| import Mathlib |
| import OAI.Geometry.Cannon.Energy.RawFunction |
| import OAI.Geometry.Cannon.Energy.StripCrossing |
| import OAI.Geometry.Cannon.Modulus.ModulusEnergy |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
| section |
| open Set Topology |
| open Set Topology Filter Metric unitInterval |
| open scoped InnerProductSpace |
| open Set unitInterval |
| open Set Metric Topology |
| open Set Topology unitInterval |
|
|
| namespace CannonCharts |
|
|
| open CannonCells |
|
|
| theorem rectangle_modulus_localization {X : Type*} [MetricSpace X] [CompactSpace X] |
| (φ : X ≃ₜ SphereTwo) (d : ℝ) (hd : 0 < d) |
| (ι : ℕ → Type*) [∀ n, Fintype (ι n)] |
| (C : (n : ℕ) → ι n → Set X) (hC : ∀ n x, ∃ i, x ∈ C n i) : |
| ∃ e : Plane → X, IsOpenEmbedding e ∧ diam (e '' box 8) < d ∧ |
| ∃ c : ℝ, 0 < c ∧ ∃ N : ℕ → ℕ, StrictMono N ∧ ∀ n, |
| c * modulus (C (N n)) (macroFamily (X := X) d) ≤ |
| modulus (C (N n)) (stripFamily e) := by |
| classical |
| obtain ⟨J,e,hJ,he,hdiam,hsub⟩ := finite_strip_localization φ d hd |
| let : Nonempty J := hJ.coe_sort |
| let H := fun j : J × Fin 4 => stripFamily (e j) |
| let K : ℝ := (Fintype.card (J × Fin 4) : ℝ)^2 |
| have hK : 0 < K := pow_pos (by exact_mod_cast Fintype.card_pos) _ |
| have hlocal (n : ℕ) : ∃ j : J × Fin 4, |
| modulus (C n) (macroFamily (X := X) d) ≤ K*modulus (C n) (H j) := by |
| exact exists_modulus_localization (C n) (hC n) (macroFamily d) H |
| (fun j => stripFamily_nonempty (e j)) |
| (fun _ _ h => stripFamily_member_nonempty h) hsub |
| obtain ⟨j,N,hN,hNj⟩ := fixed_localization_subsequence |
| (fun n => modulus (C n) (macroFamily (X := X) d)) |
| (fun j n => modulus (C n) (H j)) K hlocal |
| refine ⟨e j,he j,hdiam j,K⁻¹,inv_pos.mpr hK,N,hN,?_⟩ |
| intro n |
| calc |
| K⁻¹ * modulus (C (N n)) (macroFamily d) ≤ |
| K⁻¹ * (K * modulus (C (N n)) (H j)) := |
| mul_le_mul_of_nonneg_left (hNj n) (inv_nonneg.mpr hK.le) |
| _ = modulus (C (N n)) (stripFamily (e j)) := by rw [← mul_assoc,inv_mul_cancel₀ hK.ne',one_mul] |
|
|
| end CannonCharts |
|
|
| namespace CannonCharts |
|
|
| def squareMap (z : CannonGrid.Square) : Plane := (1+(z.1:ℝ),12*(z.2:ℝ)-6) |
|
|
| lemma continuous_squareMap : Continuous squareMap := by |
| unfold squareMap |
| fun_prop |
|
|
| lemma squareMap_injective : Function.Injective squareMap := by |
| intro z w h |
| have hx := congrArg Prod.fst h |
| have hy := congrArg Prod.snd h |
| apply Prod.ext |
| · apply Subtype.ext |
| dsimp [squareMap] at hx |
| linarith |
| · apply Subtype.ext |
| dsimp [squareMap] at hy |
| linarith |
|
|
| lemma squareMap_embedding : IsEmbedding squareMap := |
| (continuous_squareMap.isClosedEmbedding squareMap_injective).isEmbedding |
|
