| 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 | |