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0dd3af4 | 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 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.Amenability.BarrierSignalSystem
import OAI.GroupTheory.SimpleAmenable.Amenability.ChartLineTransport
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.ChartBoundaryEndpoints
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section ComponentChartContainment
open Classical Set
theorem barrier_component_cut_ordinary {a : ℕ} {B : Set (ℝ×ℝ)} {j : Fin 4} {c : CutRing}
(hline : ∀x ∈ squareInterior,cutForm a j x=ordinary c → x ∈ B)
{p x : ℝ×ℝ} (hp : p ∈ barrierRegion B)
(hx : x ∈ connectedComponentIn (barrierRegion B) p) :
cutForm a j x ≠ ordinary c ∧ (cutForm a j x < ordinary c ↔ cutForm a j p < ordinary c) := by
have hn (y : ℝ×ℝ) (hy : y ∈ connectedComponentIn (barrierRegion B) p) : cutForm a j y ≠ ordinary c := by
intro he
have hr := connectedComponentIn_subset (barrierRegion B) p hy
exact hr.2 (hline y hr.1 he)
refine ⟨hn x hx,?_⟩
have hh := continuous_nonzero_sign isPreconnected_connectedComponentIn
((cutForm_continuous a j).sub continuous_const).continuousOn
(fun y hy => sub_ne_zero.mpr (hn y hy)) hx (mem_connectedComponentIn hp)
simpa only [Pi.sub_apply,sub_lt_zero] using hh
theorem barrier_component_openChart {a : ℕ} {B : Set (ℝ×ℝ)}
(T : Finset PlaneCut) (p : GenericSquare a)
(hp : p.val ∈ barrierRegion B)
(hline : ∀l ∈ T,∀x ∈ squareInterior,cutForm a l.1 x=ordinary l.2 → x ∈ B) :
connectedComponentIn (barrierRegion B) p.val ⊆ openSquareChart T p := by
intro x hx
refine ⟨(connectedComponentIn_subset (barrierRegion B) p.val hx).1,?_⟩
intro l
obtain ⟨hne,hs⟩ := barrier_component_cut_ordinary (hline l.val l.property) hp hx
by_cases hh : cutForm a l.val.1 p.val < ordinary l.val.2
· simpa [squareSign,hh] using hs.mpr hh
· simpa [squareSign,hh] using lt_of_le_of_ne (le_of_not_gt (fun h => hh (hs.mp h))) hne.symm
theorem barrier_component_translatedChart {a : ℕ} {B : Set (ℝ×ℝ)}
(T : Finset PlaneCut) (p q : GenericSquare a) (u : CutRing×CutRing)
(hqp : q.val=p.val+(ordinary u.1,ordinary u.2))
(hq : q.val ∈ barrierRegion B)
(hline : ∀l ∈ T ∪ squareBoundaryCuts,∀x ∈ squareInterior,
cutForm a l.1 x=ordinary (l.2+integralCutForm a l.1 u) → x ∈ B) :
∀x ∈ connectedComponentIn (barrierRegion B) q.val,
x-(ordinary u.1,ordinary u.2) ∈ openSquareChart T p := by
intro x hx
have hsign (l : PlaneCut) (hl : l ∈ T ∪ squareBoundaryCuts) :
cutForm a l.1 (x-(ordinary u.1,ordinary u.2)) ≠ ordinary l.2 ∧
(cutForm a l.1 (x-(ordinary u.1,ordinary u.2)) < ordinary l.2 ↔
cutForm a l.1 p.val < ordinary l.2) := by
obtain ⟨hne,hs⟩ := barrier_component_cut_ordinary (hline l hl) hq hx
rw [hqp,cutForm_add,cutForm_ordinary,map_add] at hs
rw [map_add] at hne
rw [cutForm_sub,cutForm_ordinary]
