282 lines
10 KiB
Plaintext
282 lines
10 KiB
Plaintext
import Mathlib.Analysis.Complex.TaylorSeries
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import Mathlib.MeasureTheory.Integral.DivergenceTheorem
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import Mathlib.MeasureTheory.Integral.IntervalIntegral
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import Mathlib.MeasureTheory.Function.LocallyIntegrable
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import Nevanlinna.cauchyRiemann
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import Nevanlinna.partialDeriv
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theorem MeasureTheory.integral2_divergence₃
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
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(f g : ℝ × ℝ → E)
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(h₁f : ContDiff ℝ 1 f)
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(h₁g : ContDiff ℝ 1 g)
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(a₁ : ℝ)
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(a₂ : ℝ)
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(b₁ : ℝ)
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(b₂ : ℝ) :
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∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ((fderiv ℝ f) (x, y)) (1, 0) + ((fderiv ℝ g) (x, y)) (0, 1) = (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) - ∫ (y : ℝ) in a₂..b₂, f (a₁, y) := by
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apply integral2_divergence_prod_of_hasFDerivWithinAt_off_countable f g (fderiv ℝ f) (fderiv ℝ g) a₁ a₂ b₁ b₂ ∅
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exact Set.countable_empty
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-- ContinuousOn f (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂)
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exact h₁f.continuous.continuousOn
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--
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exact h₁g.continuous.continuousOn
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--
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rw [Set.diff_empty]
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intro x _
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exact DifferentiableAt.hasFDerivAt ((h₁f.differentiable le_rfl) x)
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--
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rw [Set.diff_empty]
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intro y _
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exact DifferentiableAt.hasFDerivAt ((h₁g.differentiable le_rfl) y)
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--
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apply ContinuousOn.integrableOn_compact
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apply IsCompact.prod
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exact isCompact_uIcc
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exact isCompact_uIcc
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apply ContinuousOn.add
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apply Continuous.continuousOn
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exact Continuous.clm_apply (ContDiff.continuous_fderiv h₁f le_rfl) continuous_const
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apply Continuous.continuousOn
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exact Continuous.clm_apply (ContDiff.continuous_fderiv h₁g le_rfl) continuous_const
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theorem integral_divergence₄
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
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(f g : ℂ → E)
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(h₁f : ContDiff ℝ 1 f)
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(h₁g : ContDiff ℝ 1 g)
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(a₁ : ℝ)
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(a₂ : ℝ)
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(b₁ : ℝ)
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(b₂ : ℝ) :
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∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ((fderiv ℝ f) ⟨x, y⟩ ) 1 + ((fderiv ℝ g) ⟨x, y⟩) Complex.I = (((∫ (x : ℝ) in a₁..b₁, g ⟨x, b₂⟩) - ∫ (x : ℝ) in a₁..b₁, g ⟨x, a₂⟩) + ∫ (y : ℝ) in a₂..b₂, f ⟨b₁, y⟩) - ∫ (y : ℝ) in a₂..b₂, f ⟨a₁, y⟩ := by
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let fr : ℝ × ℝ → E := f ∘ Complex.equivRealProdCLM.symm
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let gr : ℝ × ℝ → E := g ∘ Complex.equivRealProdCLM.symm
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have sfr {x y : ℝ} : f { re := x, im := y } = fr (x, y) := by exact rfl
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have sgr {x y : ℝ} : g { re := x, im := y } = gr (x, y) := by exact rfl
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repeat (conv in f { re := _, im := _ } => rw [sfr])
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repeat (conv in g { re := _, im := _ } => rw [sgr])
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have sfr' {x y : ℝ} {z : ℂ} : (fderiv ℝ f { re := x, im := y }) z = fderiv ℝ fr (x, y) (Complex.equivRealProdCLM z) := by
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rw [fderiv.comp]
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rw [Complex.equivRealProdCLM.symm.fderiv]
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tauto
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apply Differentiable.differentiableAt
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exact h₁f.differentiable le_rfl
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exact Complex.equivRealProdCLM.symm.differentiableAt
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conv in ⇑(fderiv ℝ f { re := _, im := _ }) _ => rw [sfr']
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have sgr' {x y : ℝ} {z : ℂ} : (fderiv ℝ g { re := x, im := y }) z = fderiv ℝ gr (x, y) (Complex.equivRealProdCLM z) := by
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rw [fderiv.comp]
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rw [Complex.equivRealProdCLM.symm.fderiv]
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tauto
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apply Differentiable.differentiableAt
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exact h₁g.differentiable le_rfl
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exact Complex.equivRealProdCLM.symm.differentiableAt
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conv in ⇑(fderiv ℝ g { re := _, im := _ }) _ => rw [sgr']
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apply MeasureTheory.integral2_divergence₃ fr gr _ _ a₁ a₂ b₁ b₂
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-- ContDiff ℝ 1 fr
