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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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.CentralCovers.RectangularAtlas
import OAI.GroupTheory.SimpleAmenable.CentralCovers.GlobalSignControl
namespace OAI
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
theorem signQuadrant_bounds (s : CutRing) (hs : 0 ≤ ordinary s)
(d : Fin 2) (positive : Bool) (j : Fin 2) :
-ordinary s ≤ ordinary (signQuadrantLower s d positive j) ∧
ordinary (signQuadrantUpper s d positive j) ≤ ordinary s := by
cases positive <;> by_cases h : j=d <;>
simp [signQuadrantLower,signQuadrantUpper,h] <;> linarith
namespace InitialCoverSystem
namespace RectangularAtlas
variable {a m M : ℕ} {r : CutRing} {hm : 2 ≤ m}
{B : InitialCoverSystem a r m hm M}
[Group.IsPerfect (alternatingGroup (Fin (m+1)))] (A : B.RectangularAtlas)
theorem inactive_quadrant_action (hlarge : 20 ≤ m+1)
(hr : 0 < ordinary r ∧ ordinary r < 1/2)
(d : Fin 2) (z z' : CutRing × CutRing) (positive : Bool)
(hline : integralCutForm a (slopeDirection d) z=integralCutForm a (slopeDirection d) z')
(L V : Fin 2 → CutRing) (hLV : ∀ j, ordinary (L j) ≤ ordinary (V j))
(hz : ∀ j, -ordinary r ≤ ordinary (L j)-ordinary (pointCoordinate z j) ∧
ordinary (V j)-ordinary (pointCoordinate z j) ≤ ordinary r)
(hlo : ∀ j, ordinary (pointCoordinate z' j)+ordinary (signQuadrantLower A.radius d positive j) ≤ ordinary (L j))
(hup : ∀ j, ordinary (V j) ≤ ordinary (pointCoordinate z' j)+ordinary (signQuadrantUpper A.radius d positive j))
(n : ℕ) (q : Fin 2 → ℤ)
(hW : ResolvedBy (fun i => (primitiveTests (a := a) (r := r)
(coordinateWindowPrimitives n q) i).val) (coordinateRectangle a L V).val)
(f : TrackStar (Fin (m+1)) →* BoundedRelationCover M (alternatingGenerator a r m hm))
(hf : B.AlignedSmallSupported f)
(hc : SmallControlled B.c f (B.windowSector (by omega) n (A.rectangles n) q (coordinateRectangle a L V)))
(I : ControlAlphabet (Fin (m+1))) (s : UniversalExtension (alternatingGroup I.val))
(t : BoundedRelationCover M (alternatingGenerator a r m hm)) (ht : t ∈ f.range) :
A.slope (by omega) d z (universalMap (subtypeAlternatingHom I.val) s)*t*
(A.slope (by omega) d z (universalMap (subtypeAlternatingHom I.val) s))⁻¹ =
(if positive then B.c (subtypeAlternatingHom I.val (universalProjection (alternatingGroup I.val) s)) else 1)*t*
(if positive then B.c (subtypeAlternatingHom I.val (universalProjection (alternatingGroup I.val) s)) else 1)⁻¹ := by
have hbound (j : Fin 2) := signQuadrant_bounds A.radius A.radius_pos.le d positive j
have hdiff (j : Fin 2) :
|ordinary (L j)-ordinary (pointCoordinate z' j)| ≤ ordinary A.radius ∧
|ordinary (V j)-ordinary (pointCoordinate z' j)| ≤ ordinary A.radius := by
have hb := hbound j
have hl := hlo j
have hu := hup j
have hlv := hLV j
constructor <;> apply abs_le.mpr <;> constructor <;> linarith
have hz' (j : Fin 2) : -ordinary r ≤ ordinary (L j)-ordinary (pointCoordinate z' j) ∧
ordinary (V j)-ordinary (pointCoordinate z' j) ≤ ordinary r := by
have hh := hdiff j
have h₁ := (abs_le.mp hh.1).1
have h₂ := (abs_le.mp hh.2).2
constructor <;> linarith [A.radius_small]
have he := A.slope_line_action hlarge hr d z z' hline L V hLV hz hz'
(ordinary z'.1,ordinary z'.2) (by rw [cutForm_ordinary,hline])
(fun j => by
have hh := hdiff j
have hh' : realCoordinate (ordinary z'.1,ordinary z'.2) j=ordinary (pointCoordinate z' j) := by
fin_cases j <;> rfl
rw [hh']
constructor <;> linarith [A.radius_small]) n q hW f hf hc I s t ht
rw [he]
have hlab (j : Fin 2) :
symmetricWindowStart A.radius j ≤ endpointLabel (signQuadrantLower A.radius d positive j) ∧
endpointLabel (signQuadrantLower A.radius d positive j) <
symmetricWindowStart A.radius j+symmetricWindowLength A.radius := by
unfold signQuadrantLower
split <;> split <;> first | exact (symmetricWindow_labels A.radius j).1 | exact (symmetricWindow_labels A.radius j).2.1
have hvab (j : Fin 2) :
symmetricWindowStart A.radius j ≤ endpointLabel (signQuadrantUpper A.radius d positive j) ∧
endpointLabel (signQuadrantUpper A.radius d positive j) <
symmetricWindowStart A.radius j+symmetricWindowLength A.radius := by
unfold signQuadrantUpper
split <;> split <;> first | exact (symmetricWindow_labels A.radius j).2.1 | exact (symmetricWindow_labels A.radius j).2.2
have hside := signQuadrant_side (a := a) r A.radius hr
⟨A.radius_pos,by linarith [A.radius_small]⟩ d positive
have hinc : coordinateRectangle a L V ≤ spatialTranslate z'
(coordinateRectangle a (signQuadrantLower A.radius d positive) (signQuadrantUpper A.radius d positive)) := by
rw [spatialTranslate_coordinateRectangle]
apply coordinateRectangle_mono L V _ _
· intro j; simpa only [map_add,pointCoordinate,add_comm] using hlo j
· exact hLV
· intro j; simpa only [map_add,pointCoordinate,add_comm] using hup j
· intro j
have hb := hbound j
simp only [map_add,add_sub_add_right_eq_sub]
linarith [A.radius_small,hr.2]
have hc' := B.all_rectangle_control_to_sign hlarge A.rectangles _ _ A.u A.quadrants
d z' positive _ _ hlab hvab hside n q _ hW hinc f hf hc
have ht' := B.tangentSign_constant_action hlarge _ _ A.u A.quadrants d z' positive f hf hc' I s t ht
have hs := B.tangentSign_eq (by omega) A.k _ A.p _ A.u A.u A.charts A.quadrants d z' true
simpa only [slope,hs] using ht'
end RectangularAtlas
end InitialCoverSystem
end SimpleAmenable
end OAI
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