", "graphWalkPath_toWalk_range" ], "OAI.Combinatorics.Crossing.GraphWalkTrace": [ "polygonalArcPath_piece_subset_range", "graphWalkPath_dart_subset_range", "walk_dart_at", "graphWalkPath_range_subset_of_darts_subset", "graphWalkPath_bypass_range_subset" ], "OAI.Combinatorics.Crossing.HalfOrderInvertibility": [ "rankDetector_halfOrder_isUnit", "lemma4_2" ], "OAI.Combinatorics.Crossing.HalfOrderReduction": [ "strictLowerMatrix", "halfOrder_decomposition", "triangleCoefficient", "rowCut", "lower_triangle_decomposition", "triangleCoefficient_nat_nonzero", "triangleCoefficient_ne_zero_iff", "rankDetector_halfOrder_reduction" ], "OAI.Combinatorics.Crossing.IncidentSpokePorts": [ "IncidentEdges", "incident_edges_adjacent", "drawingSpokePort", "drawingSpokePort_interior", "drawingSpokePort_circle", "drawingSpokePort_injective" ], "OAI.Combinatorics.Crossing.IndexedMatrixRankSum": [ "matrix_rank_sum_bound_indexed", "matrix_rank_sum_bound_product" ], "OAI.Combinatorics.Crossing.InnerDrawingPassages": [ "InnerIndex", "innerMark", "incomingArc", "outgoingArc", "incomingPort", "outgoingPort", "incoming_node", "outgoing_node", "incomingPort_mark", "outgoingPort_mark", "incomingPort_point", "outgoingPort_point", "inner_parameter_interior", "innerMark_is_crossing", "crossingChord", "crossingChord_injective" ], "OAI.Combinatorics.Crossing.Interpolation": [ "bounded_sum", "eq_zero_of_node_roots", "nodeEvaluation", "nodeEvaluation_injective", "vectorInterpolant", "eval_vectorInterpolant", "bounded_vectorInterpolant", "nodeEvaluation_surjective", "nodeEvaluationEquiv", "finrank_constrained_zero", "constrained_finrank_mono", "sum_rankIncrement", "interpolation_identity" ], "OAI.Combinatorics.Crossing.IntervalSegmentStrict": [ "intervalSegment_strict_between" ], "OAI.Combinatorics.Crossing.IntervalSegments": [ "intervalSegment", "intervalSegment_between" ], "OAI.Combinatorics.Crossing.LocalLineBends": [ "segment_direction_of_local_line", "bend_det_zero_of_local_line" ], "OAI.Combinatorics.Crossing.LocalSegmentTransversality": [ "segment_direction_of_local_line_at_interior", "det2_ne_zero_of_parallel_nonzero" ], "OAI.Combinatorics.Crossing.LocallyStraightNormalization": [ "LocallyStraightAt", "LocallyStraightCrossings", "assembled_locallyStraightCrossings", "polygonal_normalization_locally_straight" ], "OAI.Combinatorics.Crossing.LocallyTransverseNormalization": [ "LocallyTransverseCrossings", "DrawingPassages.crossingChord_meeting_det", "PolygonalExterior.assembled_locallyTransverseCrossings", "polygonal_no_adjacent_locally_transverse" ], "OAI.Combinatorics.Crossing.Main": [ "mainTarget_proof", "ordinary_crossing_number_eq", "ordinary_crossing_number_attained" ], "OAI.Combinatorics.Crossing.MarkedCurveDisks": [ "closedRoundDisk_mono", "finite_curve_mark_isolation" ], "OAI.Combinatorics.Crossing.MatrixGraphs": [ "graphKernelEquiv", "matrix_graph_finrank_add_rank", "matrix_graph_disjoint" ], "OAI.Combinatorics.Crossing.MatrixRankBounds": [ "matrix_rank_eq_zero_iff", "matrix_rank_add_le", "matrix_rank_smul_le", "matrix_rank_sum_le" ], "OAI.Combinatorics.Crossing.MatrixRankSum": [ "offDiagonalRankSum", "matrix_rank_sum_bound" ], "OAI.Combinatorics.Crossing.MiddleEdgeIncidence": [ "middle_exterior_edge_eq", "middle_spoke_edge_eq", "middle_middle_meeting_chords" ], "OAI.Combinatorics.Crossing.MiddlePathInjective": [ "middlePath_injective", "middlePath_range_iff" ], "OAI.Combinatorics.Crossing.MiddleSpokeMeeting": [ "vertex_spoke_range_in_disk", "vertex_spokes_disjoint_of_vertices_ne", "middle_spoke_meeting", "anchoredMiddle", "anchoredMiddle_range", "anchoredMiddle_injective" ], "OAI.Combinatorics.Crossing.MinimumChainGeometry": [ "adjacent_segments_meet_only_at_bend", "minimum_piece_ne_zero", "minimum_bend_det_ne_zero", "minimum_adjacent_intersection" ], "OAI.Combinatorics.Crossing.MinimumChainVertexPieces": [ "PolygonalChain.minimum_vertex_piece_index", "drawing_chain_piece_vertex" ], "OAI.Combinatorics.Crossing.MovingDiskClearance": [ "eventually_moving_segment_outside_disk", "exists_shifted_segment_disk_radius" ], "OAI.Combinatorics.Crossing.MovingSegmentStability": [ "moving_endpoints_tendsto", "eventually_moving_segment_pair", "eventually_moving_disjoint_segments" ], "OAI.Combinatorics.Crossing.NoAdjacentLocalNormalization": [ "polygonal_no_adjacent_locally_straight" ], "OAI.Combinatorics.Crossing.NoAdjacentNormalization": [ "exists_no_adjacent_assembled", "polygonal_normalization_no_adjacent" ], "OAI.Combinatorics.Crossing.NodeMoments": [ "nodePolynomial", "derivativeAtNode", "derivativeAtNode_eq_prod", "derivativeAtNode_ne_zero", "node_polynomial_moment", "node_power_moment" ], "OAI.Combinatorics.Crossing.NodeRankLoss": [ "off_node_projected_rank", "node_rank_loss" ], "OAI.Combinatorics.Crossing.NormalOffsetExterior": [ "dot2_rotateQuarter_self", "squaredLength_normal_offset", "normal_offset_radial_exterior", "normal_offset_segment_exterior" ], "OAI.Combinatorics.Crossing.NormalizedPolygonalDrawing": [ "polygonal_normalization_subdivided", "PolygonalParametrization.pieces_pos" ], "OAI.Combinatorics.Crossing.OddCrossingLowerBound": [ "odd_crossing_lower_bound" ], "OAI.Combinatorics.Crossing.OddNormalizedLowerBound": [ "odd_normalized_crossing_lower_bound" ], "OAI.Combinatorics.Crossing.OffsetBendSegments": [ "offset_bend_no_reverse_contact", "incoming_shifted_outgoing_disjoint", "outgoing_shifted_incoming_disjoint" ], "OAI.Combinatorics.Crossing.OffsetSegments": [ "offset_segment_parameterization", "offset_segment_side", "offset_segment_no_contact", "offset_bend_no_contact" ], "OAI.Combinatorics.Crossing.OffsetTrack": [ "trackDirection", "trackOffset", "shiftedTrack", "trackOffset_incident_levels", "shiftedTrack_stretch_no_contact", "exists_forward_shiftedTrack_radius" ], "OAI.Combinatorics.Crossing.OffsetVectors": [ "rotateQuarter", "squaredLength", "squaredLength_pos", "normalShift", "det2_normalShift", "cornerShift", "det2_cornerShift_left", "det2_cornerShift_right" ], "OAI.Combinatorics.Crossing.OpenSegmentEndpoints": [ "openSegmentTrace_subset_segment", "openSegmentTrace_ne_endpoints" ], "OAI.Combinatorics.Crossing.OrderShift": [ "orderSign_shift_of_separated", "orderSign_opposite_average", "shifted_node_ne" ], "OAI.Combinatorics.Crossing.OrderedAffineSubdivision": [ "strictMono_no_between_successors", "HasFiniteAffineCover.nonempty_polygonalParametrization" ], "OAI.Combinatorics.Crossing.OrderedCrossingPorts": [ "middlePath_contains_chord", "assembledEdge_contains_chord", "assembled_crossing_ports_ordered", "crossing_port_walk_indices" ], "OAI.Combinatorics.Crossing.OrderedExteriorPaths": [ "orderedArc", "orderedArc_val", "orderedArc_inj", "orderedArc_first", "orderedArc_last", "orderedInner", "orderedInner_incoming", "orderedInner_outgoing", "orderedInner_inj", "orderedPath", "orderedBridge", "orderedBridge_apply", "orderedBridge_range", "middlePath" ], "OAI.Combinatorics.Crossing.OrderedKnotCover": [ "ordered_knots_cover", "PolygonalParametrization.knot_cover" ], "OAI.Combinatorics.Crossing.OrderedNodeIndices": [ "OrderedNode", "extendOrderedNode", "extendOrderedNode_injOn", "triangle_nodes_mem", "triangleNodePair", "triangleNodePair_injective", "orderedTriangle", "triangleNodeEquiv", "triangleMatrix_ordered_entry" ], "OAI.Combinatorics.Crossing.OrderedPassages": [ "diskPassage_separated", "retainedSubarc", "ordered_passage_enter_strictMono", "ordered_passage_leave_strictMono", "retained_interval_no_mark", "retained_interval_pair_disjoint" ], "OAI.Combinatorics.Crossing.OrderedPathParts": [ "OrderedPathPart", "OrderedPathPart.refl", "OrderedPathPart.trans", "OrderedPathPart.left", "OrderedPathPart.right", "OrderedPathPart.congr_right", "OrderedPathPart.endpoints_ordered", "alternatingPath_contains_path", "alternatingPath_contains_bridge" ], "OAI.Combinatorics.Crossing.OrientedSimilarity": [ "similarityPoint", "similarityPoint_injective", "similarityPoint_segment", "similarityPoint_det", "segmentsCrossProperly_similarity", "segmentCrossingSign_similarity" ], "OAI.Combinatorics.Crossing.ParallelDirection": [ "dot2", "planar_vector_decomposition", "vector_parallel_of_det_zero", "offset_direction_det_zero", "exists_forward_offset_radius" ], "OAI.Combinatorics.Crossing.ParameterNeighborhoods": [ "parameterNeighborhood", "parameterNeighborhood_open", "parameterNeighborhood_between", "parameterNeighborhood_ordered", "finite_parameter_neighborhoods" ], "OAI.Combinatorics.Crossing.PathConcatenationInjective": [ "path_meeting_parameters", "path_trans_injective" ], "OAI.Combinatorics.Crossing.PathEndpointNeighborhoods": [ "path_near_endpoints" ], "OAI.Combinatorics.Crossing.PathSubarc": [ "intervalSegment_injective", "intervalSegment_surjective_interval", "pathSubarc", "pathSubarc_injective", "pathSubarc_range" ], "OAI.Combinatorics.Crossing.PeriodicPolygon": [ "cyclicIndex", "periodicPolygon", "cyclicIndex_of_lt", "cyclicIndex_period", "periodicPolygon_closed", "cyclicIndex_succ_ne", "periodicPolygon_near" ], "OAI.Combinatorics.Crossing.PhysicalCircleAveraging": [ "physical_spoke_average_in_similarity", "actual_circle_spoke_averaging" ], "OAI.Combinatorics.Crossing.PhysicalSpokeAveraging": [ "exists_physical_spoke_average" ], "OAI.Combinatorics.Crossing.PhysicalVertexChords": [ "shifted_chords_transverse", "chord_connector_disjoint" ], "OAI.Combinatorics.Crossing.PlanarDeterminant": [ "det2_add_left", "det2_add_right", "det2_sub_left", "det2_sub_right", "det2_smul_left", "det2_smul_right" ], "OAI.Combinatorics.Crossing.PlanarSegments": [ "det2", "segmentPoint", "openSegmentTrace", "SegmentsCrossProperly", "segmentCrossingSign", "segmentPoint_reverse", "det2_swap", "det2_neg_left", "det2_neg_right", "segment_intersection_parameters_unique" ], "OAI.Combinatorics.Crossing.PolygonComposition": [ "joinedSequence", "joinedSequence_left", "joinedSequence_right", "polygonIntersection_join_left", "polygonIntersection_join_right", "reversedSequence", "polygonIntersection_reverse_left" ], "OAI.Combinatorics.Crossing.PolygonStability": [ "exists_polygon_intersection_stability_radius" ], "OAI.Combinatorics.Crossing.PolygonTransversalityStability": [ "exists_polygon_transversality_radius" ], "OAI.Combinatorics.Crossing.PolygonalArcInjective": [ "polygonalArcPath_range_piece", "polygonalArcPath_injective", "minimum_path_injective", "exists_simple_polygonalChain_of_path" ], "OAI.Combinatorics.Crossing.PolygonalBoundaryReplacement": [ "PolygonalChain.widen", "simple_path_interior_of_range_insert", "polygonal_replacement_with_spurs" ], "OAI.Combinatorics.Crossing.PolygonalChainComposition": [ "reverse", "intersection_join_left", "intersection_join_right", "intersection_reverse_left", "intersection_reverse_right" ], "OAI.Combinatorics.Crossing.PolygonalChainExistence": [ "polygonalArcPath_range_subset", "path_range_subset", "exists_minimum_polygonalChain", "exists_minimum_polygonalChain_of_path" ], "OAI.Combinatorics.Crossing.PolygonalChainOperations": [ "vertex_mem", "single", "take", "drop", "join" ], "OAI.Combinatorics.Crossing.PolygonalChainShortcuts": [ "withEndpoints", "minimum_shortcut_bound", "minimum_vertices_ne", "minimum_nonadjacent_disjoint", "minimum_two_step_not_subset" ], "OAI.Combinatorics.Crossing.PolygonalChainTransversality": [ "Transverse", "transverse_swap", "transverse_join_left", "transverse_join_right", "closed_intersection_zero" ], "OAI.Combinatorics.Crossing.PolygonalChains": [ "polygonalArcPath", "PolygonalChain", "point", "append", "path", "nonempty_polygonalChain_of_reach", "nonempty_polygonalChain_of_path" ], "OAI.Combinatorics.Crossing.PolygonalEndpointCasts": [ "castEndpoints", "castEndpoints_pieces", "castEndpoints_vertices" ], "OAI.Combinatorics.Crossing.PolygonalExterior": [ "PolygonalExterior", "nonempty_polygonalExterior", "outside_open_disks", "avoids_new_vertices", "parameter_endpoint_of_mem_disk" ], "OAI.Combinatorics.Crossing.PolygonalNormalization": [ "alternatingPath_hasFiniteAffineCover", "middlePath_hasFiniteAffineCover", "assembledEdge_hasFiniteAffineCover", "finite_intersection_normalization", "polygonal_normalization" ], "OAI.Combinatorics.Crossing.PolygonalParametrization": [ "PolygonalParametrization", "vertices", "vertices_at", "vertices_first", "vertices_last", "segment_formula", "direction_ne_zero", "segments_disjoint_before", "nonadjacent_segments_disjoint" ], "OAI.Combinatorics.Crossing.PolygonalParametrizationCover": [ "PolygonalParametrization.hasFiniteAffineCover" ], "OAI.Combinatorics.Crossing.PolygonalReach": [ "SegmentReach", "segmentReach_mem", "isOpen_segmentReach", "isOpen_non_segmentReach", "segmentReach_of_path" ], "OAI.Combinatorics.Crossing.PolygonalWalks": [ "polygonalPath", "polygonIntersection", "polygonIntersection_swap", "telescoping_real_sum", "triangleSegmentIntersection", "polygon_cone_cancellation" ], "OAI.Combinatorics.Crossing.PolynomialCoordinates": [ "scalarPolynomial", "scalarPolynomial_coeff", "scalarPolynomial_eval", "scalarPolynomial_natDegree_le", "polynomialColumnMatrix", "polynomialColumnMatrix_eval", "exists_polynomial_minor" ], "OAI.Combinatorics.Crossing.PolynomialFiltration": [ "evaluationFiltration", "evaluationFiltration_mono", "evaluationFiltration_zero", "columnIncrement", "columnIncrement_zero", "columnIncrement_succ", "sum_columnIncrement", "exists_adapted_polynomials", "exists_adapted_minor" ], "OAI.Combinatorics.Crossing.PolynomialKernel": [ "Bounded", "bounded_zero", "bounded_mono", "bounded_add", "bounded_sub", "bounded_single", "bounded_zero_iff", "mulRoot", "mulRoot_coeff_zero", "mulRoot_coeff_succ", "eval_mulRoot", "bounded_mulRoot", "eq_constant_of_bounded_one", "exists_mulRoot_of_eval_zero", "eq_zero_of_mulRoot_eq_zero", "mulRootLinear", "constrained", "constrainedEval", "toEvalKernel", "toEvalKernel_injective", "toEvalKernel_surjective", "evalKernelEquiv", "constrainedCoefficients", "constrainedCoefficients_injective", "rankIncrement", "finrank_eval_range" ], "OAI.Combinatorics.Crossing.PolynomialLinearMatrix": [ "polynomialLinearMatrix", "polynomialLinearMatrix_eval", "polynomialLinearMatrix_rank" ], "OAI.Combinatorics.Crossing.PolynomialMinors": [ "natDegree_det_le_column_sum", "pow_card_dvd_det_of_columns_vanish", "prod_pow_roots_dvd", "sum_multiplicities_le_natDegree", "sum_multiplicities_le_column_sum" ], "OAI.Combinatorics.Crossing.ProperCrossingLocality": [ "chart_image_mem_iff", "chart_restr_image_of_local_eq", "properCrossing_of_local_eq" ], "OAI.Combinatorics.Crossing.ProperSegmentCrossing": [ "properCrossing_openSegments" ], "OAI.Combinatorics.Crossing.RadialBoundaryPorts": [ "radialPort", "squaredLength_smul", "radialPort_sub", "radialPort_on_circle", "radialPort_parameter", "radialPort_parameter_between", "radialPort_mem_openSegment", "radialPort_ne_center" ], "OAI.Combinatorics.Crossing.RadialExterior": [ "squaredLength_add_smul", "squaredLength_segment_translated", "exterior_segment_of_dot", "radial_spur_exterior", "radial_tip_dot" ], "OAI.Combinatorics.Crossing.RankDetector": [ "detectorVector", "detectorWeight", "detectorAtom", "rankDetector", "detectorAtom_self", "detectorWeight_ne_zero", "detectorAtom_entry", "rankDetector_diagonal" ], "OAI.Combinatorics.Crossing.RankMultiplicity": [ "rank_deficiency_pow_dvd_det", "pow_dvd_det_of_rank_le" ], "OAI.Combinatorics.Crossing.RankSpecialization": [ "exists_row_minor", "exists_rank_minor", "polynomial_rank_specialization" ], "OAI.Combinatorics.Crossing.ReconnectedIntersectionCases": [ "reconnected_paths_meeting_cases", "reconnected_paths_meeting_old_edges" ], "OAI.Combinatorics.Crossing.ReconnectedPaths": [ "reconnectedForwardPath", "reconnectedBackwardPath", "reconnectedForwardPath_injective", "reconnectedBackwardPath_injective", "reconnectedForwardPath_hasFiniteAffineCover", "reconnectedBackwardPath_hasFiniteAffineCover", "reconnectedForwardPath_trace", "reconnectedBackwardPath_trace" ], "OAI.Combinatorics.Crossing.ReconnectedPointRemoval": [ "reconnected_paths_avoid_crossing" ], "OAI.Combinatorics.Crossing.ReconnectedVertexAvoidance": [ "path_interior_avoids_vertices_of_range", "reconnectedForwardPath_vertex_eq", "reconnectedBackwardPath_vertex_eq", "reconnectedForwardPath_interior_avoids_vertices", "reconnectedBackwardPath_interior_avoids_vertices" ], "OAI.Combinatorics.Crossing.ReconnectedWalks": [ "surgery_forward_adj", "surgery_backward_adj", "reconnectedForwardWalk", "reconnectedBackwardWalk", "reconnectedForwardWalk_trace", "reconnectedBackwardWalk_trace" ], "OAI.Combinatorics.Crossing.ReconnectionAvoidance": [ "segment_avoids_of_affine_positive", "crossing_reconnections_avoid_point", "DrawingPassages.reconnection_avoids_chord_meeting" ], "OAI.Combinatorics.Crossing.ReconnectionChords": [ "reconnectionStart", "reconnectionEnd", "reconnection_endpoints_ne", "crossingChord_meeting_segmentsCrossProperly", "reconnection_chords_disjoint", "reconnection_endpoints_on_circle", "reconnection_open_in_disk" ], "OAI.Combinatorics.Crossing.ReconnectionOppositeCuts": [ "forward_chord_disjoint_opposite_prefix", "forward_chord_disjoint_opposite_suffix", "backward_chord_disjoint_opposite_prefix", "backward_chord_disjoint_opposite_suffix" ], "OAI.Combinatorics.Crossing.ReconnectionOtherEdges": [ "segment_disjoint_of_open_containment", "assembled_port_not_mem_other_edge", "reconnection_other_edge_disjoint" ], "OAI.Combinatorics.Crossing.ReconnectionVertexAvoidance": [ "reconnection_segment_in_disk", "reconnection_segment_avoids_vertices" ], "OAI.Combinatorics.Crossing.RectangleAlgebra": [ "edgePairing", "edgePairing_single", "edgePairing_add_left", "edgePairing_add_right", "edgePairing_sub_left", "edgePairing_sub_right", "rectangle_additive" ], "OAI.Combinatorics.Crossing.ReducedCrossingCorners": [ "det2_of_common_openSegment", "LocallyStraightAt.exists_trace_line", "minimum_bend_not_locallyStraight", "minimum_bend_not_crossing", "minimum_vertices_not_crossing", "minimum_piece_crossing_open" ], "OAI.Combinatorics.Crossing.ReducedCrossingTransversality": [ "LocallyTransverseCrossings.exists_trace_lines", "minimum_piece_det_at_crossing" ], "OAI.Combinatorics.Crossing.ReducedPolygonalPresentation": [ "PolygonalParametrization.containedChain", "PolygonalChain.pieces_pos_of_ne", "exists_reduced_polygonalChain" ], "OAI.Combinatorics.Crossing.ReplaceDrawingEdges": [ "replaceTwoRoutes", "replaceTwoRoutes_first", "replaceTwoRoutes_second", "replaceTwoRoutes_other", "replaceTwoEdges" ], "OAI.Combinatorics.Crossing.ReplaceDrawingRangeCases": [ "replaceTwoRoutes_range_cases" ], "OAI.Combinatorics.Crossing.RetainedArcDisjointness": [ "arcs_pairwise_disjoint" ], "OAI.Combinatorics.Crossing.RetainedDrawingArcs": [ "ArcIndex", "node_mark", "arcStartMark", "arcEndMark", "arcStart", "arcEnd", "arc", "arc_gap", "arc_injective", "arc_endpoints_ne", "arc_range", "arc_avoids_marks", "retained_parameter_outside", "arc_interior_outside", "arc_start_on_circle", "arc_end_on_circle" ], "OAI.Combinatorics.Crossing.RoundCircleChords": [ "round_circle_line_two_points", "round_circle_orient_ne_zero", "round_circle_chords_transverse", "round_chord_in_open_disk" ], "OAI.Combinatorics.Crossing.RoundDiskChords": [ "round_chord_subset_closed_disk", "round_chord_boundary_endpoints" ], "OAI.Combinatorics.Crossing.RoundDiskClearance": [ "closedRoundDisk", "dist_le_of_mem_closedRoundDisk", "exists_round_disk_clearance", "exists_disjoint_round_disks" ], "OAI.Combinatorics.Crossing.RoundDiskVisits": [ "continuous_path_squared_distance", "path_round_disk_visits", "path_initial_disk_last_visit" ], "OAI.Combinatorics.Crossing.RowCrossingCount": [ "NoAdjacentCrossings", "pairCrossings_eq_empty_of_adjacent", "rowCrossingCount", "rowCrossingCount_self", "sum_rowCrossingCount", "sum_offDiagonal_rowCrossingCount" ], "OAI.Combinatorics.Crossing.SegmentAffineForms": [ "segmentPoint_start_form", "segmentPoint_end_form" ], "OAI.Combinatorics.Crossing.SegmentAugmentation": [ "segment_endpoint_or_open", "segmentAugmentedGraph", "le_segmentAugmentedGraph", "segmentAugmentedGraph_added_adj", "exterior_segment_chord_meeting", "segmentAugmentedGraph_embedding" ], "OAI.Combinatorics.Crossing.SegmentClosedIncidence": [ "closedSegmentsMeet_iff_nonempty_inter", "not_closedSegmentsMeet_of_disjoint" ], "OAI.Combinatorics.Crossing.SegmentContinuity": [ "ClosedSegmentsMeet", "closedSegmentsMeet_of_proper", "isClosed_closedSegmentsMeet", "eventually_segmentCrossingSign_of_proper", "SegmentPairTransverse", "eventually_segmentCrossingSign" ], "OAI.Combinatorics.Crossing.SegmentCrossingCoordinates": [ "inverseCoordinates_first_segment", "inverseCoordinates_second_segment", "crossingCoordinates_first_segment", "crossingCoordinates_second_segment" ], "OAI.Combinatorics.Crossing.SegmentIntersectionUnique": [ "transverse_segment_intersection_subsingleton" ], "OAI.Combinatorics.Crossing.SegmentParameters": [ "firstIntersectionParameter", "secondIntersectionParameter", "intersection_parameters_eq", "segmentsCrossProperly_iff_parameters" ], "OAI.Combinatorics.Crossing.SegmentSides": [ "orient2", "orient2_self_segment", "orient2_segment", "orient2_difference", "orient2_other_difference", "segmentCrossingSign_of_positive_sides", "not_segmentsCrossProperly_of_same_side" ], "OAI.Combinatorics.Crossing.SegmentSymmetries": [ "segmentsCrossProperly_swap", "segmentsCrossProperly_reverse_left", "nonzero_orientation_sign_neg", "segmentCrossingSign_swap", "segmentCrossingSign_reverse_left", "openSegment_intersection_subsingleton" ], "OAI.Combinatorics.Crossing.SegmentTraversalReversal": [ "segmentPairTransverse_of_not_meet", "segmentCrossingSign_eq_zero_of_not_meet", "segmentPairTransverse_reverse_left", "segmentPairTransverse_reverse_right", "transverse_reverse_left", "transverse_reverse_right" ], "OAI.Combinatorics.Crossing.SharedTrackAvoidance": [ "exists_shared_track_avoidance_radius" ], "OAI.Combinatorics.Crossing.ShiftedEdgeIntersections": [ "exists_complete_edge_intersection_radius" ], "OAI.Combinatorics.Crossing.ShiftedRadialPieces": [ "shifted_first_piece_exterior", "shifted_last_piece_exterior" ], "OAI.Combinatorics.Crossing.ShiftedSegmentStability": [ "shifted_endpoints_tendsto", "eventually_shifted_segment_pair", "closedSegmentsMeet_symm", "eventually_shifted_disjoint_segments", "exists_shifted_segment_pairs_radius" ], "OAI.Combinatorics.Crossing.ShiftedTrackDisks": [ "exists_nonincident_shift_radius", "exists_all_vertex_disk_radius" ], "OAI.Combinatorics.Crossing.ShortVertexRadius": [ "exists_short_vertex_radius" ], "OAI.Combinatorics.Crossing.SimilarityCoordinates": [ "similarityCoordinates", "similarityPoint_coordinates", "similarityCoordinates_point", "similarityCoordinates_length", "similarityPoint_cut", "circle_similarity_parameter" ], "OAI.Combinatorics.Crossing.SimultaneousExteriorArcs": [ "finite_exterior_arc_replacements" ], "OAI.Combinatorics.Crossing.SimultaneousPolygonalReplacement": [ "exists_simple_polygonalChain", "finite_disjoint_polygonal_replacements" ], "OAI.Combinatorics.Crossing.SortedMarkedParameters": [ "exists_sorted_unitInterval_finset", "exists_sorted_drawing_parameters" ], "OAI.Combinatorics.Crossing.SortedSideEvents": [ "exists_sorted_permutation", "sum_affine_side_events_unordered", "prod_positiveIndicator" ], "OAI.Combinatorics.Crossing.SpokeEndpointSeparation": [ "continuous_shiftedPort", "shiftedPort_zero", "shiftedPort_ne_original", "exists_endpoint_separation_radius" ], "OAI.Combinatorics.Crossing.SpokePieceIsolation": [ "spokePort_interior", "spokePort_piece_unique" ], "OAI.Combinatorics.Crossing.SpokeTrackDirections": [ "drawingSpokeDirection", "drawingSpokeDirection_ne_zero", "drawingSpokeDirection_port_parallel", "drawingSpokeScale_ne_zero" ], "OAI.Combinatorics.Crossing.StageDegreeBound": [ "polynomial_columns_rank_le", "stage_degree_bound" ], "OAI.Combinatorics.Crossing.StepEvents": [ "stepAt", "stepEvent", "sum_stepEvent" ], "OAI.Combinatorics.Crossing.StraightChainIntersections": [ "intersection_single_right", "intersection_single_left", "transverse_single_right", "transverse_single_left", "intersection_single_right_zero", "intersection_single_single" ], "OAI.Combinatorics.Crossing.StraightChainTransversality": [ "transverse_single_single", "intersection_single_left_zero" ], "OAI.Combinatorics.Crossing.SubdivisionPointEndpoints": [ "subdivision_point_endpoint", "drawingSubdivisionGraph_point_endpoint" ], "OAI.Combinatorics.Crossing.SubdivisionSegments": [ "pieceTrace", "pieceTrace_parameter", "pieceTrace_subset_range", "pieceTrace_knot_endpoint", "pieceTrace_other_piece_endpoint", "subdivision_mark_endpoint" ], "OAI.Combinatorics.Crossing.SublevelExtrema": [ "sublevel_passage_extrema" ], "OAI.Combinatorics.Crossing.SubspaceIntersection": [ "subspace_intersection_bound", "lemma2_1" ], "OAI.Combinatorics.Crossing.SubspaceProjection": [ "finrank_map_add_le", "finrank_map_of_disjoint" ], "OAI.Combinatorics.Crossing.SurgeryCrossingContainment": [ "adjacentSurgeryDrawing_crossings_subset" ], "OAI.Combinatorics.Crossing.SurgeryCutWalks": [ "surgeryPrefixWalk", "surgerySuffixWalk", "surgeryPrefixWalk_trace", "surgerySuffixWalk_trace" ], "OAI.Combinatorics.Crossing.SurgeryFiniteDrawing": [ "adjacentSurgeryFiniteDrawing", "adjacentSurgeryFiniteDrawing_count_lt" ], "OAI.Combinatorics.Crossing.SurgeryNoTriples": [ "adjacentSurgeryDrawing_no_triples" ], "OAI.Combinatorics.Crossing.SurgeryPathWitnesses": [ "walkThroughChord", "walkThroughChord_range", "exists_simple_path_through_chord", "surgery_mixed_endpoints_ne" ], "OAI.Combinatorics.Crossing.SurgeryRouteIntersections": [ "reconnectedForwardPath_meeting_other", "reconnectedBackwardPath_meeting_other", "adjacentSurgeryRoutes_meeting_other", "adjacentSurgeryRoutes_meeting_each_other" ], "OAI.Combinatorics.Crossing.SurgeryStrictPointCount": [ "crossingChord_meeting_is_assembled_crossing", "adjacentSurgeryDrawing_strict_crossing_ncard" ], "OAI.Combinatorics.Crossing.TangentTrackIdentity": [ "det2_rotateQuarter_self", "normalShift_eq_circleTrackScale", "similarity_circleTangentOffset", "tangentOffset_eq_normalShift", "similarityPoint_squaredLength", "physical_circleConnector_strictly_outside" ], "OAI.Combinatorics.Crossing.TrackChordSeparation": [ "track_in_disk_is_port", "track_shifted_chord_disjoint" ], "OAI.Combinatorics.Crossing.TrackEndpointConnections": [ "leftIncident", "rightIncident", "vertices_first", "vertices_last", "spokePort_left", "spokePort_right", "shiftedTrack_first", "shiftedTrack_last" ], "OAI.Combinatorics.Crossing.TransversalityStability": [ "segmentPairTransverse_of_proper", "segmentPairTransverse_swap", "eventually_segmentPairTransverse" ], "OAI.Combinatorics.Crossing.TransversePolygonIntersection": [ "segmentCrossingSign_self_left", "polygon_intersection_zero_transverse" ], "OAI.Combinatorics.Crossing.TriangleEntryExit": [ "triangleStart", "triangleEnd", "triangleIndicator", "triangle_area_cycle", "affine_side_factor", "triangle_segment_entry_exit" ], "OAI.Combinatorics.Crossing.TriangleFactorization": [ "triangleDiagonalWeight", "triangleDiagonalWeight_ne_zero", "rankDetector_triangle_sum", "rankDetector_triangle_factorization", "rankDetector_ordered_isUnit" ], "OAI.Combinatorics.Crossing.TriangleOrientations": [ "orient2_reverse_edge", "det2_reverse_edge", "triangle_position_reverse", "triangle_intersection_reverse", "orientedTriangleIndicator", "oriented_triangle_entry_exit", "triangle_closed_walk_intersection_zero" ], "OAI.Combinatorics.Crossing.TriangleSideEvent": [ "triangle_side_at_first_edge", "triangle_other_side_at_first_edge", "triangle_edge_side_tests", "segmentCrossingSign_triangle_event" ], "OAI.Combinatorics.Crossing.TriangleTransversality": [ "TriangleSegmentPosition", "triangle_consecutive_roots_ne", "second_parameter_ne_zero", "second_parameter_ne_one", "triangle_roots_injective", "triangle_segment_entry_exit_of_position" ], "OAI.Combinatorics.Crossing.TruncatedChainDirections": [ "truncatedScale", "truncated_start_formula", "truncated_end_formula", "truncated_direction", "truncatedScale_pos", "truncated_direction_ne_zero" ], "OAI.Combinatorics.Crossing.TruncatedChainPath": [ "truncated", "truncated_segments_disjoint", "truncated_path_injective" ], "OAI.Combinatorics.Crossing.TruncatedChainVertices": [ "radialPort_parameter_half", "startFraction", "endFraction", "truncatedVertices", "startPort_eq_fraction", "endPort_eq_fraction", "truncatedVertices_first", "truncatedVertices_last", "truncated_piece_subset" ], "OAI.Combinatorics.Crossing.TruncatedCrossingCorners": [ "truncated_vertices_not_crossing" ], "OAI.Combinatorics.Crossing.TruncatedCrossingRetention": [ "crossing_mem_truncated_edge" ], "OAI.Combinatorics.Crossing.TruncatedDrawingDisks": [ "truncated_piece_vertex_disk_cases", "truncated_path_vertex_disk_cases" ], "OAI.Combinatorics.Crossing.TruncatedDrawingInteriors": [ "truncated_range_subset_edgeInterior", "truncated_edge_intersections_eq" ], "OAI.Combinatorics.Crossing.TruncatedRadialDots": [ "dot2_scaled_self", "start_radial_dot_pos", "end_radial_dot_neg", "bend_ne_zero", "exists_forward_radius" ], "OAI.Combinatorics.Crossing.TruncatedRadialEnds": [ "radial_segment_of_coefficients", "truncated_first_exterior", "truncated_last_exterior" ], "OAI.Combinatorics.Crossing.TruncatedSegmentTransversality": [ "segment_parameter_strict_of_avoids_endpoints", "truncated_segment_pair_transverse" ], "OAI.Combinatorics.Crossing.TruncatedSignedIntersections": [ "piece_subset_range", "signedIntersection", "signedIntersection_self", "signedIntersection_swap", "signedIntersection_zero_of_pairCrossings_empty", "pairCrossings_pos_of_signedIntersection_ne_zero", "signedIntersection_adjacent_zero" ], "OAI.Combinatorics.Crossing.TruncatedSpokeCoordinates": [ "TruncatedSpokeCoordinates", "nonempty_truncatedSpokeCoordinates", "shiftedPort", "tangentPort", "tangentPort_eq", "shiftedPort_level", "shiftedPort_circle", "connector_strictly_outside", "exists_vertex_average_radius" ], "OAI.Combinatorics.Crossing.TruncatedSpokeDirections": [ "spokePort", "spokeIndex", "spokeDirection", "spokeIndex_lt", "spokeDirection_ne_zero", "spokeDirection_eq_scale", "spokePort_circle", "spokePort_ne_center", "spokeDirection_parallel", "spokeScale", "spokeScale_ne_zero", "spokePort_injective" ], "OAI.Combinatorics.Crossing.TruncatedTraceDecomposition": [ "center_segment_in_disk", "drawing_edge_trace_decomposition" ], "OAI.Combinatorics.Crossing.TruncatedTrackAvoidance": [ "truncated_bend_ne_zero", "truncated_offset_avoidance" ], "OAI.Combinatorics.Crossing.TruncatedTrackStability": [ "exists_distinct_edge_shift_radius" ], "OAI.Combinatorics.Crossing.TwoTriangleBasis": [ "TriangleIndex", "triangleCoordEquiv", "monomialVector", "triangleMatrix", "triangleMatrix_eval", "triangleMatrix_isUnit", "twoTriangleBasis", "twoTriangleBasis_apply" ], "OAI.Combinatorics.Crossing.TwoTriangleIndices": [ "lowerTriangle", "upperTriangle", "twoTriangle", "mem_lowerTriangle", "mem_upperTriangle", "mem_twoTriangle", "disjoint_triangles", "card_lowerTriangle", "card_upperTriangle", "adjacent_choose_two", "card_twoTriangle", "triangleRelabel", "triangleRelabel_injective_on", "triangleRelabel_range", "relabel_twoTriangle" ], "OAI.Combinatorics.Crossing.TwoTriangleUnisolvence": [ "eq_zero_of_roots_injOn", "twoTriangle_unisolvent", "twoTriangle_evaluation_injective" ], "OAI.Combinatorics.Crossing.UniformCrossTypeGeometry": [ "exists_cross_type_radius" ], "OAI.Combinatorics.Crossing.UniformSpokeRadii": [ "port_injective", "exists_uniform_endpoint_radius", "exists_uniform_scale_radius", "exists_uniform_average_radius" ], "OAI.Combinatorics.Crossing.UniformTrackPerturbation": [ "exists_all_edge_shift_radius", "exists_uniform_track_intersections" ], "OAI.Combinatorics.Crossing.UniqueWalkPassage": [ "UniqueWalkVisit", "walk_dart_parameters", "unique_walk_visit_passage", "bypass_unique_walk_visit_passage" ], "OAI.Combinatorics.Crossing.VertexChordIntersections": [ "chordChain_intersection", "exists_vertex_chord_chain_radius" ], "OAI.Combinatorics.Crossing.VertexDiskSpokes": [ "spokes_meet_only_at_center", "spoke_in_open_disk", "exists_vertex_disk_point" ], "OAI.Combinatorics.Crossing.VertexDisksCrossingFree": [ "vertex_disks_crossing_free" ], "OAI.Combinatorics.Crossing.VertexEndpointPorts": [ "firstArc", "lastArc", "lastArc_node", "firstArc_mark", "lastArc_mark", "startPortAt", "endPortAt" ], "OAI.Combinatorics.Crossing.VertexOrderDetector": [ "leftNode", "leftNode_injective", "leftHalfOrder_eq_halfOrder", "diagonal_detector_isUnit", "offDiagonal_detector", "additive_detector_relation" ], "OAI.Combinatorics.Crossing.VertexPassagePairing": [ "vertexChordPairing", "passagePairing", "exists_vertex_pairing_radius", "exists_passage_pairing_radius" ], "OAI.Combinatorics.Crossing.VertexPieceClearance": [ "exists_vertex_piece_clearance" ], "OAI.Combinatorics.Crossing.VertexSpokeBounds": [ "exists_small_squared_radius", "exists_vertex_spoke_bound" ], "OAI.Combinatorics.Crossing.VertexTraceDisks": [ "exists_vertex_trace_disks" ], "OAI.Combinatorics.Crossing.VertexTruncation": [ "VertexTruncation", "nonempty_vertexTruncation", "pieces", "vertices", "startPort", "endPort", "pieces_pos", "weak_short", "path", "path_injective", "path_hasFiniteAffineCover", "range_subset_edgeInterior", "direction_ne_zero" ], "OAI.Combinatorics.Crossing.VertexTruncationGeometry": [ "offset_avoidance", "edge_intersections_eq", "pair_transverse", "crossing_free_disks", "vertices_not_crossing", "vertex_disk_cases" ], "OAI.Combinatorics.Crossing.WalkVectorSum": [ "walkEdgeChain_sum_steps", "walkEdgeChain_sum_cyclic_darts", "edgePairing_sum_left", "edgePairing_sum_right", "edgePairing_smul_single", "edgePairing_walk_dart_sum" ], "OAI.Combinatorics.Crossing.WeightedGram": [ "weightedGram_eq_sum", "weightedGram_isUnit" ] } }