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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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.VaryingCellClassification
import OAI.GroupTheory.SimpleAmenable.CentralCovers.LocalArrangementAssignments
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.InwardPlanarGerms
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.CompactFineGrid
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section ConcurrentInwardModel
namespace ConcurrentGeometry
variable {a : ℕ} {r : CutRing} (C : ConcurrentGeometry a r)
noncomputable def inwardGate (t : VertexType (commonVertexDenominator a))
(u : CutRing × CutRing) (z : ℝ × ℝ) (k : Fin 2) : polygonAlgebra a :=
if realCoordinate z k=0 then C.positiveDecision t u (axisDirection k)
else if realCoordinate z k=1 then (C.positiveDecision t u (axisDirection k))ᶜ
else wholePolygon a
noncomputable def inwardMargin (t : VertexType (commonVertexDenominator a))
(u : CutRing × CutRing) (z : ℝ × ℝ) : polygonAlgebra a :=
C.marginPolygon t u ⊓ C.inwardGate t u z 0 ⊓ C.inwardGate t u z 1
noncomputable def localDecision (t : VertexType (commonVertexDenominator a))
(u : CutRing × CutRing) (z : ℝ × ℝ) (j : Fin 4) (c : CutRing) : polygonAlgebra a :=
if cutForm a j z=ordinary c then (C.positiveDecision t u j)ᶜ
else if cutForm a j z<ordinary c then ⊤ else ⊥
theorem localDecision_assignment_realized {ι : Type*} [Finite ι]
(hr : 0<ordinary r ∧ ordinary r<1/2)
(t : VertexType (commonVertexDenominator a)) (u : CutRing × CutRing)
(j : ι → Fin 4) (c : ι → CutRing) (z : ℝ × ℝ)
(hz₁ : z.1 ∈ Set.Icc (0:ℝ) 1) (hz₂ : z.2 ∈ Set.Icc (0:ℝ) 1)
(hline : ∀ i, cutForm a (j i) z=ordinary (c i) →
integralCutForm a (j i) (u+C.anchors t (j i))=c i)
(hboundary : ∀ k, realCoordinate z k=0 ∨ realCoordinate z k=1 →
ordinary (pointCoordinate (u+C.anchors t (axisDirection k)) k)=realCoordinate z k)
(hnear : ∀ i, cutForm a (j i) z=ordinary (c i) → ∀ y : ℝ × ℝ,
(∀ k, ordinary ((C.margins t).lower k+pointCoordinate u k)≤realCoordinate y k ∧
realCoordinate y k≤ordinary ((C.margins t).upper k+pointCoordinate u k)) →
∀ k, |realCoordinate y k-ordinary (pointCoordinate (u+C.anchors t (j i)) k)|<3*C.epsilon)
(hnearB : ∀ d, realCoordinate z d=0 ∨ realCoordinate z d=1 → ∀ y : ℝ × ℝ,
(∀ k, ordinary ((C.margins t).lower k+pointCoordinate u k)≤realCoordinate y k ∧
realCoordinate y k≤ordinary ((C.margins t).upper k+pointCoordinate u k)) →
∀ k, |realCoordinate y k-ordinary (pointCoordinate (u+C.anchors t (axisDirection d)) k)|<3*C.epsilon)
(p : GenericSquare a) (hp : p ∈ (C.inwardMargin t u z).val)
(N : Set (ℝ × ℝ)) (hN : IsOpen N) (hz : z ∈ N) :
∃ q : GenericSquare a, q.val ∈ N ∧
polygonAssignment (fun i => cutPolygon a (j i) (c i)) q=
polygonAssignment (fun i => C.localDecision t u z (j i) (c i)) p := by
classical
have hpM : p ∈ (C.marginPolygon t u).val := hp.1.1
have hpG (k : Fin 2) : p ∈ (C.inwardGate t u z k).val := by
fin_cases k
· exact hp.1.2
· exact hp.2
obtain ⟨y,hy,ha,hcell⟩ := C.marginPolygon_lift hr.2 t u p hpM
have hzero : ∀ k, realCoordinate z k=0 → 0<realCoordinate y k := by
intro k hk
have h := hpG k
simp only [inwardGate,hk,ite_eq_left] at h
have hd := (C.positiveDecision_planar hr t u (axisDirection k) (hnearB k (Or.inl hk)) p y hy
(fun k => ⟨(hcell k).1.le,(hcell k).2.le⟩)).mp h
rw [integralCutForm_axis,hboundary k (Or.inl hk),hk] at hd
have haxis : cutForm a (axisDirection k) y=realCoordinate y k := by fin_cases k <;> rfl
rw [haxis] at hd
apply lt_of_le_of_ne hd
intro he
have hn := ha (axisDirection k) 0
simp only [map_zero,haxis] at hn
exact hn he.symm
have hone : ∀ k, realCoordinate z k=1 → realCoordinate y k<1 := by
intro k hk
have h := hpG k
have hn : realCoordinate z k≠0 := by rw [hk]; norm_num
simp only [inwardGate,ite_eq_right hn,ite_eq_left hk] at h
have hd := (not_congr (C.positiveDecision_planar hr t u (axisDirection k) (hnearB k (Or.inr hk)) p y hy
(fun k => ⟨(hcell k).1.le,(hcell k).2.le⟩))).mp h
