File size: 4,280 Bytes
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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.RandomFields.PrimitiveCovariance
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.SmallControlTransfer
import OAI.GroupTheory.SimpleAmenable.CentralCovers.CoordinateFamilyCompare
import OAI.GroupTheory.SimpleAmenable.CentralCovers.ControlCalculus
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section ChartControlChain
variable {α H : Type*} [Fintype α] [DecidableEq α] [Group H]
[Group.IsPerfect (alternatingGroup α)]
theorem small_control_action_chain (c : alternatingGroup α →* H)
(f : TrackStar α →* H) (g : ℕ → TrackStar α →* H) (n : ℕ)
(hstart : SmallControlled c f (g 0))
(hstep : ∀ k, k < n → ∀ (I : ControlAlphabet α) s, ∀ x ∈ f.range,
g k (universalMap (subtypeAlternatingHom I.val) s)*x*
(g k (universalMap (subtypeAlternatingHom I.val) s))⁻¹ =
g (k+1) (universalMap (subtypeAlternatingHom I.val) s)*x*
(g (k+1) (universalMap (subtypeAlternatingHom I.val) s))⁻¹) :
SmallControlled c f (g n) := by
have h (k : ℕ) (hk : k ≤ n) : SmallControlled c f (g k) := by
induction k with
| zero => exact hstart
| succ k ih =>
intro I s x hx
dsimp only [MonoidHom.comp_apply]
rw [← hstep k (by omega) I s x hx]
exact ih (by omega) I s x hx
exact h n le_rfl
namespace InitialCoverSystem
variable {a m M : ℕ} {r : CutRing} {hm : 2 ≤ m}
(B : InitialCoverSystem a r m hm M) {ι κ : Type*} [Finite ι] [Finite κ]
[Group.IsPerfect (alternatingGroup (Fin (m+1)))]
theorem chartCoordinateSector_eq_window (hlarge : 15 < m+1)
(n : ℕ) (h : B.CoordinateWindowLaw n) (q : Fin 2 → ℤ)
(P : ι → Fin 5 × (CutRing × CutRing))
(hP : ∀ I, I.card ≤ 15 → ∀ b hb, B.PrimitiveFamilyLaw I b hb P)
(v : κ → ι) (j : κ → Fin 2) (u : κ → CutRing × CutRing)
(hv : ∀ i, P (v i) = (coordinateTestIndex (j i),u i))
(hs : ∀ i, q (j i) ≤ endpointLabel (if j i=0 then (u i).1 else (u i).2))
(he : ∀ i, endpointLabel (if j i=0 then (u i).1 else (u i).2) < q (j i)+(n-1:ℕ))
(V : polygonAlgebra a)
(hV : ResolvedBy (fun i => (spatialTranslate (u i) (coordinatePrimitive a (j i))).val) V.val) :
B.fullGeometricSector hlarge P hP V = B.windowSector hlarge n h q V := by
let Q := fun i => (coordinateTestIndex (j i),u i)
have heq : P ∘ v = Q := funext hv
have hQ : ∀ I, I.card ≤ 15 → ∀ b hb, B.PrimitiveFamilyLaw I b hb Q := by
intro I hI b hb
rw [← heq]
exact PrimitiveFamilyLaw.reindex B I b hb P (hP I hI b hb) v
have hVQ : ResolvedBy (fun i => (primitiveTests (a := a) (r := r) Q i).val) V.val := by
simpa only [Q,primitiveTests,primitiveFamilyTests,initialTest_coordinate] using hV
rw [← B.fullGeometricSector_subfamily hlarge P Q v heq hP hQ V hVQ]
exact B.coordinateFamilySector_eq_window hlarge n h q j u hs he hQ V hV
theorem chart_chain_disjoint_control (hlarge : 20 ≤ m+1)
(f g : TrackStar (Fin (m+1)) →* BoundedRelationCover M (alternatingGenerator a r m hm))
(hf : B.AlignedSmallSupported f) (hg : B.AlignedSmallSupported g)
(c : ℕ → TrackStar (Fin (m+1)) →* BoundedRelationCover M (alternatingGenerator a r m hm))
(n : ℕ) (hc : B.AlignedSmallSupported (c n))
(d : TrackStar (Fin (m+1)) →* BoundedRelationCover M (alternatingGenerator a r m hm))
(hstart : SmallControlled B.c f (c 0))
(hstep : ∀ k, k < n → ∀ (I : ControlAlphabet (Fin (m+1))) s, ∀ x ∈ f.range,
c k (universalMap (subtypeAlternatingHom I.val) s)*x*
(c k (universalMap (subtypeAlternatingHom I.val) s))⁻¹ =
c (k+1) (universalMap (subtypeAlternatingHom I.val) s)*x*
(c (k+1) (universalMap (subtypeAlternatingHom I.val) s))⁻¹)
(hfinish : SmallControlled B.c g d) (hsplit : ∀ s t, Commute (c n s) (d t)) :
∀ s t, Commute (f s) (g t) := by
apply small_control_disjoint _ B.c B.constant_aligned B.disjoint_aligned
(by simpa using hlarge) f g (c n) d hf hg hc
(small_control_action_chain B.c f c n hstart hstep) hfinish hsplit
end InitialCoverSystem
end ChartControlChain
end SimpleAmenable
end
end
end OAI
|