| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.StarAtomLaw |
| import OAI.GroupTheory.SimpleAmenable.Configurations.SymmetricCharts |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.UniformTangentTransport |
|
|
| namespace OAI |
|
|
| section |
| section |
| open scoped symmDiff |
| namespace SimpleAmenable |
| open scoped commutatorElement |
| open scoped commutatorElement |
| section FiniteNestedLaw |
| variable {E H Q : Type*} [Group E] [Group H] [Group Q] |
|
|
| theorem prefix_range_le_differences (f : E →* H) (l h : ℕ → E →* 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)) |
| (hl : l 0 = 1) (N : ℕ) (j : ℕ) (hj : j ≤ N) : |
| (l j).range ≤ ⨆ i : Fin N, (cutDifference f l h hf hc i.val).range := by |
| let K : Subgroup H := ⨆ i : Fin N, (cutDifference f l h hf hc i.val).range |
| rintro x ⟨s,rfl⟩ |
| have he := cutDifference_telescope f l h hf hc j s |
| rw [hl] at he |
| simp only [MonoidHom.one_apply,inv_one,one_mul] at he |
| rw [← he] |
| apply K.list_prod_mem |
| intro y hy |
| obtain ⟨i,hi,rfl⟩ := List.mem_map.mp hy |
| have hi' : i < j := List.mem_range.mp hi |
| exact (le_iSup (fun i : Fin N => (cutDifference f l h hf hc i.val).range) |
| ⟨i,by omega⟩) ⟨s,rfl⟩ |
|
|
| theorem centralOn_of_nested_star_cuts (q : H →* Q) |
| (N : ℕ) (v : (Fin N → E) →* Q) (hv : Function.Injective v) |
| (f : UniversalExtension E →* H) (l h : ℕ → UniversalExtension E →* 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)) |
| (hl : l 0 = 1) |
| (hproj : ∀ i : Fin N, q.comp (cutDifference f l h hf hc i.val) = |
| (v.comp (sectorMask {i})).comp (universalProjection E)) : |
| CentralOn q (⨆ j : Fin (N+1), (l j.val).range) := by |
| have hpair : Pairwise fun i j : Fin N => ∀ s t, |
| Commute (cutDifference f l h hf hc i.val s) (cutDifference f l h hf hc j.val t) := by |
| intro i j hij s t |
| rcases lt_or_gt_of_ne hij with hij | hij |
| · exact cutDifference_pairwise f l h hf hc i.val j.val hij s t |
| · exact (cutDifference_pairwise f l h hf hc j.val i.val hij t s).symm |
| have hcentral := centralOn_of_star_atoms q v hv (fun i : Fin N => cutDifference f l h hf hc i.val) hpair hproj |
| apply hcentral.mono |
| apply iSup_le |
| intro j |
| exact prefix_range_le_differences f l h hf hc hl N j.val (by omega) |
|
|
| end FiniteNestedLaw |
|
|
| section CommonContainingWindows |
|
|
| theorem source_chart_containing_window (n t : ℕ) (p z v : ℤ) (b d : ℕ) |
| (hv : p ≤ v ∧ v < p+t) |
| (hz : (z-v).natAbs ≤ d) |
| (hsize : t+2*d+2*b+2 ≤ n) : |
| p ⊓ (z-(b:ℤ)) ≤ p ∧ p+t ≤ (p ⊓ (z-(b:ℤ)))+n ∧ |
| p ⊓ (z-(b:ℤ)) ≤ z-(b:ℤ) ∧ |
| z-(b:ℤ)+(2*b+2:ℕ) ≤ (p ⊓ (z-(b:ℤ)))+n := by |
| have hh : |z-v| ≤ (d:ℤ) := by |
| simpa only [Int.natCast_natAbs] using (show ((z-v).natAbs:ℤ) ≤ (d:ℤ) by exact_mod_cast hz) |
| have ha := abs_le.mp hh |
| have hn : (t:ℤ)+2*(d:ℤ)+2*(b:ℤ)+2 ≤ (n:ℤ) := by exact_mod_cast hsize |
| constructor |
| · exact inf_le_left |
| constructor |
| · rw [min_def] |
| split_ifs <;> omega |
| constructor |
| · exact inf_le_right |
| · rw [min_def] |
| split_ifs <;> omega |
|
|
| theorem fixed_chart_margin_eventually (b : ℕ) : |
| ∃ N : ℕ, ∀ n : ℕ, N ≤ n → n*9/10+2*b+2 ≤ n := by |
| refine ⟨20*b+20,?_⟩ |
| intro n hn |
| have hh := Nat.mul_div_le (n*9) 10 |
| omega |
|
|
| theorem transverse_chart_margin_eventually (b : ℕ) : |
| ∃ N : ℕ, ∀ n : ℕ, N ≤ n → n/4+2*(n/10+1)+2*b+2 ≤ n := by |
| refine ⟨20*b+40,?_⟩ |
| intro n hn |
| have h₀ := Nat.mul_div_le n 4 |
| have h₁ := Nat.mul_div_le n 10 |
| omega |
|
|
| end CommonContainingWindows |
|
|
| section QuadrantControl |
|
|
| def signQuadrantLower (s : CutRing) (d : Fin 2) (positive : Bool) (j : Fin 2) : CutRing := |
| if positive then (if j=d then -s else 0) else (if j=d then 0 else -s) |
|
|
| def signQuadrantUpper (s : CutRing) (d : Fin 2) (positive : Bool) (j : Fin 2) : CutRing := |
| if positive then (if j=d then 0 else s) else (if j=d then s else 0) |
|
|
| theorem signQuadrant_side {a : ℕ} (r s : CutRing) |
| (hr : 0 < ordinary r ∧ ordinary r < 1/2) |
| (hs : 0 < ordinary s ∧ ordinary s < ordinary r) (d : Fin 2) (positive : Bool) : |
| coordinateRectangle a (signQuadrantLower s d positive) (signQuadrantUpper s d positive) ≤ |
| if positive then clippedSlopePrimitive a r (slopeDirection d) |
| else (clippedSlopePrimitive a r (slopeDirection d))ᶜ := by |
| have hSlope : 0 ≤ Real.goldenRatio^a := pow_nonneg Real.goldenRatio_pos.le _ |
| cases positive with |
| | true => |
| apply coordinateRectangle_sign r (slopeDirection d) hr |
| · intro j; fin_cases d <;> fin_cases j <;> simp [signQuadrantLower,signQuadrantUpper] <;> linarith [hs.1] |
| · intro j; fin_cases d <;> fin_cases j <;> simp [signQuadrantLower] <;> linarith [hs.2,hr.1] |
| · intro j; fin_cases d <;> fin_cases j <;> simp [signQuadrantUpper] <;> linarith [hs.2,hr.1] |
| · intro t hx hy |
| fin_cases d |
| · simp [signQuadrantLower,signQuadrantUpper] at hx hy |
| change 0 ≤ t.2-Real.goldenRatio^a*t.1 |
| exact sub_nonneg.mpr ((mul_nonpos_of_nonneg_of_nonpos hSlope hx.2.le).trans hy.1) |
| · simp [signQuadrantLower,signQuadrantUpper] at hx hy |
| change 0 ≤ t.1-Real.goldenRatio^a*t.2 |
| exact sub_nonneg.mpr ((mul_nonpos_of_nonneg_of_nonpos hSlope hy.2.le).trans hx.1) |
| | false => |
| apply coordinateRectangle_negative_sign r (slopeDirection d) hr |
| · intro j; fin_cases d <;> fin_cases j <;> simp [signQuadrantLower,signQuadrantUpper] <;> linarith [hs.1] |
| · intro j; fin_cases d <;> fin_cases j <;> simp [signQuadrantLower] <;> linarith [hs.2,hr.1] |
| · intro j; fin_cases d <;> fin_cases j <;> simp [signQuadrantUpper] <;> linarith [hs.2,hr.1] |