refine ⟨fun h => hne (by linarith),?_⟩
constructor
· intro h
have hh := hs.mp (by linarith : cutForm a l.1 x < ordinary l.2+ordinary (integralCutForm a l.1 u))
linarith
· intro h
have hh := hs.mpr (by linarith : cutForm a l.1 p.val+ordinary (integralCutForm a l.1 u) < ordinary l.2+ordinary (integralCutForm a l.1 u))
linarith
have hp := generic_mem_squareInterior p
have hx₀ := hsign (0,0) (Finset.mem_union_right _ (by simp [squareBoundaryCuts]))
have hx₁ := hsign (0,1) (Finset.mem_union_right _ (by simp [squareBoundaryCuts]))
have hy₀ := hsign (1,0) (Finset.mem_union_right _ (by simp [squareBoundaryCuts]))
have hy₁ := hsign (1,1) (Finset.mem_union_right _ (by simp [squareBoundaryCuts]))
simp only [cutForm,Matrix.cons_val_zero,Matrix.cons_val_one,
map_zero,map_one,Prod.fst_sub,Prod.snd_sub] at hx₀ hx₁ hy₀ hy₁
refine ⟨⟨⟨?_,?_⟩,⟨?_,?_⟩⟩,?_⟩
· exact lt_of_le_of_ne (le_of_not_gt (fun h => not_lt_of_gt hp.1.1 (hx₀.2.mp h))) hx₀.1.symm
· exact hx₁.2.mpr hp.1.2
· exact lt_of_le_of_ne (le_of_not_gt (fun h => not_lt_of_gt hp.2.1 (hy₀.2.mp h))) hy₀.1.symm
· exact hy₁.2.mpr hp.2.2
· intro l
obtain ⟨hne,hs⟩ := hsign l.val (Finset.mem_union_left _ l.property)
by_cases hh : cutForm a l.val.1 p.val < ordinary l.val.2
· simpa [squareSign,hh] using hs.mpr hh
· simpa [squareSign,hh] using lt_of_le_of_ne (le_of_not_gt (fun h => hh (hs.mp h))) hne.symm
end ComponentChartContainment
section ComponentTransport
open Classical Set
namespace BarrierSignalSystem
variable {a : ℕ} {ha : 0 < a} {Q B : ℝ} (S : BarrierSignalSystem a ha Q B)
theorem component_transport {m : ℕ} (g : polygonFullGroup a m) (i k : Fin m)
(T : Finset PlaneCut)
(hT : ∀p q,(∀l ∈ T,squareSign l p=squareSign l q) →
fullGroupAffineData g i p=fullGroupAffineData g i q) (p q : GenericSquare a)
(u : CutRing×CutRing) (hdata : fullGroupAffineData g i p=(k,u))
(hqp : q.val=p.val+(ordinary u.1,ordinary u.2))
(hQ : 0 ≤ Q) (huQ : ∀j,|conjugate (integralCutForm a j u)| ≤ Q)
(hB : ∀l ∈ T ∪ squareBoundaryCuts,|conjugate l.2| ≤ B ∧
|conjugate (l.2+integralCutForm a l.1 u)| ≤ B)
{N : ℝ} (hN : 1 ≤ N) (hQN : Q < N) (hBN : B ≤ N)
(b₁ b₂ b₁' b₂' : ∀j,FlagSite a m (commonVertexDenominator a) (barrierFlagDirection ha j) → Bool)
(hs : ∀j,signalSuccess (flagMeanSignal S.first.signal N) (b₁ j))
(hs' : ∀j,signalSuccess (flagMeanSignal S.first.signal N) (b₁' j))
(ht : ∀j,signalSuccess (flagMeanSignal (S.second j).signal N) (b₂ j))
(ht' : ∀j,signalSuccess (flagMeanSignal (S.second j).signal N) (b₂' j))
(hb₁ : ∀j z,b₁' j (flagSiteAction (commonVertexDenominator_pos ha) g z)=b₁ j z)
(hb₂ : ∀j z,b₂' j (flagSiteAction (commonVertexDenominator_pos ha) g z)=b₂ j z) :
(fun x => x+(ordinary u.1,ordinary u.2)) '' connectedComponentIn
(barrierRegion (sampledBarriers ha i N
(fun j => S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j (b₁ j)) b₂)) p.val =
