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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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.FiniteNestedLaw
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section OldRectangles
structure LabelledRectangle (n : ℕ) where
length : Fin 2 → ℕ
length_le : ∀ j, length j ≤ n
start : Fin 2 → ℤ
lower : Fin 2 → CutRing
upper : Fin 2 → CutRing
ordered : ∀ j, ordinary (lower j) ≤ ordinary (upper j)
lower_label : ∀ j, start j ≤ endpointLabel (lower j) ∧ endpointLabel (lower j) < start j+length j
upper_label : ∀ j, start j ≤ endpointLabel (upper j) ∧ endpointLabel (upper j) < start j+length j
namespace LabelledRectangle
variable {n : ℕ} (R : LabelledRectangle n)
noncomputable def polygon (a : ℕ) : polygonAlgebra a := coordinateRectangle a R.lower R.upper
theorem resolved (a : ℕ) (r : CutRing) :
ResolvedBy (fun i => (InitialCoverSystem.primitiveTests (a := a) (r := r)
(coordinateWindowPrimitives n R.start) i).val) (R.polygon a).val := by
have hl (j : Fin 2) : R.start j ≤ endpointLabel (R.lower j) ∧ endpointLabel (R.lower j) < R.start j+n := by
have hh := R.lower_label j; have hh' := R.length_le j; omega
have hv (j : Fin 2) : R.start j ≤ endpointLabel (R.upper j) ∧ endpointLabel (R.upper j) < R.start j+n := by
have hh := R.upper_label j; have hh' := R.length_le j; omega
intro x y he
exact and_congr (coordinateLabelWindow_resolved n R.start 0 _ _ (hl 0) (hv 0) x y he)
(coordinateLabelWindow_resolved n R.start 1 _ _ (hl 1) (hv 1) x y he)
structure ChartFit (s : CutRing) (z : CutRing × CutRing) where
window : Fin 2 → ℤ
source_start : ∀ j, window j ≤ R.start j
source_end : ∀ j, R.start j+R.length j ≤ window j+n
chart_start : ∀ j, window j ≤ pointLabel z j+symmetricWindowStart s j
chart_end : ∀ j, pointLabel z j+symmetricWindowStart s j+symmetricWindowLength s ≤ window j+n
theorem chartFit_of_near_labels (s : CutRing) (z : CutRing × CutRing)
(v : Fin 2 → ℤ) (d : Fin 2 → ℕ)
(hv : ∀ j, R.start j ≤ v j ∧ v j < R.start j+R.length j)
(hz : ∀ j, (pointLabel z j-v j).natAbs ≤ d j)
(hsize : ∀ j, R.length j+2*d j+2*(endpointLabel s).natAbs+2 ≤ n) :
Nonempty (R.ChartFit s z) := by
have hh (j : Fin 2) := source_chart_containing_window n (R.length j) (R.start j)
(pointLabel z j) (v j) (endpointLabel s).natAbs (d j) (hv j) (hz j) (hsize j)
refine ⟨⟨fun j => R.start j ⊓ (pointLabel z j-((endpointLabel s).natAbs:ℤ)),
fun j => (hh j).1,fun j => (hh j).2.1,?_,?_⟩⟩
· intro j
simpa only [symmetricWindowStart,sub_eq_add_neg] using (hh j).2.2.1
· intro j
simpa only [symmetricWindowStart,symmetricWindowLength,sub_eq_add_neg] using (hh j).2.2.2
end LabelledRectangle
namespace InitialCoverSystem
variable {a m M : ℕ} {r : CutRing} {hm : 2 ≤ m}
(B : InitialCoverSystem a r m hm M)
[Group.IsPerfect (alternatingGroup (Fin (m+1)))]
noncomputable def rectangleCopy (hlarge : 15 < m+1) {n : ℕ}
(g : B.CoordinateWindowLaw n) (R : LabelledRectangle n) :
TrackStar (Fin (m+1)) →* BoundedRelationCover M (alternatingGenerator a r m hm) :=
B.windowSector hlarge n g R.start (R.polygon a)
theorem rectangleCopy_supported (hlarge : 15 < m+1) {n : ℕ}
(g : B.CoordinateWindowLaw n) (R : LabelledRectangle n) :
B.AlignedSmallSupported (B.rectangleCopy hlarge g R) :=
B.fullGeometricSector_supported hlarge _ _ _ (R.resolved a r)
theorem rectangleCopy_axes_control (hlarge : 15 < m+1) {n : ℕ}
(g : B.CoordinateWindowLaw n) (R : LabelledRectangle n) (j : Fin 2) :
SmallControlled B.c (B.rectangleCopy hlarge g R)
(B.axisSector hlarge n g j (R.length j) (R.length_le j) (R.start j)
(coordinateInterval a j (R.lower j) (R.upper j))) := by
rw [B.axisSector_in_window hlarge n g j _ (R.length_le j) _ R.start le_rfl
(by have hh := R.length_le j; omega) _
(axisInterval_resolved j _ _ _ _ (R.lower_label j) (R.upper_label j))]
apply B.fullGeometricSector_small_control
fin_cases j
· exact inf_le_left
· exact inf_le_right
theorem rectangleCopy_quadrant_control (hlarge : 20 ≤ m+1)
(hr : 0 < ordinary r ∧ ordinary r < 1/2)
(s u : CutRing) (hs : 0 < ordinary s ∧ ordinary s < ordinary r)
(h : B.TangentChartLaws (symmetricWindowLength s) (symmetricWindowStart s) u)
{n : ℕ} (g : B.CoordinateWindowLaw n) (R : LabelledRectangle n)
(d : Fin 2) (z : CutRing × CutRing) (positive : Bool) (fit : R.ChartFit s z)
(hlo : ∀ j, ordinary (signQuadrantLower s d positive j)+ordinary (pointCoordinate z j) ≤ ordinary (R.lower j))
(hup : ∀ j, ordinary (R.upper j) ≤ ordinary (signQuadrantUpper s d positive j)+ordinary (pointCoordinate z j)) :
SmallControlled B.c (B.rectangleCopy (by omega) g R)
(B.tangentSign (by omega) (symmetricWindowLength s) (symmetricWindowStart s) u h d z positive) :=
B.tangentSign_quadrant_control hlarge hr s u hs h n g R.length R.length_le R.start
fit.window fit.source_start fit.source_end d z positive fit.chart_start fit.chart_end
R.lower R.upper R.ordered R.lower_label R.upper_label hlo hup _
(B.rectangleCopy_supported (by omega) g R) (B.rectangleCopy_axes_control (by omega) g R)
end InitialCoverSystem
end OldRectangles
end SimpleAmenable
end
end
end OAI
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