| import OAI.Analysis.CircleDomains.Topology.RetainedJordanConvergence |
| import OAI.Analysis.CircleDomains.Topology.PhysicalCollarCoordinates |
|
|
| namespace OAI |
|
|
| /-! Uniform convergence on a source circuit gives a vanishing numerical |
| error bound. No regularity of the finitely many early models is needed. -/ |
|
|
| open Set Function Filter Metric |
| open scoped Topology |
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| theorem exists_circuit_uniform_error |
| (f : ℕ → ℝ → ℂ) (g : ℝ → ℂ) |
| (hf : TendstoUniformlyOn f g atTop (Icc (0 : ℝ) 1)) : |
| ∃ e : ℕ → ℝ, (∀ n, 0 ≤ e n) ∧ Tendsto e atTop (𝓝 0) ∧ |
| ∀ᶠ n in atTop, ∀ t ∈ Icc (0 : ℝ) 1, dist (f n t) (g t) ≤ e n := by |
| classical |
| let D (n : ℕ) := (fun t => dist (f n t) (g t)) '' Icc (0 : ℝ) 1 |
| have hD (n) : (D n).Nonempty := ⟨dist (f n 0) (g 0), 0, ⟨le_rfl, zero_le_one⟩, rfl⟩ |
| let e (n : ℕ) := if h : BddAbove (D n) then sSup (D n) else 0 |
| have he (n) : 0 ≤ e n := by |
| by_cases h : BddAbove (D n) |
| · exact (dist_nonneg : 0 ≤ dist (f n 0) (g 0)).trans |
| (by simp only [e, dite_eq_left h]; exact le_csSup h ⟨0, ⟨le_rfl, zero_le_one⟩, rfl⟩) |
| · simp only [e, dite_eq_right h, le_refl] |
| have hε (δ : ℝ) (hδ : 0 < δ) : |
| ∀ᶠ n in atTop, ∀ t ∈ Icc (0 : ℝ) 1, dist (f n t) (g t) < δ := by |
| simpa only [dist_comm] using (Metric.tendstoUniformlyOn_iff.mp hf δ hδ) |
| have hbounded {n : ℕ} {δ : ℝ} |
| (h : ∀ t ∈ Icc (0 : ℝ) 1, dist (f n t) (g t) < δ) : BddAbove (D n) := by |
| refine ⟨δ, ?_⟩ |
| rintro _ ⟨t, ht, rfl⟩ |
| exact (h t ht).le |
| refine ⟨e, he, ?_, ?_⟩ |
| · apply Metric.tendsto_nhds.mpr |
| intro δ hδ |
| filter_upwards [hε (δ / 2) (half_pos hδ)] with n hn |
| have hnB := hbounded hn |
| have hle : e n ≤ δ / 2 := by |
| simp only [e, dite_eq_left hnB] |
| exact csSup_le (hD n) (by rintro _ ⟨t, ht, rfl⟩; exact (hn t ht).le) |
| simpa only [dist_zero_right, Real.norm_eq_abs, abs_of_nonneg (he n)] using |
| (hle.trans_lt (half_lt_self hδ)) |
| · filter_upwards [hε 1 zero_lt_one] with n hn t ht |
| simp only [e, dite_eq_left (hbounded hn)] |
| exact le_csSup (hbounded hn) ⟨t, ht, rfl⟩ |
|
|
| theorem tendstoUniformlyOn_retained_circuit |
| (f : ℕ → Sphere → Sphere) (F : ℂ → ℂ) {G : Set Sphere} |
| (hf : TendstoLocallyUniformlyOn (fun n (z : ℂ) => finiteCoord (f n (z : Sphere))) F atTop |
| (((↑) : ℂ → Sphere) ⁻¹' G)) |
| (H : ℂ ≃ₜ ℂ) (γ : ℝ → ℂ) (hγ : PositiveJordanParametrization H γ) |
| (hG : ((↑) : ℂ → Sphere) '' frontier (jordanInterior H) ⊆ G) : |
| TendstoUniformlyOn (fun n t => finiteCoord (f n (γ t : Sphere))) (F ∘ γ) atTop |
| (Icc (0 : ℝ) 1) := by |
| apply (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact isCompact_Icc).mp |
| exact hf.comp γ (fun t _ => hG (mem_image_of_mem _ (hγ.mem_frontier t))) |
| hγ.smooth.continuous.continuousOn |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
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|
