File size: 3,596 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 69 70 71 72 73 74 75 76 | import OAI.Combinatorics.Crossing.AssembledEdge
namespace OAI
noncomputable section
namespace Zarankiewicz.DrawingPassages
variable {m n : ℕ} {D : FiniteIntersectionDrawing m n}
theorem vertex_spoke_vertex_eq (C : DrawingPassages D)
(v : BipartiteVertex m n → Plane) (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2)
(j k : BipartiteVertex m n) (p : C.PortsAt (C.vertexMark j))
(hk : v k ∈ Set.range (Path.segment (v j) (C.portPoint p.val))) : k=j := by
by_contra hkj
have hd := C.disks_disjoint _ _ (C.vertexMark_injective.ne (Ne.symm hkj))
exact Set.disjoint_left.mp hd (C.vertex_spoke_range_in_disk v hv j p hk) (hv k).le
end DrawingPassages
namespace PolygonalExterior
variable {m n : ℕ} {D : FiniteIntersectionDrawing m n} {C : DrawingPassages D}
theorem middlePath_avoids_vertices (E : PolygonalExterior C) (v : BipartiteVertex m n → Plane)
(hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) (e : BipartiteEdge m n)
(j : BipartiteVertex m n) : v j ∉ Set.range (E.middlePath e) := by
intro hx
rcases (E.middlePath_range_iff e (v j)).mp hx with ⟨i,hi,⟨t,ht⟩⟩ | ⟨p,hp,hx⟩
· exact E.avoids_new_vertices v hv i t j ht
· exact Set.disjoint_left.mp (C.crossingChord_avoids_vertex_disks p j) hx (hv j).le
theorem assembledEdge_vertex_eq (E : PolygonalExterior C) (v : BipartiteVertex m n → Plane)
(hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) (e : BipartiteEdge m n)
(j : BipartiteVertex m n) (hj : v j ∈ Set.range (E.assembledEdge v e)) :
j=.inl e.1 ∨ j=.inr e.2 := by
rw [assembledEdge,Path.trans_range,Path.trans_range] at hj
rcases hj with (hj | hj) | hj
· exact Or.inl (C.vertex_spoke_vertex_eq v hv _ j (C.startPortAt e) hj)
· exact False.elim (E.middlePath_avoids_vertices v hv e j hj)
· rw [endSpoke_range] at hj
exact Or.inr (C.vertex_spoke_vertex_eq v hv _ j (C.endPortAt e) hj)
theorem assembledEdge_interior_avoids_vertices (E : PolygonalExterior C)
(v : BipartiteVertex m n → Plane) (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2)
(e : BipartiteEdge m n) (t : unitInterval) (ht : t ∈ Set.Ioo (0 : unitInterval) 1)
(j : BipartiteVertex m n) : E.assembledEdge v e t ≠ v j := by
intro he
rcases E.assembledEdge_vertex_eq v hv e j ⟨t,he⟩ with rfl | rfl
· have hh := E.assembledEdge_injective v hv e (he.trans (E.assembledEdge v e).source.symm)
exact ht.1.ne hh.symm
· have hh := E.assembledEdge_injective v hv e (he.trans (E.assembledEdge v e).target.symm)
exact ht.2.ne hh
def assembledDrawing (E : PolygonalExterior C) (v : BipartiteVertex m n → Plane)
(hvinj : Function.Injective v) (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) :
ContinuousDrawing m n where
vertex := v
vertex_injective := hvinj
edge := E.assembledEdge v
edge_injective := E.assembledEdge_injective v hv
interior_avoids_vertices := E.assembledEdge_interior_avoids_vertices v hv
theorem exists_assembledDrawing (D : FiniteIntersectionDrawing m n) :
∃ (C : DrawingPassages D) (E : PolygonalExterior C)
(v : BipartiteVertex m n → Plane) (hvinj : Function.Injective v)
(hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2),
∀ e, Function.Injective ((E.assembledDrawing v hvinj hv).edge e) := by
obtain ⟨C⟩ := nonempty_drawingPassages D
obtain ⟨E⟩ := nonempty_polygonalExterior C
obtain ⟨v,hvinj,hv,hne,hdet⟩ := C.exists_drawing_vertex_placement
exact ⟨C,E,v,hvinj,hv,E.assembledEdge_injective v hv⟩
end PolygonalExterior
end Zarankiewicz
end
end OAI
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