| import OAI.Combinatorics.Crossing.StraightChainTransversality |
| import OAI.Combinatorics.Crossing.UniformCrossTypeGeometry |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace Zarankiewicz.TruncatedSpokeCoordinates |
| variable {m n : ℕ} {D : AdmissibleDrawing m n} {V : VertexTruncation D.toContinuousDrawing} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem exists_chord_edge_radius (hnorm : NoAdjacentCrossings D) : |
| ∃ δ : ℝ, 0<δ ∧ ∀ ε : ℝ, ε ≠ 0 → |ε|<δ → |
| (∀ j (a b : IncidentEdges j) e, |
| (V.chordChain j a b).Transverse (C.shiftedEdgeChain e ε) ∧ |
| polygonIntersection (V.chordChain j a b).vertices (C.shiftedEdgeChain e ε).vertices |
| (V.chordChain j a b).pieces (C.shiftedEdgeChain e ε).pieces=0) ∧ |
| (∀ j (a b : IncidentEdges j) e, |
| (V.trackChain e).Transverse (C.shiftedChordChain j a b ε) ∧ |
| polygonIntersection (V.trackChain e).vertices (C.shiftedChordChain j a b ε).vertices |
| (V.trackChain e).pieces (C.shiftedChordChain j a b ε).pieces=0) := by |
| obtain ⟨δ,hδ,hh⟩ := C.exists_cross_type_radius hnorm |
| refine ⟨δ,hδ,?_⟩ |
| intro ε hε hsmall |
| obtain ⟨hTC,hTChord,hChordT,hChordC,hChordChord,hOther⟩ := hh ε hε hsmall |
| constructor |
| · intro j a b e |
| have hconn (k : BipartiteVertex m n) (f : IncidentEdges k) : |
| (V.chordChain j a b).Transverse (C.connectorChain k f ε) := by |
| apply PolygonalChain.transverse_single_single |
| exact segmentPairTransverse_of_not_meet _ _ _ _ (hChordC j k a b f) |
| have hmid : (V.chordChain j a b).Transverse (C.shiftedTrackChain e ε) := by |
| apply PolygonalChain.transverse_single_left |
| intro i |
| exact segmentPairTransverse_of_not_meet _ _ _ _ (hChordT j a b e i) |
| have hzconn (k : BipartiteVertex m n) (f : IncidentEdges k) : |
| polygonIntersection (V.chordChain j a b).vertices (C.connectorChain k f ε).vertices 1 1=0 := by |
| rw [VertexTruncation.chordChain,connectorChain,PolygonalChain.intersection_single_single] |
| exact segmentCrossingSign_eq_zero_of_not_meet _ _ _ _ (hChordC j k a b f) |
| have hzmid : polygonIntersection (V.chordChain j a b).vertices |
| (shiftedTrack (V.vertices e) (V.pieces e) ε) 1 (V.pieces e)=0 := by |
| apply (C.shiftedTrackChain e ε).intersection_single_left_zero |
| intro i |
| exact hChordT j a b e i |
| constructor |
| · change (V.chordChain j a b).Transverse |
| (((C.connectorChain (Sum.inl e.1) (V.leftIncident e) ε).join (C.shiftedTrackChain e ε)).join |
| (C.connectorChain (Sum.inr e.2) (V.rightIncident e) ε).reverse) |
| apply PolygonalChain.transverse_join_right |
| · exact PolygonalChain.transverse_join_right _ _ _ (hconn _ _) hmid |
| · exact (V.chordChain j a b).transverse_reverse_right _ (hconn _ _) |
| · change polygonIntersection (V.chordChain j a b).vertices (C.shiftedEdgeChain e ε).vertices 1 _=0 |
| rw [C.shiftedEdgeChain_intersection_right,hzconn,hzconn,hzmid] |
| norm_num |
| · intro j a b e |
| constructor |
| · apply PolygonalChain.transverse_single_right |
| intro i |
| exact segmentPairTransverse_of_not_meet _ _ _ _ (hTChord j a b e i) |
| · apply PolygonalChain.intersection_single_right_zero |
| intro i |
| exact hTChord j a b e i |
|
|
| end Zarankiewicz.TruncatedSpokeCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