| · intro t hx hy |
| fin_cases d |
| · simp [signQuadrantLower,signQuadrantUpper] at hx hy |
| change t.2-Real.goldenRatio^a*t.1 < 0 |
| exact sub_neg.mpr (hy.2.trans_le (mul_nonneg hSlope hx.1)) |
| · simp [signQuadrantLower,signQuadrantUpper] at hx hy |
| change t.1-Real.goldenRatio^a*t.2 < 0 |
| exact sub_neg.mpr (hx.2.trans_le (mul_nonneg hSlope hy.1)) |
|
|
| theorem coordinateRectangle_mono {a : ℕ} (L V l v : Fin 2 → CutRing) |
| (hL : ∀ j, ordinary (l j) ≤ ordinary (L j)) |
| (hLV : ∀ j, ordinary (L j) ≤ ordinary (V j)) |
| (hV : ∀ j, ordinary (V j) ≤ ordinary (v j)) |
| (hlen : ∀ j, ordinary (v j)-ordinary (l j) < 1) : |
| coordinateRectangle a L V ≤ coordinateRectangle a l v := by |
| exact inf_le_inf (coordinateInterval_mono 0 _ _ _ _ (hL 0) (hLV 0) (hV 0) (hlen 0)) |
| (coordinateInterval_mono 1 _ _ _ _ (hL 1) (hLV 1) (hV 1) (hlen 1)) |
|
|
| 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)))] |
|
|
| theorem rectangle_control_from_axes (hlarge : 15 < m+1) |
| (n : ℕ) (g : B.CoordinateWindowLaw n) |
| (t : Fin 2 → ℕ) (ht : ∀ j, t j ≤ n) (p q : Fin 2 → ℤ) |
| (hp : ∀ j, q j ≤ p j) (hq : ∀ j, p j+t j ≤ q j+n) |
| (L V : Fin 2 → CutRing) |
| (hL : ∀ j, p j ≤ endpointLabel (L j) ∧ endpointLabel (L j) < p j+t j) |
| (hV : ∀ j, p j ≤ endpointLabel (V j) ∧ endpointLabel (V j) < p j+t j) |
| (f : TrackStar (Fin (m+1)) →* BoundedRelationCover M (alternatingGenerator a r m hm)) |
| (hc : ∀ j, SmallControlled B.c f |
| (B.axisSector hlarge n g j (t j) (ht j) (p j) (coordinateInterval a j (L j) (V j)))) : |
| SmallControlled B.c f (B.windowSector hlarge n g q (coordinateRectangle a L V)) := by |
| have hsmall (j : Fin 2) := axisInterval_resolved (a := a) (r := r) |
| j (t j) (p j) (L j) (V j) (hL j) (hV j) |
| have hlargeL (j : Fin 2) : q j ≤ endpointLabel (L j) ∧ endpointLabel (L j) < q j+n := by |
| have hh := hL j; have hh' := hp j; have hh'' := hq j; omega |
| have hlargeV (j : Fin 2) : q j ≤ endpointLabel (V j) ∧ endpointLabel (V j) < q j+n := by |
| have hh := hV j; have hh' := hp j; have hh'' := hq j; omega |
| have he (j : Fin 2) := B.axisSector_in_window hlarge n g j (t j) (ht j) (p j) q |
| (hp j) (hq j) (coordinateInterval a j (L j) (V j)) (hsmall j) |
| change SmallControlled B.c f (B.fullGeometricSector hlarge _ _ |
| (coordinateInterval a 0 (L 0) (V 0) ⊓ coordinateInterval a 1 (L 1) (V 1))) |
| apply B.fullGeometricSector_intersection_control hlarge _ _ _ _ |
| (coordinateInterval_window n q 0 _ _ (hlargeL 0) (hlargeV 0)) |
| (coordinateInterval_window n q 1 _ _ (hlargeL 1) (hlargeV 1)) f |
| · change SmallControlled B.c f (B.windowSector hlarge n g q (coordinateInterval a 0 (L 0) (V 0))) |
| rw [← he 0]; exact hc 0 |
| · change SmallControlled B.c f (B.windowSector hlarge n g q (coordinateInterval a 1 (L 1) (V 1))) |
