| import OAI.Combinatorics.Crossing.AssembledCrossings |
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| namespace OAI |
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| noncomputable section |
| namespace Zarankiewicz.PolygonalExterior |
| variable {m n : ℕ} {D : FiniteIntersectionDrawing m n} {C : DrawingPassages D} |
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| theorem assembled_no_triples (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) |
| (e f g : BipartiteEdge m n) (hef : e ≠ f) (heg : e ≠ g) (hfg : f ≠ g) : |
| edgeInterior (E.assembledDrawing v hvinj hv) e ∩ |
| edgeInterior (E.assembledDrawing v hvinj hv) f ∩ |
| edgeInterior (E.assembledDrawing v hvinj hv) g=∅ := by |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| rintro x ⟨⟨⟨s,hs,he⟩,⟨t,ht,hf⟩⟩,⟨u,hu,hg⟩⟩ |
| have ha : ∀ j, x ≠ v j := fun j heq => |
| E.assembledEdge_interior_avoids_vertices v hv e s hs j (he.trans heq) |
| obtain ⟨p,q,hp,hq,hxp,hxq⟩ := E.assembled_meeting_chords v hv hdet e f hef x ha ⟨s,he⟩ ⟨t,hf⟩ |
| obtain ⟨r,z,hr,hz,hxr,hxz⟩ := E.assembled_meeting_chords v hv hdet e g heg x ha ⟨s,he⟩ ⟨u,hg⟩ |
| apply C.innerMark_no_triples p q z |
| · intro heq |
| exact hef (hp.symm.trans ((congrArg Sigma.fst heq).trans hq)) |
| · intro heq |
| exact heg (hp.symm.trans ((congrArg Sigma.fst heq).trans hz)) |
| · intro heq |
| exact hfg (hq.symm.trans ((congrArg Sigma.fst heq).trans hz)) |
| · exact C.chord_meeting_marks_eq p q hxp hxq |
| · exact C.chord_meeting_marks_eq p z hxp hxz |
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| end Zarankiewicz.PolygonalExterior |
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| end |
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| end OAI |
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