File size: 3,312 Bytes
d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | 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
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