| import OAI.Combinatorics.Crossing.CrossingDiskNeighborhood |
| import OAI.Combinatorics.Crossing.ProperSegmentCrossing |
| import OAI.Combinatorics.Crossing.ProperCrossingLocality |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace Zarankiewicz.DrawingPassages |
| variable {m n : ℕ} {D : FiniteIntersectionDrawing m n} |
|
|
| theorem crossingChord_meeting_proper (C : DrawingPassages D) (p q : C.InnerIndex) |
| (hpq : p ≠ q) (x : Plane) (hxp : x ∈ Set.range (C.crossingChord p)) |
| (hxq : x ∈ Set.range (C.crossingChord q)) : |
| x ∈ C.crossingDisk p ∧ |
| ProperCrossing |
| (openSegmentTrace (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort p))) |
| (openSegmentTrace (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort q))) x := by |
| have hm := C.chord_meeting_marks_eq p q hxp hxq |
| obtain ⟨s,hs⟩ := hxp |
| obtain ⟨t,ht⟩ := hxq |
| have hst : segmentPoint (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort p)) s.val = |
| segmentPoint (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort q)) t.val := by |
| simpa only [crossingChord,Path.segment_apply,AffineMap.lineMap_apply_module,segmentPoint] using hs.trans ht.symm |
| obtain ⟨hd,hsi,hti⟩ := C.crossing_chords_transverse_at_mark p q hpq hm s.val s.property t.val t.property hst |
| refine ⟨?_,?_⟩ |
| · rw [← hs] |
| exact C.crossingChord_interior_in_disk p s hsi |
| · apply properCrossing_openSegments _ _ _ _ x hd |
| · exact ⟨s.val,hsi,by simpa only [crossingChord,Path.segment_apply,AffineMap.lineMap_apply_module,segmentPoint] using hs⟩ |
| · exact ⟨t.val,hti,by simpa only [crossingChord,Path.segment_apply,AffineMap.lineMap_apply_module,segmentPoint] using ht⟩ |
|
|
| end DrawingPassages |
| namespace PolygonalExterior |
| variable {m n : ℕ} {D : FiniteIntersectionDrawing m n} {C : DrawingPassages D} |
|
|
| theorem assembled_crossings_proper (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 : BipartiteEdge m n) (hef : e ≠ f) (x : Plane) |
| (hx : x ∈ edgeInterior (E.assembledDrawing v hvinj hv) e) |
| (hy : x ∈ edgeInterior (E.assembledDrawing v hvinj hv) f) : |
| ProperCrossing (edgeInterior (E.assembledDrawing v hvinj hv) e) |
| (edgeInterior (E.assembledDrawing v hvinj hv) f) x := by |
| obtain ⟨s,hs,he⟩ := hx |
| obtain ⟨t,ht,hf⟩ := hy |
| obtain ⟨p,q,hp,hq,hxp,hxq⟩ := E.assembled_meeting_chords v hv hdet e f hef x |
| (fun j heq => E.assembledEdge_interior_avoids_vertices v hv e s hs j (he.trans heq)) ⟨s,he⟩ ⟨t,hf⟩ |
| have hpq : p ≠ q := fun heq => hef (hp.symm.trans ((congrArg Sigma.fst heq).trans hq)) |
| have hm := C.chord_meeting_marks_eq p q hxp hxq |
| obtain ⟨hxin,hproper⟩ := C.crossingChord_meeting_proper p q hpq x hxp hxq |
| apply properCrossing_of_local_eq _ _ _ _ x hproper (C.crossingDisk p) (C.crossingDisk_isOpen p) hxin |
| · intro z hz |
| simpa only [hp] using (E.assembled_trace_in_crossingDisk v hvinj hv p z hz).symm |
| · intro z hz |
| have hzq : z ∈ C.crossingDisk q := by |
| change squaredLength (z-(C.innerMark q).val) < C.radius^2 |
| rw [← hm] |
| exact hz |
| simpa only [hq] using (E.assembled_trace_in_crossingDisk v hvinj hv q z hzq).symm |
|
|
| def assembledAdmissible (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) : AdmissibleDrawing m n where |
| toContinuousDrawing := E.assembledDrawing v hvinj hv |
| crossings_finite := E.assembled_crossings_finite v hvinj hv hdet |
| crossings_proper := E.assembled_crossings_proper v hvinj hv hdet |
| no_triples := E.assembled_no_triples v hvinj hv hdet |
|
|
| theorem assembled_crossingCount_le (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) : |
| crossingCount (E.assembledAdmissible v hvinj hv hdet) ≤ finiteCrossingCount D := by |
| have h := Set.ncard_le_ncard (E.assembled_crossings_subset v hvinj hv hdet) C.chordCrossings_finite |
| have hcount : crossingCount (E.assembledAdmissible v hvinj hv hdet)= |
| (crossingPoints (E.assembledDrawing v hvinj hv)).ncard := by |
| rw [Set.ncard_eq_toFinset_card _ (E.assembled_crossings_finite v hvinj hv hdet)] |
| rfl |
| rw [hcount] |
| exact h.trans C.chordCrossings_ncard_le |
|
|
| end PolygonalExterior |
| end Zarankiewicz |
|
|
| end |
|
|
| end OAI |
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|