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exact (ContinuousLinearEquiv.contDiff_comp_iff (ContinuousLinearEquiv.symm Complex.equivRealProdCLM)).mpr h₁f
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-- ContDiff ℝ 1 gr
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exact (ContinuousLinearEquiv.contDiff_comp_iff (ContinuousLinearEquiv.symm Complex.equivRealProdCLM)).mpr h₁g
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theorem integral_divergence₅
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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(F : ℂ → E)
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(hF : Differentiable ℂ F)
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(lowerLeft upperRight : ℂ) :
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(∫ (x : ℝ) in lowerLeft.re..upperRight.re, F ⟨x, lowerLeft.im⟩) + Complex.I • ∫ (x : ℝ) in lowerLeft.im..upperRight.im, F ⟨upperRight.re, x⟩ =
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(∫ (x : ℝ) in lowerLeft.re..upperRight.re, F ⟨x, upperRight.im⟩) + Complex.I • ∫ (x : ℝ) in lowerLeft.im..upperRight.im, F ⟨lowerLeft.re, x⟩ := by
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let h₁f : ContDiff ℝ 1 F := (hF.contDiff : ContDiff ℂ 1 F).restrict_scalars ℝ
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let h₁g : ContDiff ℝ 1 (-Complex.I • F) := by
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have : -Complex.I • F = fun x ↦ -Complex.I • F x := by rfl
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rw [this]
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apply ContDiff.comp
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exact contDiff_const_smul _
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exact h₁f
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let A := integral_divergence₄ (-Complex.I • F) F h₁g h₁f lowerLeft.re upperRight.im upperRight.re lowerLeft.im
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have {z : ℂ} : fderiv ℝ F z Complex.I = partialDeriv ℝ _ F z := by rfl
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conv at A in (fderiv ℝ F _) _ => rw [this]
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have {z : ℂ} : fderiv ℝ (-Complex.I • F) z 1 = partialDeriv ℝ _ (-Complex.I • F) z := by rfl
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conv at A in (fderiv ℝ (-Complex.I • F) _) _ => rw [this]
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conv at A =>
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left
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arg 1
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intro x
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arg 1
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intro y
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rw [CauchyRiemann₄ hF]
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rw [partialDeriv_smul'₂]
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simp
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simp at A
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have {t₁ t₂ t₃ t₄ : E} : 0 = (t₁ - t₂) + t₃ + t₄ → t₁ + t₃ = t₂ - t₄ := by
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intro hyp
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calc
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t₁ + t₃ = t₁ + t₃ - 0 := by rw [sub_zero (t₁ + t₃)]
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_ = t₁ + t₃ - (t₁ - t₂ + t₃ + t₄) := by rw [hyp]
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_ = t₂ - t₄ := by abel
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let B := this A
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repeat
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rw [intervalIntegral.integral_symm lowerLeft.im upperRight.im] at B
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simp at B
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exact B
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noncomputable def primitive
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] :
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ℂ → (ℂ → E) → (ℂ → E) := by
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intro z₀
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intro f
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exact fun z ↦ (∫ (x : ℝ) in z₀.re..z.re, f ⟨x, z₀.im⟩) + Complex.I • ∫ (x : ℝ) in z₀.im..z.im, f ⟨z.re, x⟩
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theorem primitive_zeroAtBasepoint
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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(f : ℂ → E)
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(z₀ : ℂ) :
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(primitive z₀ f) z₀ = 0 := by
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unfold primitive
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simp
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theorem primitive_lem1
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] [IsScalarTower ℝ ℂ E]
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(v : E) :
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HasDerivAt (primitive 0 (fun _ ↦ v)) v 0 := by
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unfold primitive
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simp
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have : (fun (z : ℂ) => z.re • v + Complex.I • z.im • v) = (fun (y : ℂ) => ((fun w ↦ w) y) • v) := by
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funext z
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rw [smul_comm]
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rw [← smul_assoc]
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simp
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have : z.re • v = (z.re : ℂ) • v := by exact rfl
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rw [this, ← add_smul]
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simp
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rw [this]
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have hc : HasDerivAt (fun (w : ℂ) ↦ w) 1 0 := by
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apply hasDerivAt_id'
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nth_rewrite 2 [← (one_smul ℂ v)]
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exact HasDerivAt.smul_const hc v
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theorem primitive_fderivAtBasepoint
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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(f : ℂ → E)
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(hf : Continuous f) :
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HasDerivAt (primitive 0 f) (f 0) 0 := by
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unfold primitive
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simp
