openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /PolygonGeometry /AxisCutSequences.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.AxisSplitting | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.SeparatedAxisSectors | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section AxisCutSequences | |
| structure AxisCutSequence (n K : ℕ) where | |
| cut : Fin (K+1) → CutRing | |
| strictMono : StrictMono (fun i => ordinary (cut i)) | |
| length : Fin (K+1) → ℕ | |
| start : Fin (K+1) → ℤ | |
| length_le : ∀ i, length i ≤ n*9/10 | |
| lower_label : ∀ i, start i ≤ endpointLabel (cut 0) ∧ endpointLabel (cut 0) < start i+length i | |
| upper_label : ∀ i, start i ≤ endpointLabel (cut (Fin.last K)) ∧ endpointLabel (cut (Fin.last K)) < start i+length i | |
| cut_label : ∀ i, start i ≤ endpointLabel (cut i) ∧ endpointLabel (cut i) < start i+length i | |
| namespace AxisCutSequence | |
| variable {n K : ℕ} (D : AxisCutSequence n K) | |
| def index (_D : AxisCutSequence n K) (i : ℕ) : Fin (K+1) := ⟨min i K,by omega⟩ | |
| @[simp] theorem index_zero : D.index 0 = 0 := by ext; simp [index] | |
| @[simp] theorem index_fin (i : Fin (K+1)) : D.index i.val = i := by ext; simp [index]; omega | |
| theorem index_mono : Monotone D.index := by | |
| intro i j hij | |
| exact min_le_min_right K hij | |
| theorem ordered (i : Fin (K+1)) : | |
| ordinary (D.cut 0) ≤ ordinary (D.cut i) ∧ ordinary (D.cut i) ≤ ordinary (D.cut (Fin.last K)) := | |
| ⟨D.strictMono.monotone (Fin.zero_le i),D.strictMono.monotone (Fin.le_last i)⟩ | |
| theorem length_old (i : Fin (K+1)) : D.length i ≤ n := by have hh := D.length_le i; omega | |
| variable {a m M : ℕ} {r : CutRing} {hm : 2 ≤ m} | |
| (B : InitialCoverSystem a r m hm M) | |
| [Group.IsPerfect (alternatingGroup (Fin (m+1)))] | |
| (hlarge : 15 < m+1) (g : B.CoordinateWindowLaw n) (d : Fin 2) | |
| noncomputable def anchorCopy : TrackStar (Fin (m+1)) →* | |
| BoundedRelationCover M (alternatingGenerator a r m hm) := | |
| B.axisSector hlarge n g d (D.length 0) (D.length_old 0) (D.start 0) | |
| (coordinateInterval a d (D.cut 0) (D.cut (Fin.last K))) | |
| noncomputable def prefixCopy (i : ℕ) : TrackStar (Fin (m+1)) →* | |
| BoundedRelationCover M (alternatingGenerator a r m hm) := | |
| B.axisSector hlarge n g d (D.length (D.index i)) (D.length_old _) (D.start (D.index i)) | |
| (coordinateInterval a d (D.cut 0) (D.cut (D.index i))) | |
| noncomputable def suffix (i : ℕ) : TrackStar (Fin (m+1)) →* | |
| BoundedRelationCover M (alternatingGenerator a r m hm) := | |
| B.axisSector hlarge n g d (D.length (D.index i)) (D.length_old _) (D.start (D.index i)) | |
| (coordinateInterval a d (D.cut (D.index i)) (D.cut (Fin.last K))) | |
| theorem anchorCopy_eq (i : Fin (K+1)) : | |
| D.anchorCopy B hlarge g d = | |
| B.axisSector hlarge n g d (D.length i) (D.length_old i) (D.start i) | |
| (coordinateInterval a d (D.cut 0) (D.cut (Fin.last K))) := | |
| B.axisInterval_copy_eq hlarge n g d _ _ _ _ _ _ _ _ | |
| (D.lower_label 0) (D.upper_label 0) (D.lower_label i) (D.upper_label i) | |
| theorem prefix_suffix (hlen : ordinary (D.cut (Fin.last K))-ordinary (D.cut 0) < 1) (i : ℕ) | |
| (s : TrackStar (Fin (m+1))) : | |
| D.anchorCopy B hlarge g d s = D.prefixCopy B hlarge g d i s*D.suffix B hlarge g d i s := by | |
| rw [D.anchorCopy_eq B hlarge g d (D.index i)] | |
| exact B.axisSector_split hlarge n g d _ _ _ _ _ _ | |
| (D.ordered _).1 (D.ordered _).2 hlen (D.lower_label _) (D.cut_label _) s | |
| @[simp] theorem prefix_zero : D.prefixCopy B hlarge g d 0 = 1 := by | |
| simp only [prefixCopy,index_zero,InitialCoverSystem.axisSector_self] | |
