| import Mathlib |
| import OAI.Geometry.Cannon.ScalarLimits.CompactVariationRegion |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
| namespace CannonRelease |
|
|
| open CannonGraph CannonVisual CannonCells CannonCharts Set Metric |
|
|
| universe v |
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| variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
|
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| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
|
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| local instance : Setoid (BasedRay D) := raySetoid D |
|
|
| theorem boundary_modulus_subexponential (φ : Boundary D ≃ₜ SphereTwo) : |
| let _ := visualMetricSpace D hδ |
| let a := Real.exp (2*visualExponent δ) |
| let LE := 8*Real.exp (visualExponent δ*(4*(8*δ+6):ℕ)) |
| ∀ d : ℝ, 0 < d → d < diam (univ : Set (Boundary D)) → d ≤ 1/(LE*3^2) → |
| ∀ (ι : ℕ → Type*) [∀ n, Fintype (ι n)] (p : ∀ n, ι n → Boundary D), |
| (∀ n i j, i ≠ j → (a^n)⁻¹ ≤ dist (p n i) (p n j)) → |
| (∀ n x, ∃ i, dist (p n i) x ≤ (a^n)⁻¹) → |
| ∀ lam : ℝ, 1 < lam → ∃ B : ℝ, 0 < B ∧ ∀ n, |
| modulus (fun i => ball (p n i) (2*(a^n)⁻¹)) (macroFamily d) ≤ B*lam^n := by |
| classical |
| let _ := visualMetricSpace D hδ |
| dsimp only |
| intro d hd hdia hdc ι hι p hsep hnet lam hlam |
| have hn := boundary_exponential_records_impossible D hδ φ d hd hdia hdc ι p hsep hnet lam hlam |
| have hbnd : ∃ B : ℝ, ∀ n, |
| modulus (fun i => ball (p n i) (2*(Real.exp (2*visualExponent δ)^n)⁻¹)) |
| (macroFamily d)/lam^n ≤ B := by |
| by_contra h |
| apply hn |
| intro B |
| by_contra hB |
| apply h |
| refine ⟨B,fun n => ?_⟩ |
| exact le_of_not_gt (fun hh => hB ⟨n,hh⟩) |
| obtain ⟨B,hB⟩ := hbnd |
| refine ⟨max B 1,lt_of_lt_of_le zero_lt_one (le_max_right ..),fun n => ?_⟩ |
| have hh := hB n |
| have hl : 0 < lam^n := pow_pos (lt_trans zero_lt_one hlam) n |
| exact (div_le_iff₀ hl).mp (hh.trans (le_max_left ..)) |
|
|
| end CannonRelease |
|
|
| end |
|
|
| section |
|
|
| open Set Topology Filter Metric unitInterval |
| open scoped InnerProductSpace |
| open Set unitInterval |
|
|
| namespace CannonScalarLocal |
|
|
| open CannonGrid CannonCells CannonRegularization CannonChart |
|
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| variable {X ι : Type*} [MetricSpace X] [Fintype ι] |
|
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| local instance : LocallyPathConnectedSpace unitInterval := |
| (isQuotientMap_projIcc (a := (0 : ℝ)) (b := 1) (h := zero_le_one)).locallyPathConnectedSpace |
|
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| local instance : PathConnectedSpace Square := |
| pathConnectedSpace_iff_connectedSpace.mpr inferInstance |
|
|
| end CannonScalarLocal |
|
|
| namespace CannonApprox |
|
|
| open CannonGrid CannonRegularization CannonLevel CannonCells CannonBands |
|
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| variable {ι : Type*} [Fintype ι] |
|
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| local instance : PathConnectedSpace (Strip height (1 / 3) (2 / 3)) := |
| strip_pathConnected _ _ (by norm_num) (by norm_num) (by norm_num) |
|
|
| end CannonApprox |
|
|
| open Set Metric Topology |
| open Set Topology unitInterval |
| open Set Filter Topology |
| open scoped BigOperators |
| open Set unitInterval |
| open Set MeasureTheory |
| open scoped NNReal ENNReal |
|
|
| namespace CannonAtomic |
|
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| attribute [local instance] Classical.propDecidable |
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| variable {X ι : Type*} [MeasurableSpace X] [Fintype ι] |
|
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| end CannonAtomic |
|
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| namespace CannonAtomic |
|
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| open Set MeasureTheory |
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| attribute [local instance] Classical.propDecidable |
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| end CannonAtomic |
|
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| open Set Filter Metric Topology |
| open Set Filter Topology |
| open Set Filter MeasureTheory Topology |
| open scoped NNReal ENNReal |
|
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| namespace CannonScalarLimit |
|
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| open CannonAtomic CannonAnnularCompactness CannonScales Set Filter Metric Topology MeasureTheory |
|
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| attribute [local instance] Classical.propDecidable |
|
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| end CannonScalarLimit |
|
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| namespace CannonScalarLimit |
|
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| open CannonAtomic CannonAnnularCompactness Set Filter Metric Topology MeasureTheory |
| open scoped NNReal ENNReal |
|
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| attribute [local instance] Classical.propDecidable |
|
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| end CannonScalarLimit |
|
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| open Set Filter Topology TopologicalSpace Metric |
|
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| end |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