| rw [← he 1]; exact hc 1 |
|
|
| theorem tangentSign_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) |
| (t : Fin 2 → ℕ) (ht : ∀ j, t j ≤ n) (p q : Fin 2 → ℤ) |
| (hp : ∀ j, q j ≤ p j) (hq : ∀ j, p j+t j ≤ q j+n) |
| (d : Fin 2) (z : CutRing × CutRing) (positive : Bool) |
| (hstart : ∀ j, q j ≤ pointLabel z j+symmetricWindowStart s j) |
| (hend : ∀ j, pointLabel z j+symmetricWindowStart s j+symmetricWindowLength s ≤ q j+n) |
| (L V : Fin 2 → CutRing) |
| (hLV : ∀ j, ordinary (L j) ≤ ordinary (V j)) |
| (hL : ∀ j, p j ≤ endpointLabel (L j) ∧ endpointLabel (L j) < p j+t j) |
| (hV : ∀ j, p j ≤ endpointLabel (V j) ∧ endpointLabel (V j) < p j+t j) |
| (hlo : ∀ j, ordinary (signQuadrantLower s d positive j)+ordinary (pointCoordinate z j) ≤ ordinary (L j)) |
| (hup : ∀ j, ordinary (V j) ≤ ordinary (signQuadrantUpper s d positive j)+ordinary (pointCoordinate z j)) |
| (f : TrackStar (Fin (m+1)) →* BoundedRelationCover M (alternatingGenerator a r m hm)) |
| (hf : B.AlignedSmallSupported f) |
| (hc : ∀ j, SmallControlled B.c f |
| (B.axisSector (by omega) n g j (t j) (ht j) (p j) (coordinateInterval a j (L j) (V j)))) : |
| SmallControlled B.c f (B.tangentSign (by omega) (symmetricWindowLength s) |
| (symmetricWindowStart s) u h d z positive) := by |
| have hlab (j : Fin 2) : |
| symmetricWindowStart s j ≤ endpointLabel (signQuadrantLower s d positive j) ∧ |
| endpointLabel (signQuadrantLower s d positive j) < symmetricWindowStart s j+symmetricWindowLength s := by |
| unfold signQuadrantLower |
| split <;> split <;> first | exact (symmetricWindow_labels s j).1 | exact (symmetricWindow_labels s j).2.1 |
| have hvab (j : Fin 2) : |
| symmetricWindowStart s j ≤ endpointLabel (signQuadrantUpper s d positive j) ∧ |
| endpointLabel (signQuadrantUpper s d positive j) < symmetricWindowStart s j+symmetricWindowLength s := by |
| unfold signQuadrantUpper |
| split <;> split <;> first | exact (symmetricWindow_labels s j).2.1 | exact (symmetricWindow_labels s j).2.2 |
| apply B.tangentSign_control_from_rectangle hlarge _ _ u h n g q d z positive hstart hend |
| _ _ hlab hvab (signQuadrant_side r s hr hs d positive) (coordinateRectangle a L V) ?_ f hf |
| (B.rectangle_control_from_axes (by omega) n g t ht p q hp hq L V hL hV f hc) |
| rw [spatialTranslate_coordinateRectangle] |
| apply coordinateRectangle_mono L V _ _ |
| · intro j; simpa only [map_add,pointCoordinate] using hlo j |
| · exact hLV |
| · intro j; simpa only [map_add,pointCoordinate] using hup j |
| · intro j |
| simp only [map_add,add_sub_add_right_eq_sub] |
| unfold signQuadrantUpper signQuadrantLower |
| split <;> split <;> simp only [map_zero,map_neg] <;> linarith [hs.2,hr.2] |
|
|
| end InitialCoverSystem |
| end QuadrantControl |
|
|
| end SimpleAmenable |
| end |
| end |
|
|
| end OAI |
|
|