| theorem prefix_last : D.prefixCopy B hlarge g d K = D.anchorCopy B hlarge g d := by | |
| have hi : D.index K = Fin.last K := by ext; simp [index] | |
| simp only [prefixCopy,hi] | |
| exact (D.anchorCopy_eq B hlarge g d (Fin.last K)).symm | |
| theorem prefix_projection (i : ℕ) : | |
| (coverMap M (alternatingGenerator a r m hm)).comp (D.prefixCopy B hlarge g d i) = | |
| (conditionalAlternatingHom (coordinateInterval a d (D.cut 0) (D.cut (D.index i)))).comp | |
| (universalProjection (alternatingGroup (Fin (m+1)))) := | |
| B.axisSector_projection hlarge n g d _ _ _ _ _ (D.lower_label _) (D.cut_label _) | |
| theorem suffix_projection (i : ℕ) : | |
| (coverMap M (alternatingGenerator a r m hm)).comp (D.suffix B hlarge g d i) = | |
| (conditionalAlternatingHom (coordinateInterval a d (D.cut (D.index i)) (D.cut (Fin.last K)))).comp | |
| (universalProjection (alternatingGroup (Fin (m+1)))) := | |
| B.axisSector_projection hlarge n g d _ _ _ _ _ (D.cut_label _) (D.upper_label _) | |
| theorem prefix_small_supported (i : ℕ) : | |
| B.AlignedSmallSupported (D.prefixCopy B hlarge g d i) := | |
| B.axisSector_supported hlarge n g d _ _ _ _ | |
| (axisInterval_resolved d _ _ _ _ (D.lower_label _) (D.cut_label _)) | |
| theorem anchorCopy_small_supported : B.AlignedSmallSupported (D.anchorCopy B hlarge g d) := | |
| B.axisSector_supported hlarge n g d _ _ _ _ | |
| (axisInterval_resolved d _ _ _ _ (D.lower_label 0) (D.upper_label 0)) | |
| theorem prefix_small_controlled (hlen : ordinary (D.cut (Fin.last K))-ordinary (D.cut 0) < 1) (i : ℕ) : | |
| SmallControlled B.c (D.prefixCopy B hlarge g d i) (D.anchorCopy B hlarge g d) := by | |
| rw [D.anchorCopy_eq B hlarge g d (D.index i)] | |
| exact B.fullGeometricSector_small_control hlarge _ _ _ _ | |
| (coordinateInterval_mono d _ _ _ _ le_rfl (D.ordered _).1 (D.ordered _).2 hlen) | |
| theorem prefix_suffix_self_commute (hlen : ordinary (D.cut (Fin.last K))-ordinary (D.cut 0) < 1) | |
| (i : ℕ) (s t : TrackStar (Fin (m+1))) : | |
| Commute (D.prefixCopy B hlarge g d i s) (D.suffix B hlarge g d i t) := | |
| B.axisSector_split_commute hlarge n g d _ _ _ _ _ _ | |
| (D.ordered _).1 (D.ordered _).2 hlen (D.lower_label _) (D.cut_label _) s t | |
| end AxisCutSequence | |
| end AxisCutSequences | |
| section NestedSupport | |
| variable {α H Q : Type*} [Fintype α] [DecidableEq α] [Group H] [Group Q] | |
| [Group.IsPerfect (alternatingGroup α)] | |
| variable (L : Finset α → Subgroup H) (c : alternatingGroup α →* H) | |
| (f : TrackStar α →* H) (l h : ℕ → TrackStar α →* H) | |
| (hf : ∀ i s, f s = l i s*h i s) | |
| (hc : ∀ i j, i ≤ j → ∀ s t, Commute (l i s) (h j t)) | |
| theorem cutDifference_small_supported (hs : ∀ i, SmallSupported L (l i)) (i : ℕ) : | |
| SmallSupported L (cutDifference f l h hf hc i) := by | |
| intro I x hx | |
| obtain ⟨s,rfl⟩ := hx | |
| change (l i (universalMap (subtypeAlternatingHom I.val) s))⁻¹ * | |
| l (i+1) (universalMap (subtypeAlternatingHom I.val) s) ∈ L I.val | |
| exact (L I.val).mul_mem ((L I.val).inv_mem (hs i I ⟨s,rfl⟩)) (hs (i+1) I ⟨s,rfl⟩) | |
| theorem cutDifference_small_controlled (hs : ∀ i, SmallControlled c (l i) f) (i : ℕ) : | |
| SmallControlled c (cutDifference f l h hf hc i) f := by | |
| intro I t x hx | |
| obtain ⟨s,rfl⟩ := hx | |
| apply (same_conj_iff _ _ _).mpr | |
| change Commute _ ((l i s)⁻¹*l (i+1) s) | |
| exact ((same_conj_iff _ _ _).mp (hs i I t _ ⟨s,rfl⟩)).inv_right.mul_right | |
| ((same_conj_iff _ _ _).mp (hs (i+1) I t _ ⟨s,rfl⟩)) | |
| end NestedSupport | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |