openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Uniformization /ComplexRectangleLocalLimit.lean
| import OAI.Analysis.CircleDomains.Uniformization.ComplexRectangleLimit | |
| namespace OAI | |
| /-! A local rectangular Morera passage, allowing the smooth approximants' | |
| poles outside the fixed open set and allowing a finite initial segment. -/ | |
| open Set Function Filter MeasureTheory Complex | |
| open scoped Topology Interval | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| open CannonBeltrami | |
| lemma locally_uniform_curve_integral_on {U : Set ℂ} (hU : IsOpen U) | |
| {f : ℕ → ℂ → ℂ} {g : ℂ → ℂ} (hf : ∀ᶠn in atTop,ContinuousOn (f n) U) | |
| (hl : TendstoLocallyUniformlyOn f g atTop U) (γ : ℝ → ℂ) (hγ : Continuous γ) | |
| (a b : ℝ) (hsub : MapsTo γ (uIcc a b) U) : | |
| Tendsto (fun n => ∫t in a..b,f n (γ t)) atTop (𝓝 (∫t in a..b,g (γ t))) := by | |
| have ht := (tendstoLocallyUniformlyOn_iff_forall_isCompact hU).mp hl | |
| (γ '' uIcc a b) (image_subset_iff.mpr hsub) (isCompact_uIcc.image hγ) | |
| have hc : TendstoUniformlyOn (fun n t => f n (γ t)) (fun t => g (γ t)) atTop (uIcc a b) := | |
| (ht.comp γ).mono (fun t ht => mem_image_of_mem γ ht) | |
| exact hc.tendsto_intervalIntegral_of_continuousOn | |
| (hf.mono fun n hn => hn.comp hγ.continuousOn hsub) | |
| lemma complexRectangleFlux_tendsto_on {U : Set ℂ} (hU : IsOpen U) | |
| {f : ℕ → ℂ → ℂ} {g : ℂ → ℂ} (hf : ∀ᶠn in atTop,ContinuousOn (f n) U) | |
| (hl : TendstoLocallyUniformlyOn f g atTop U) (z w : ℂ) (hzw : Rectangle z w⊆U) : | |
| Tendsto (fun n => complexRectangleFlux (f n) z w) atTop (𝓝 (complexRectangleFlux g z w)) := by | |
| have h₁ := locally_uniform_curve_integral_on hU hf hl (fun x => x+z.im*I) | |
| (by fun_prop) z.re w.re (fun x hx => hzw (by change _ ∈ _ ∧ _ ∈ _; simpa using And.intro hx (left_mem_uIcc : z.im∈uIcc z.im w.im))) | |
| have h₂ := locally_uniform_curve_integral_on hU hf hl (fun x => x+w.im*I) | |
| (by fun_prop) z.re w.re (fun x hx => hzw (by change _ ∈ _ ∧ _ ∈ _; simpa using And.intro hx (right_mem_uIcc : w.im∈uIcc z.im w.im))) | |
| have h₃ := locally_uniform_curve_integral_on hU hf hl (fun y => w.re+y*I) | |
| (by fun_prop) z.im w.im (fun y hy => hzw (by change _ ∈ _ ∧ _ ∈ _; simpa using And.intro (right_mem_uIcc : w.re∈uIcc z.re w.re) hy)) | |
| have h₄ := locally_uniform_curve_integral_on hU hf hl (fun y => z.re+y*I) | |
| (by fun_prop) z.im w.im (fun y hy => hzw (by change _ ∈ _ ∧ _ ∈ _; simpa using And.intro (left_mem_uIcc : z.re∈uIcc z.re w.re) hy)) | |
| exact ((h₁.sub h₂).add (h₃.const_mul I)).sub (h₄.const_mul I) | |
| lemma complexRectangleFlux_eq_bar_integral_on {U : Set ℂ} (hU : IsOpen U) | |
| {f : ℂ → ℂ} (hf : ContDiffOn ℝ 1 f U) (z w : ℂ) (hzw : Rectangle z w⊆U) : | |
| complexRectangleFlux f z w=(2*I)*(∫x : ℝ in z.re..w.re, | |
| ∫y : ℝ in z.im..w.im,partialBar f (x+y*I)) := by | |
| have hd := hf.continuousOn_fderiv_of_isOpen hU (by norm_num) | |
| have hi : IntegrableOn (fun p => I • fderiv ℝ f p 1-fderiv ℝ f p I) (Rectangle z w) := | |
| ((hd.clm_apply continuousOn_const).const_smul I |>.sub | |
| (hd.clm_apply continuousOn_const)).mono hzw |>.integrableOn_compact | |
| (isCompact_uIcc.reProdIm isCompact_uIcc) | |
| have he := integral_boundary_rect_of_differentiableOn_real f z w | |
| ((hf.differentiableOn (by norm_num)).mono hzw) hi | |
| have hid (p : ℂ) : I • fderiv ℝ f p 1-fderiv ℝ f p I=(2*I)*partialBar f p := by | |
| simp only [smul_eq_mul,partialBar] | |
| ring_nf | |
| simp [I_sq] | |
| ring | |
| change complexRectangleFlux f z w=_ at he | |
| rw [he] | |
| simp_rw [hid,intervalIntegral.integral_const_mul] | |
| lemma differentiableOn_of_local_rectangle_errors {U : Set ℂ} (hU : IsOpen U) | |
| {f : ℕ → ℂ → ℂ} {g : ℂ → ℂ} | |
| (hf : ∀ᶠn in atTop,ContDiffOn ℝ 1 (f n) U) | |
| (hl : TendstoLocallyUniformlyOn f g atTop U) | |
| (he : ∀z w,Rectangle z w⊆U → | |
| Tendsto (fun n => ∫x : ℝ in z.re..w.re, | |
| ∫y : ℝ in z.im..w.im,partialBar (f n) (x+y*I)) atTop (𝓝 0)) : | |
| DifferentiableOn ℂ g U := by | |
| apply (isConservativeOn_and_continuousOn_iff_isDifferentiableOn hU).mp | |
| refine ⟨?_,hl.continuousOn (hf.frequently.mono fun n hn => hn.continuousOn)⟩ | |
| intro z w hzw | |
| have hz : complexRectangleFlux g z w=0 := by | |
| apply tendsto_nhds_unique (complexRectangleFlux_tendsto_on hU | |
| (hf.mono fun n hn => hn.continuousOn) hl z w hzw) | |
| have ht := (he z w hzw).const_mul (2*I) | |
| simp only [mul_zero] at ht | |
| apply ht.congr' | |
| filter_upwards [hf] with n hn | |
| exact (complexRectangleFlux_eq_bar_integral_on hU hn z w hzw).symm | |
| rw [←add_eq_zero_iff_eq_neg,wedgeIntegral_add_wedgeIntegral_eq] | |
| exact hz | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |