| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.IntersectionControl |
|
|
| namespace OAI |
|
|
| section |
| section |
| open scoped symmDiff |
| namespace SimpleAmenable |
| open scoped commutatorElement |
| open scoped commutatorElement |
| section ChartQuadrants |
|
|
| theorem coordinateRectangle_lift {a : ℕ} (u v : Fin 2 → CutRing) |
| (huv : ∀ j, ordinary (u j) ≤ ordinary (v j)) |
| (hlen : ∀ j, ordinary (v j)-ordinary (u j) < 1) |
| (p : GenericSquare a) (hp : p ∈ (coordinateRectangle a u v).val) : |
| ∃ t : ℝ × ℝ, p.val = (Int.fract t.1,Int.fract t.2) ∧ |
| (ordinary (u 0) ≤ t.1 ∧ t.1 < ordinary (v 0)) ∧ |
| (ordinary (u 1) ≤ t.2 ∧ t.2 < ordinary (v 1)) := by |
| change p ∈ coordinateBetween a 0 (u 0) (v 0) ∧ |
| p ∈ coordinateBetween a 1 (u 1) (v 1) at hp |
| obtain ⟨k,hk⟩ := (mem_coordinateBetween_iff 0 (u 0) (v 0) (huv 0) (hlen 0) p).mp hp.1 |
| obtain ⟨l,hl⟩ := (mem_coordinateBetween_iff 1 (u 1) (v 1) (huv 1) (hlen 1) p).mp hp.2 |
| refine ⟨(coordinate 0 p+(k:ℝ),coordinate 1 p+(l:ℝ)),?_,hk,hl⟩ |
| simp only [Int.fract_add_intCast,Int.fract_eq_self.mpr (coordinate_bounds 0 p), |
| Int.fract_eq_self.mpr (coordinate_bounds 1 p)] |
| rfl |
|
|
| theorem coordinateRectangle_sign {a : ℕ} (r : CutRing) (j : Fin 4) |
| (hr : 0 < ordinary r ∧ ordinary r < 1/2) (u v : Fin 2 → CutRing) |
| (huv : ∀ k, ordinary (u k) ≤ ordinary (v k)) |
| (hu : ∀ k, -ordinary r < ordinary (u k)) |
| (hv : ∀ k, ordinary (v k) < ordinary r) |
| (hside : ∀ t : ℝ × ℝ, |
| (ordinary (u 0) ≤ t.1 ∧ t.1 < ordinary (v 0)) → |
| (ordinary (u 1) ≤ t.2 ∧ t.2 < ordinary (v 1)) → 0 ≤ cutForm a j t) : |
| coordinateRectangle a u v ≤ clippedSlopePrimitive a r j := by |
| intro p hp |
| obtain ⟨t,ht,hx,hy⟩ := coordinateRectangle_lift u v huv |
| (fun k => by have h1 := hu k; have h2 := hv k; linarith [hr.2]) p hp |
| apply (mem_clippedSlope_lift r j hr p t ?_ ht).mpr |
| · exact hside t hx hy |
| · exact ⟨⟨(hu 0).trans_le hx.1,hx.2.trans (hv 0)⟩, |
| ⟨(hu 1).trans_le hy.1,hy.2.trans (hv 1)⟩⟩ |
|
|
| theorem coordinateRectangle_negative_sign {a : ℕ} (r : CutRing) (j : Fin 4) |
| (hr : 0 < ordinary r ∧ ordinary r < 1/2) (u v : Fin 2 → CutRing) |
| (huv : ∀ k, ordinary (u k) ≤ ordinary (v k)) |
| (hu : ∀ k, -ordinary r < ordinary (u k)) |
| (hv : ∀ k, ordinary (v k) < ordinary r) |
| (hside : ∀ t : ℝ × ℝ, |
| (ordinary (u 0) ≤ t.1 ∧ t.1 < ordinary (v 0)) → |
| (ordinary (u 1) ≤ t.2 ∧ t.2 < ordinary (v 1)) → cutForm a j t < 0) : |
| coordinateRectangle a u v ≤ (clippedSlopePrimitive a r j)ᶜ := by |
| intro p hp hpos |
| obtain ⟨t,ht,hx,hy⟩ := coordinateRectangle_lift u v huv |
| (fun k => by have h1 := hu k; have h2 := hv k; linarith [hr.2]) p hp |
| have hR : (-ordinary r < t.1 ∧ t.1 < ordinary r) ∧ |
| (-ordinary r < t.2 ∧ t.2 < ordinary r) := |
