Update holomorphic.primitive.lean
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Stefan Kebekus
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@ -4,67 +4,6 @@ import Mathlib.MeasureTheory.Function.LocallyIntegrable
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import Nevanlinna.cauchyRiemann
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import Nevanlinna.partialDeriv
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/-
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variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
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variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] [CompleteSpace F]
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variable {G : Type*} [NormedAddCommGroup G] [NormedSpace ℂ G] [CompleteSpace G]
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noncomputable def Complex.primitive
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(f : ℂ → F) : ℂ → F :=
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fun z ↦ ∫ t : ℝ in (0)..1, z • f (t * z)
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-/
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theorem MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable₁
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
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(f : ℝ × ℝ → E)
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(g : ℝ × ℝ → E)
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(f' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E)
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(g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E)
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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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(s : Set (ℝ × ℝ))
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(hs : s.Countable)
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(Hcf : ContinuousOn f (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂))
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(Hcg : ContinuousOn g (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂))
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(Hdf : ∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt f (f' x) x)
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(Hdg : ∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt g (g' x) x)
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(Hi : MeasureTheory.IntegrableOn (fun (x : ℝ × ℝ) => (f' x) (1, 0) + (g' x) (0, 1)) (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂) MeasureTheory.volume) :
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∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, (f' (x, y)) (1, 0) + (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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exact
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integral2_divergence_prod_of_hasFDerivWithinAt_off_countable f g f' g' a₁ a₂ b₁ b₂ s hs Hcf Hcg
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Hdf Hdg Hi
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theorem MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable₂
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
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(f : ℝ × ℝ → E)
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(g : ℝ × ℝ → E)
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(f' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E)
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(g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E)
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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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(Hcf : ContinuousOn f (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂))
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(Hcg : ContinuousOn g (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂))
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(Hdf : ∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂), HasFDerivAt f (f' x) x)
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(Hdg : ∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂), HasFDerivAt g (g' x) x)
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(Hi : MeasureTheory.IntegrableOn (fun (x : ℝ × ℝ) => (f' x) (1, 0) + (g' x) (0, 1)) (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂) MeasureTheory.volume) :
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∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, (f' (x, y)) (1, 0) + (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
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integral2_divergence_prod_of_hasFDerivWithinAt_off_countable f g f' g' a₁ a₂ b₁ b₂ ∅
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exact Set.countable_empty
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assumption
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assumption
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rwa [Set.diff_empty]
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rwa [Set.diff_empty]
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assumption
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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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@ -149,49 +88,46 @@ theorem integral_divergence₄
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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 → ℂ)
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(F : ℂ → E)
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(hF : Differentiable ℂ F)
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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, Complex.I • F { re := x, im := b₂ }) +
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(∫ (y : ℝ) in a₂..b₂, F { re := b₁, im := y })
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=
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(∫ (x : ℝ) in a₁..b₁, Complex.I • F { re := x, im := a₂ }) +
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(∫ (y : ℝ) in a₂..b₂, F { re := a₁, im := y }) := by
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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 g := Complex.I • F
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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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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 Complex.I
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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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let A := integral_divergence₄ F g h₁f h₁g a₁ a₂ b₁ b₂
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have {z : ℂ} : fderiv ℝ F z 1 = partialDeriv ℝ 1 F z := by rfl
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conv at A in (fderiv ℝ F _) 1 => rw [this]
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have {z : ℂ} : fderiv ℝ g z Complex.I = partialDeriv ℝ Complex.I g z := by rfl
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conv at A in (fderiv ℝ g _) Complex.I => rw [this]
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have : Differentiable ℂ g := by sorry
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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 this]
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rw [CauchyRiemann₄ hF]
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rw [partialDeriv_smul'₂]
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rw [← smul_assoc]
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simp
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simp at A
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sorry
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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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