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apply hasDerivAt_iff_isLittleO.2
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simp
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rw [Asymptotics.isLittleO_iff]
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intro c hc
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have {z : ℂ} {e : E} : z • e = (∫ (x : ℝ) in (0)..(z.re), e) + Complex.I • ∫ (x : ℝ) in (0)..(z.im), e:= by
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simp
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rw [smul_comm]
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rw [← smul_assoc]
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simp
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have : z.re • e = (z.re : ℂ) • e := by exact rfl
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rw [this, ← add_smul]
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simp
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conv =>
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left
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intro x
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left
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arg 1
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arg 2
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rw [this]
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have {A B C D :E} : (A + B) - (C + D) = (A - C) + (B - D) := by
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abel
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have t₀ {r : ℝ} : IntervalIntegrable (fun x => f { re := x, im := 0 }) MeasureTheory.volume 0 r := by sorry
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have t₁ {r : ℝ} :IntervalIntegrable (fun x => f 0) MeasureTheory.volume 0 r := by sorry
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have t₂ {a b : ℝ}: IntervalIntegrable (fun x_1 => f { re := a, im := x_1 }) MeasureTheory.volume 0 b := by sorry
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have t₃ {a : ℝ} : IntervalIntegrable (fun x => f 0) MeasureTheory.volume 0 a := by sorry
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conv =>
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left
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intro x
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left
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arg 1
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rw [this]
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rw [← smul_sub]
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rw [← intervalIntegral.integral_sub t₀ t₁]
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rw [← intervalIntegral.integral_sub t₂ t₃]
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rw [Filter.eventually_iff_exists_mem]
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let s := f⁻¹' Metric.ball (f 0) c
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have h₁s : IsOpen s := IsOpen.preimage hf Metric.isOpen_ball
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have h₂s : 0 ∈ s := by
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apply Set.mem_preimage.mpr
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exact Metric.mem_ball_self hc
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obtain ⟨ε, h₁ε, h₂ε⟩ := Metric.isOpen_iff.1 h₁s 0 h₂s
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have h₃ε : ∀ y ∈ Metric.ball 0 ε, ‖(f y) - (f 0)‖ < c := by
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intro y hy
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exact mem_ball_iff_norm.mp (h₂ε hy)
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use Metric.ball 0 ε
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constructor
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· exact Metric.ball_mem_nhds 0 h₁ε
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· intro y hy
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have h₁y : |y.re| < ε := by
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sorry
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have : ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ ≤ c * |y.re - 0| := by
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apply intervalIntegral.norm_integral_le_of_norm_le_const
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intro x hx
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have h₁x : |x| < ε := by sorry
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apply le_of_lt
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apply h₃ε { re := x, im := 0 }
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simp
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have : { re := x, im := 0 } = (x : ℂ) := by rfl
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rw [this]
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rw [Complex.abs_ofReal]
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exact h₁x
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sorry
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/-
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calc ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0) + Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖
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_ ≤ ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by apply norm_add_le
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_ ≤ ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by
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simp
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rw [norm_smul]
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simp
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_ ≤ |(∫ (x : ℝ) in (0)..(y.re), ‖f { re := x, im := 0 } - f 0‖)| + |∫ (x : ℝ) in (0)..(y.im), ‖f { re := y.re, im := x } - f 0‖| := by
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apply add_le_add
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apply intervalIntegral.norm_integral_le_abs_integral_norm
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apply intervalIntegral.norm_integral_le_abs_integral_norm
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_ ≤
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-/
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sorry
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theorem primitive_additivity
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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(f : ℂ → E)
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(hf : Differentiable ℂ f)
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(z₀ z₁ : ℂ) :
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(primitive z₁ f) = (primitive z₀ f) - (fun z ↦ primitive z₀ f z₁) := by
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sorry
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