|
| lemma squareMap_in_box (z : CannonGrid.Square) : |
| squareMap z ∈ Metric.closedBall 0 8 := by |
| simp only [Metric.mem_closedBall,dist_zero_right,Prod.norm_def,Real.norm_eq_abs, |
| max_le_iff,abs_le,squareMap] |
| constructor <;> constructor <;> nlinarith [z.1.property.1,z.1.property.2,z.2.property.1,z.2.property.2] |
|
|
| theorem lift_strip_crossing (q : I → Plane) (hq : StripCrossing q) : |
| ∃ u v : CannonRectangle.Middle, u.val.1 = 0 ∧ v.val.1 = 1 ∧ |
| ∃ p : Path u v, ∀ t, squareMap (p t).val = q t := by |
| let m : I → CannonRectangle.Middle := fun t => |
| ⟨(⟨(q t).1-1,by constructor <;> linarith [(hq.2.2.2 t).1.1,(hq.2.2.2 t).1.2]⟩, |
| ⟨((q t).2+6)/12,by |
| have hy := abs_le.mp (hq.2.2.2 t).2 |
| constructor <;> linarith [hy.1,hy.2]⟩),by |
| have hy := abs_le.mp (hq.2.2.2 t).2 |
| change ((q t).2+6)/12 ∈ Icc (1/3:ℝ) (2/3) |
| constructor <;> linarith [hy.1,hy.2]⟩ |
| have hqc : Continuous q := hq.1 |
| have hm : Continuous m := by dsimp [m]; fun_prop |
| let p : Path (m 0) (m 1) := ⟨⟨m,hm⟩,rfl,rfl⟩ |
| refine ⟨m 0,m 1,?_,?_,p,?_⟩ |
| · apply Subtype.ext |
| change (q 0).1-1 = 0 |
| rw [hq.2.1]; norm_num |
| · apply Subtype.ext |
| change (q 1).1-1 = 1 |
| rw [hq.2.2.1]; norm_num |
| · intro t |
| apply Prod.ext <;> dsimp [squareMap,p,m] <;> ring |
|
|
| theorem stripFamily_eq_image {X : Type*} (e : Plane → X) : |
| stripFamily e = {A | ∃ B ∈ CannonRectangle.leftRight, A = (e ∘ squareMap) '' B} := by |
| ext A |
| constructor |
| · rintro ⟨q,hq,rfl⟩ |
| obtain ⟨u,v,hu,hv,p,hp⟩ := lift_strip_crossing q hq |
| refine ⟨Subtype.val '' range p,⟨u,v,hu,hv,p,rfl⟩,?_⟩ |
| ext x |
| constructor |
| · rintro ⟨t,rfl⟩ |
| exact ⟨(p t).val,⟨p t,⟨t,rfl⟩,rfl⟩,by dsimp; rw [hp]⟩ |
| · rintro ⟨z,⟨m,⟨t,rfl⟩,rfl⟩,rfl⟩ |
| exact ⟨t,by dsimp; rw [hp]⟩ |
| · rintro ⟨B,⟨u,v,hu,hv,p,rfl⟩,rfl⟩ |
| let q : I → Plane := fun t => squareMap (p t).val |
| have hq : Continuous q := continuous_squareMap.comp (continuous_subtype_val.comp p.continuous) |
| refine ⟨q,⟨hq,?_,?_,?_⟩,?_⟩ |
| · change 1+((p 0).val.1:ℝ) = 1 |
| rw [p.source,hu] |
| norm_num |
| · change 1+((p 1).val.1:ℝ) = 2 |
| rw [p.target,hv] |
| norm_num |
| · intro t |
| have hy := (p t).property |
| change ((p t).val.2:ℝ) ∈ Icc (1/3:ℝ) (2/3) at hy |
| change 1+((p t).val.1:ℝ) ∈ Icc 1 2 ∧ |12*((p t).val.2:ℝ)-6| ≤ 2 |
| constructor |
| · constructor <;> nlinarith [(p t).val.1.property.1,(p t).val.1.property.2] |
| · rw [abs_le] |
| constructor <;> linarith [hy.1,hy.2] |
| · ext x |
| constructor |
| · rintro ⟨_,⟨_,⟨t,rfl⟩,rfl⟩,rfl⟩ |
| exact ⟨t,rfl⟩ |
| · rintro ⟨t,rfl⟩ |
| exact ⟨(p t).val,⟨p t,⟨t,rfl⟩,rfl⟩,rfl⟩ |
|
|
| end CannonCharts |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