| ⟨⟨(hu 0).trans_le hx.1,hx.2.trans (hv 0)⟩, |
| ⟨(hu 1).trans_le hy.1,hy.2.trans (hv 1)⟩⟩ |
| have hh := (mem_clippedSlope_lift r j hr p t hR ht).mp hpos |
| exact (not_lt_of_ge hh) (hside t hx hy) |
|
|
| theorem northwest_quadrant_sign {a : ℕ} (r s : CutRing) |
| (hr : 0 < ordinary r ∧ ordinary r < 1/2) |
| (hs : 0 < ordinary s ∧ ordinary s < ordinary r) : |
| coordinateRectangle a ![-s,0] ![0,s] ≤ clippedSlopePrimitive a r 2 := by |
| apply coordinateRectangle_sign r 2 hr |
| · intro k; fin_cases k <;> simp <;> linarith [hs.1] |
| · intro k; fin_cases k <;> simp <;> linarith [hr.1,hs.2] |
| · intro k; fin_cases k <;> simp <;> linarith [hr.1,hs.2] |
| · intro t hx hy |
| change ordinary (-s) ≤ t.1 ∧ t.1 < ordinary 0 at hx |
| change ordinary 0 ≤ t.2 ∧ t.2 < ordinary s at hy |
| have hlambda : 0 ≤ Real.goldenRatio^a := pow_nonneg Real.goldenRatio_pos.le _ |
| have hx' : t.1 ≤ 0 := by simpa using hx.2.le |
| have hy' : 0 ≤ t.2 := by simpa using hy.1 |
| change 0 ≤ t.2-Real.goldenRatio^a*t.1 |
| exact sub_nonneg.mpr ((mul_nonpos_of_nonneg_of_nonpos hlambda hx').trans hy') |
|
|
| theorem southeast_quadrant_sign {a : ℕ} (r s : CutRing) |
| (hr : 0 < ordinary r ∧ ordinary r < 1/2) |
| (hs : 0 < ordinary s ∧ ordinary s < ordinary r) : |
| coordinateRectangle a ![0,-s] ![s,0] ≤ (clippedSlopePrimitive a r 2)ᶜ := by |
| apply coordinateRectangle_negative_sign r 2 hr |
| · intro k; fin_cases k <;> simp <;> linarith [hs.1] |
| · intro k; fin_cases k <;> simp <;> linarith [hr.1,hs.2] |
| · intro k; fin_cases k <;> simp <;> linarith [hr.1,hs.2] |
| · intro t hx hy |
| change ordinary 0 ≤ t.1 ∧ t.1 < ordinary s at hx |
| change ordinary (-s) ≤ t.2 ∧ t.2 < ordinary 0 at hy |
| have hlambda : 0 ≤ Real.goldenRatio^a := pow_nonneg Real.goldenRatio_pos.le _ |
| have hx' : 0 ≤ t.1 := by simpa using hx.1 |
| have hy' : t.2 < 0 := by simpa using hy.2 |
| change t.2-Real.goldenRatio^a*t.1 < 0 |
| exact sub_neg.mpr (hy'.trans_le (mul_nonneg hlambda hx')) |
|
|
| end ChartQuadrants |
|
|
| section ChartAction |
| namespace ActualLawfulTable |
| variable {E H Q Ω : Type*} [Group E] [Group H] [Group Q] [Group.IsPerfect E] |
| {q : H →* Q} {v : (Ω → E) →* Q} (T : ActualLawfulTable q v) |
|
|
| theorem sector_same_action (U V W : Set Ω) (he : U ∩ W = V ∩ W) : |
| SameActionOn (T.sector U) (T.sector V) (T.sector W).range := by |
| intro s x hx |
| obtain ⟨t,rfl⟩ := hx |
| rw [T.sector_split U W s,T.sector_split V W s] |
| rw [conj_mul_of_commute _ _ _ |
| (T.sector_commute (Set.disjoint_left.mpr (by aesop)) s t)] |
| rw [conj_mul_of_commute _ _ _ |
| (T.sector_commute (Set.disjoint_left.mpr (by aesop)) s t)] |
| rw [he] |
| end ActualLawfulTable |
|
|
| namespace InitialCoverSystem |
