File size: 3,149 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 | import OAI.Combinatorics.Crossing.MiddleSpokeMeeting
namespace OAI
noncomputable section
namespace Zarankiewicz.PolygonalExterior
variable {m n : ℕ} {D : FiniteIntersectionDrawing m n} {C : DrawingPassages D}
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 _ _
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)
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)
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
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]
end Zarankiewicz.PolygonalExterior
end
end OAI
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