| import OAI.Combinatorics.Crossing.MiddleSpokeMeeting |
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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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|
| def startSpoke (C : DrawingPassages D) (v : BipartiteVertex m n → Plane) (e : BipartiteEdge m n) : |
| Path (v (.inl e.1)) (C.arcStart (C.firstArc e)) := Path.segment _ _ |
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| def endSpoke (C : DrawingPassages D) (v : BipartiteVertex m n → Plane) (e : BipartiteEdge m n) : |
| Path (C.arcEnd (C.lastArc e)) (v (.inr e.2)) := (Path.segment _ _).symm |
|
|
| def assembledEdge (E : PolygonalExterior C) (v : BipartiteVertex m n → Plane) |
| (e : BipartiteEdge m n) : Path (v (.inl e.1)) (v (.inr e.2)) := |
| ((startSpoke C v e).trans (E.anchoredMiddle e)).trans (endSpoke C v e) |
|
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| theorem startSpoke_injective (v : BipartiteVertex m n → Plane) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) (e : BipartiteEdge m n) : |
| Function.Injective (startSpoke C v e) := C.vertex_spoke_injective v hv _ (C.startPortAt e) |
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| theorem endSpoke_range (v : BipartiteVertex m n → Plane) (e : BipartiteEdge m n) : |
| Set.range (endSpoke C v e)=Set.range (Path.segment (v (.inr e.2)) (C.portPoint (C.endPortAt e).val)) := by |
| exact Path.symm_range _ |
|
|
| theorem endSpoke_injective (v : BipartiteVertex m n → Plane) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) (e : BipartiteEdge m n) : |
| Function.Injective (endSpoke C v e) := by |
| intro s t he |
| have hh := C.vertex_spoke_injective v hv _ (C.endPortAt e) he |
| have hval := congrArg Subtype.val hh |
| apply Subtype.ext |
| change 1-s.val=1-t.val at hval |
| linarith |
|
|
| theorem assembledEdge_injective (E : PolygonalExterior C) (v : BipartiteVertex m n → Plane) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) (e : BipartiteEdge m n) : |
| Function.Injective (E.assembledEdge v e) := by |
| have hfirst : Function.Injective ((startSpoke C v e).trans (E.anchoredMiddle e)) := by |
| apply path_trans_injective _ _ (startSpoke_injective v hv e) (E.anchoredMiddle_injective e) |
| intro x hx hy |
| exact E.middle_spoke_meeting v hv e _ (C.startPortAt e) x hy hx |
| apply path_trans_injective _ _ hfirst (endSpoke_injective v hv e) |
| intro x hx hy |
| rw [Path.trans_range] at hx |
| rw [endSpoke_range] at hy |
| rcases hx with hx | hx |
| · have hd := C.vertex_spokes_disjoint_of_vertices_ne v hv (.inl e.1) (.inr e.2) |
| (by simp) (C.startPortAt e) (C.endPortAt e) |
| exact False.elim (Set.disjoint_left.mp hd hx hy) |
| · exact E.middle_spoke_meeting v hv e _ (C.endPortAt e) x hx hy |
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|
| theorem assembledEdge_range_iff (E : PolygonalExterior C) (v : BipartiteVertex m n → Plane) |
| (e : BipartiteEdge m n) (x : Plane) : |
| x ∈ Set.range (E.assembledEdge v e) ↔ |
| (x ∈ Set.range (startSpoke C v e) ∨ |
| (∃ i : C.ArcIndex, i.1=e ∧ x ∈ Set.range (E.chain i).path) ∨ |
| (∃ p : C.InnerIndex, p.1=e ∧ x ∈ Set.range (C.crossingChord p))) ∨ |
| x ∈ Set.range (endSpoke C v e) := by |
| simp only [assembledEdge,Path.trans_range,Set.mem_union,anchoredMiddle_range,E.middlePath_range_iff] |
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| end Zarankiewicz.PolygonalExterior |
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| end |
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| end OAI |
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