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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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 | 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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