| variable {a m M : ℕ} {r : CutRing} {hm : 2 ≤ m} |
| (B : InitialCoverSystem a r m hm M) {ι : Type*} [Finite ι] |
| [Group.IsPerfect (alternatingGroup (Fin (m+1)))] |
|
|
| theorem fullGeometricSector_same_action (hlarge : 15 < m+1) |
| (P : ι → Fin 5 × (CutRing × CutRing)) |
| (h : ∀ I, I.card ≤ 15 → ∀ b hb, B.PrimitiveFamilyLaw I b hb P) |
| (U V W : polygonAlgebra a) |
| (hU : ResolvedBy (fun i => (primitiveTests (a := a) (r := r) P i).val) U.val) |
| (hV : ResolvedBy (fun i => (primitiveTests (a := a) (r := r) P i).val) V.val) |
| (hW : ResolvedBy (fun i => (primitiveTests (a := a) (r := r) P i).val) W.val) |
| (he : U ⊓ W = V ⊓ W) : |
| SameActionOn (B.fullGeometricSector hlarge P h U) |
| (B.fullGeometricSector hlarge P h V) (B.fullGeometricSector hlarge P h W).range := by |
| apply (B.fullPrimitiveTable hlarge P h).sector_same_action |
| rw [← resolvedPolygonMask_inter _ U.val W.val hU hW, |
| ← resolvedPolygonMask_inter _ V.val W.val hV hW] |
| exact congrArg (resolvedPolygonMask (primitiveTests (a := a) (r := r) P)) |
| (congrArg Subtype.val he) |
|
|
| theorem chart_action_transfer (hlarge : 20 ≤ m+1) |
| (P : ι → Fin 5 × (CutRing × CutRing)) |
| (h : ∀ I, I.card ≤ 15 → ∀ b hb, B.PrimitiveFamilyLaw I b hb P) |
| (U V W : polygonAlgebra a) |
| (hU : ResolvedBy (fun i => (primitiveTests (a := a) (r := r) P i).val) U.val) |
| (hV : ResolvedBy (fun i => (primitiveTests (a := a) (r := r) P i).val) V.val) |
| (hW : ResolvedBy (fun i => (primitiveTests (a := a) (r := r) P i).val) W.val) |
| (he : U ⊓ W = V ⊓ W) |
| (f : TrackStar (Fin (m+1)) →* BoundedRelationCover M (alternatingGenerator a r m hm)) |
| (hf : B.AlignedSmallSupported f) |
| (hcontrol : SmallControlled B.c f (B.fullGeometricSector (by omega) P h W)) |
| (I : ControlAlphabet (Fin (m+1))) (s : UniversalExtension (alternatingGroup I.val)) : |
| ∀ x ∈ f.range, |
| B.fullGeometricSector (by omega) P h U (universalMap (subtypeAlternatingHom I.val) s)*x* |
| (B.fullGeometricSector (by omega) P h U (universalMap (subtypeAlternatingHom I.val) s))⁻¹ = |
| B.fullGeometricSector (by omega) P h V (universalMap (subtypeAlternatingHom I.val) s)*x* |
| (B.fullGeometricSector (by omega) P h V (universalMap (subtypeAlternatingHom I.val) s))⁻¹ := by |
| apply small_control_comparison _ B.c B.constant_aligned B.disjoint_aligned f _ hf hcontrol I.val _ _ |
| (B.fullGeometricSector_supported (by omega) P h U hU I ⟨s,rfl⟩) |
| (B.fullGeometricSector_supported (by omega) P h V hV I ⟨s,rfl⟩) |
| (by have hh := I.property.2; simp only [Fintype.card_fin]; omega) |
| intro J t |
| exact B.fullGeometricSector_same_action (by omega) P h U V W hU hV hW he _ _ ⟨_,rfl⟩ |
|
|
| end InitialCoverSystem |
| end ChartAction |
|
|
| end SimpleAmenable |
| end |
| end |
|
|
| end OAI |
|
|