connectedComponentIn (barrierRegion (sampledBarriers ha k N
(fun j => S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j (b₁' j)) b₂')) q.val := by
let v := (ordinary u.1,ordinary u.2)
let A₀ := sampledBarriers ha i N
(fun j => S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j (b₁ j)) b₂
let A₁ := sampledBarriers ha k N
(fun j => S.marks (N:=N) (lt_of_lt_of_le zero_lt_one hN) j (b₁' j)) b₂'
have hp : p.val ∈ barrierRegion A₀ := generic_mem_barrierRegion (sampledBarriers_supported ha i N _ b₂) p
have hq : q.val ∈ barrierRegion A₁ := generic_mem_barrierRegion (sampledBarriers_supported ha k N _ b₂') q
have hsource : connectedComponentIn (barrierRegion A₀) p.val ⊆ openSquareChart T p := by
apply barrier_component_openChart T p hp
intro l hl x hx hxc
have hband := ((hB l (Finset.mem_union_left _ hl)).1).trans hBN
exact S.central_line_barrier i hN b₁ b₂ hs ht
⟨by linarith, x,hx,hxc⟩ hband x hx hxc
have htarget : ∀x ∈ connectedComponentIn (barrierRegion A₁) q.val,x-v ∈ openSquareChart T p := by
apply barrier_component_translatedChart T p q u hqp hq
intro l hl x hx hxc
have hband := ((hB l hl).2).trans hBN
exact S.central_line_barrier k hN b₁' b₂' hs' ht'
⟨by linarith,x,hx,hxc⟩ hband x hx hxc
have hbar (x : ℝ×ℝ) (hx : x ∈ openSquareChart T p) : x ∈ A₀ ↔ x+v ∈ A₁ :=
S.chart_barrier_transport g i k T hT p u hdata hQ huQ hB hN hQN
b₁ b₂ b₁' b₂' hs hs' ht ht' hb₁ hb₂ hx
have hforward : (fun x => x+v) '' connectedComponentIn (barrierRegion A₀) p.val ⊆
connectedComponentIn (barrierRegion A₁) q.val := by
apply (isPreconnected_connectedComponentIn.image _ (continuous_id.add continuous_const).continuousOn).subset_connectedComponentIn
· exact ⟨p.val,mem_connectedComponentIn hp,hqp.symm⟩
· rintro x ⟨y,hy,rfl⟩
have hyC := hsource hy
have hyreg := connectedComponentIn_subset (barrierRegion A₀) p.val hy
refine ⟨?_,fun h => hyreg.2 ((hbar y hyC).mpr h)⟩
have himage := fullGroupAffineChart_square g i T hT p
rw [hdata] at himage
exact himage ⟨y,hyC,rfl⟩
have hbackward : (fun x => x-v) '' connectedComponentIn (barrierRegion A₁) q.val ⊆
connectedComponentIn (barrierRegion A₀) p.val := by
apply (isPreconnected_connectedComponentIn.image _ (continuous_id.sub continuous_const).continuousOn).subset_connectedComponentIn
· refine ⟨q.val,mem_connectedComponentIn hq,?_⟩
change q.val-v=p.val
rw [hqp,add_sub_cancel_right]
· rintro x ⟨y,hy,rfl⟩
have hyC := htarget y hy
have hyreg := connectedComponentIn_subset (barrierRegion A₁) q.val hy
refine ⟨hyC.1,?_⟩
intro h
have hh := (hbar (y-v) hyC).mp h
rw [sub_add_cancel] at hh
exact hyreg.2 hh
apply Subset.antisymm hforward
intro y hy
exact ⟨y-v,hbackward ⟨y,hy,rfl⟩,sub_add_cancel y v⟩
end BarrierSignalSystem
end ComponentTransport
end SimpleAmenable
end
end
end OAI
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