rw [integralCutForm_axis,hboundary k (Or.inr hk),hk] at hd
have haxis : cutForm a (axisDirection k) y=realCoordinate y k := by fin_cases k <;> rfl
rw [haxis] at hd
exact lt_of_not_ge hd
let σ := polygonAssignment (fun i => C.localDecision t u z (j i) (c i)) p
have hactive : ∀ i, cutForm a (j i) z=ordinary (c i) →
if σ i then cutForm a (j i) y<ordinary (c i)
else ordinary (c i)<cutForm a (j i) y := by
intro i hi
have hd := C.positiveDecision_planar hr t u (j i) (hnear i hi) p y hy
(fun k => ⟨(hcell k).1.le,(hcell k).2.le⟩)
rw [hline i hi] at hd
have hσ : σ i=decide (¬ordinary (c i)≤cutForm a (j i) y) := by
apply decide_eq_decide.mpr
change (p ∈ (C.localDecision t u z (j i) (c i)).val) ↔ _
simp only [localDecision,ite_eq_left hi]
exact not_congr hd
rw [hσ]
split_ifs with h
· exact lt_of_not_ge (of_decide_eq_true h)
· have hh := of_decide_eq_false (Bool.eq_false_iff.mpr h)
exact lt_of_le_of_ne (not_not.mp hh) (ha (j i) (c i)).symm
have hinactive : ∀ i, cutForm a (j i) z≠ordinary (c i) →
if σ i then cutForm a (j i) z<ordinary (c i)
else ordinary (c i)<cutForm a (j i) z := by
intro i hi
by_cases hlt : cutForm a (j i) z<ordinary (c i)
· simp [σ,polygonAssignment,localDecision,hi,hlt]
· simp only [σ,polygonAssignment,localDecision,ite_eq_right hi,ite_eq_right hlt]
simpa using lt_of_le_of_ne (le_of_not_gt hlt) hi.symm
obtain ⟨q,hq,hassign⟩ := finite_arrangement_germ_inward a j (fun i => ordinary (c i))
z y σ hz₁ hz₂ hzero hone hactive hinactive N hN hz
refine ⟨q,hq,?_⟩
funext i
change decide (cutForm a (j i) q.val<ordinary (c i))=σ i
have hi := hassign i
cases he : σ i
· simp only [he,Bool.false_eq_true,↓reduceIte] at hi
exact decide_eq_false (not_lt_of_gt hi)
· simp only [he,↓reduceIte] at hi
exact decide_eq_true hi
theorem exists_inward_assignment_model {ι : Type*} [Finite ι]
(ha : 0<a) (hr : 0<ordinary r ∧ ordinary r<1/2)
(j : ι → Fin 4) (c : ι → CutRing) (z : ℝ × ℝ)
(hz₁ : z.1 ∈ Set.Icc (0:ℝ) 1) (hz₂ : z.2 ∈ Set.Icc (0:ℝ) 1) :
∃ t : VertexType (commonVertexDenominator a), ∃ u : CutRing × CutRing,
(∀ y : ℝ × ℝ, dist y z<C.epsilon/2 → ∀ k,
ordinary ((C.margins t).lower k+pointCoordinate u k)<realCoordinate y k ∧
realCoordinate y k<ordinary ((C.margins t).upper k+pointCoordinate u k)) ∧
∀ p : GenericSquare a, p ∈ (C.inwardMargin t u z).val →
∀ N : Set (ℝ × ℝ), IsOpen N → z ∈ N →
∃ q : GenericSquare a, q.val ∈ N ∧
polygonAssignment (fun i => cutPolygon a (j i) (c i)) q=
polygonAssignment (fun i => C.localDecision t u z (j i) (c i)) p := by
classical
let J : Sum ι (Fin 2 × Bool) → Fin 4 := Sum.elim j (fun d => axisDirection d.1)
let K : Sum ι (Fin 2 × Bool) → CutRing := Sum.elim c (fun d => if d.2 then 1 else 0)
obtain ⟨t,u,hcell,hactive⟩ := C.simultaneous_active_neighborhood ha J K z
have hb (k : Fin 2) (hk : realCoordinate z k=0 ∨ realCoordinate z k=1) :
∃ b : Bool, cutForm a (J (.inr (k,b))) z=ordinary (K (.inr (k,b))) := by
have haxis : cutForm a (axisDirection k) z=realCoordinate z k := by fin_cases k <;> rfl
rcases hk with hk | hk
· exact ⟨false,by simpa only [J,K,Sum.elim_inr,Bool.false_eq_true,↓reduceIte,map_zero,haxis] using hk⟩
· exact ⟨true,by simpa only [J,K,Sum.elim_inr,↓reduceIte,map_one,haxis] using hk⟩
refine ⟨t,u,hcell,fun p hp N hN hz =>
C.localDecision_assignment_realized hr t u j c z hz₁ hz₂
(fun i hi => (hactive (.inl i) hi).1) ?_
(fun i hi => (hactive (.inl i) hi).2) ?_ p hp N hN hz⟩
· intro k hk
obtain ⟨b,hb⟩ := hb k hk
have he := congrArg ordinary (hactive (.inr (k,b)) hb).1
have haxis : cutForm a (axisDirection k) z=realCoordinate z k := by fin_cases k <;> rfl
change ordinary (integralCutForm a (axisDirection k) (u+C.anchors t (axisDirection k)))=_ at he
rw [integralCutForm_axis] at he
exact he.trans (by simpa only [J,Sum.elim_inr,haxis] using hb.symm)
· intro k hk
obtain ⟨b,hb⟩ := hb k hk
exact (hactive (.inr (k,b)) hb).2
end ConcurrentGeometry
end ConcurrentInwardModel
end SimpleAmenable
end
end
end OAI
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