Update holomorphic.primitive.lean
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Stefan Kebekus
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f43fd50528
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@ -38,6 +38,7 @@ theorem MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countab
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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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@ -65,7 +66,7 @@ theorem MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countab
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assumption
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theorem MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable₃
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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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@ -84,11 +85,11 @@ theorem MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countab
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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 h₁x
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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 h₁y
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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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@ -103,7 +104,7 @@ theorem MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countab
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theorem integral_divergence₄
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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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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@ -113,27 +114,76 @@ theorem integral_divergence₄
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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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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 h₁x
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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 h₁y
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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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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 → ℂ)
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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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let f := F
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let h₁f : ContDiff ℝ 1 f := by sorry
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let g := Complex.I • F
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let h₁g : ContDiff ℝ 1 g := by sorry
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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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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 [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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