| import OAI.Combinatorics.Crossing.SurgeryFiniteDrawing |
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| namespace OAI |
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| noncomputable section |
| namespace Zarankiewicz.PolygonalExterior |
| variable {m n : ℕ} {D : AdmissibleDrawing m n} {C : DrawingPassages D.toFiniteIntersectionDrawing} |
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| theorem exists_admissible_adjacent_surgery (E : PolygonalExterior C) |
| (v : BipartiteVertex m n → Plane) (hvinj : Function.Injective v) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) |
| (p q : C.InnerIndex) (hpq : p ≠ q) (hadj : p.1.1=q.1.1 ∨ p.1.2=q.1.2) |
| (x : Plane) (hxp : x ∈ Set.range (C.crossingChord p)) (hxq : x ∈ Set.range (C.crossingChord q)) : |
| ∃ A : AdmissibleDrawing m n, (∀ e, HasFiniteAffineCover (A.edge e)) ∧ |
| crossingCount A < crossingCount (E.assembledAdmissible v hvinj hv hdet) := by |
| have hm := C.chord_meeting_marks_eq p q hxp hxq |
| have hcross := C.crossingChord_meeting_segmentsCrossProperly p q hpq x hxp hxq |
| obtain ⟨P,hmark,hport⟩ := E.exists_assembled_port_subdivision v hvinj hv hdet |
| obtain ⟨Kp⟩ := E.nonempty_exteriorCrossingCut v hvinj hv hdet P hport p |
| obtain ⟨Kq⟩ := E.nonempty_exteriorCrossingCut v hvinj hv hdet P hport q |
| have hemb := crossingSurgeryGraph_embedding C P hmark hport p q hm hcross |
| let B := E.adjacentSurgeryFiniteDrawing v hvinj hv hdet P Kp Kq hpq hm hadj hemb x hxp hxq |
| obtain ⟨A,hpoly,hcount⟩ := finite_intersection_normalization B |
| exact ⟨A,hpoly,hcount.trans_lt |
| (E.adjacentSurgeryFiniteDrawing_count_lt v hvinj hv hdet P Kp Kq hpq hm hadj hemb x hxp hxq)⟩ |
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| end Zarankiewicz.PolygonalExterior |
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| end |
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| end OAI |
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