Update cauchyRiemann.lean
This commit is contained in:
Stefan Kebekus
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77d1fd2ae4
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38eaf677f9
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@ -1,15 +1,15 @@
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import Mathlib.Analysis.Calculus.Conformal.NormedSpace
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import Mathlib.Analysis.Calculus.ContDiff.Basic
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import Mathlib.Analysis.Calculus.ContDiff.Defs
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import Mathlib.Analysis.Calculus.Deriv.Linear
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import Mathlib.Analysis.Calculus.FDeriv.Basic
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import Mathlib.Analysis.Calculus.FDeriv.Comp
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import Mathlib.Analysis.Calculus.FDeriv.Linear
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import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
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--import Mathlib.Analysis.Calculus.Conformal.NormedSpace
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--import Mathlib.Analysis.Calculus.ContDiff.Basic
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--import Mathlib.Analysis.Calculus.ContDiff.Defs
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--import Mathlib.Analysis.Calculus.Deriv.Linear
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--import Mathlib.Analysis.Calculus.FDeriv.Basic
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--import Mathlib.Analysis.Calculus.FDeriv.Comp
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--import Mathlib.Analysis.Calculus.FDeriv.Linear
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--import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
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import Mathlib.Analysis.Calculus.LineDeriv.Basic
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import Mathlib.Analysis.Complex.Basic
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import Mathlib.Analysis.Complex.CauchyIntegral
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import Mathlib.Analysis.Complex.Conformal
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--import Mathlib.Analysis.Complex.Basic
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--import Mathlib.Analysis.Complex.CauchyIntegral
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--import Mathlib.Analysis.Complex.Conformal
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import Mathlib.Analysis.Complex.RealDeriv
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variable {z : ℂ} {f : ℂ → ℂ}
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@ -26,12 +26,9 @@ theorem CauchyRiemann₁ : (DifferentiableAt ℂ f z)
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have t₂a : Complex.I * (AR 1) = Complex.I • (AR 1) := by
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exact rfl
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rw [← t₂a] at t₂
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have t₂b : Complex.I • 1 = Complex.I := by
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simp
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have t₂c : AR Complex.I = Complex.I • (AR 1) := by
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simp at t₂
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exact t₂
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let C : HasFDerivAt f (ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z)) z := h.hasFDerivAt.restrictScalars ℝ
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have t₃ : fderiv ℝ f z = ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z) := by
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exact DifferentiableAt.fderiv_restrictScalars ℝ h
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rw [t₃]
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@ -54,72 +51,78 @@ theorem CauchyRiemann₂ : (DifferentiableAt ℂ f z)
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exact CauchyRiemann₁ h
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theorem CauchyRiemann₃ : (DifferentiableAt ℂ f z)
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→ lineDeriv ℝ (Complex.reCLM ∘ f) z 1 = lineDeriv ℝ (Complex.imCLM ∘ f) z Complex.I := by
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→ (lineDeriv ℝ (Complex.reCLM ∘ f) z 1 = lineDeriv ℝ (Complex.imCLM ∘ f) z Complex.I)
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∧ (lineDeriv ℝ (Complex.reCLM ∘ f) z Complex.I = -lineDeriv ℝ (Complex.imCLM ∘ f) z 1)
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:= by
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-- Proof starts here
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intro h
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have t₀₀ : DifferentiableAt ℝ f z := by
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exact h.restrictScalars ℝ
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have t₀₁ : DifferentiableAt ℝ Complex.reCLM (f z) := by
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have : ∀ w, DifferentiableAt ℝ Complex.reCLM w := by
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intro w
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refine Differentiable.differentiableAt ?h
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exact ContinuousLinearMap.differentiable (Complex.reCLM : ℂ →L[ℝ] ℝ)
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exact this (f z)
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have t₁ : DifferentiableAt ℝ (Complex.reCLM ∘ f) z := by
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apply t₀₁.comp
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exact t₀₀
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have t₂ : lineDeriv ℝ (Complex.reCLM ∘ f) z 1 = (fderiv ℝ (Complex.reCLM ∘ f) z) 1 := by
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apply DifferentiableAt.lineDeriv_eq_fderiv
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exact t₁
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rw [t₂]
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have s₀₀ : DifferentiableAt ℝ f z := by
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exact h.restrictScalars ℝ
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have s₀₁ : DifferentiableAt ℝ Complex.imCLM (f z) := by
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have : ∀ w, DifferentiableAt ℝ Complex.imCLM w := by
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intro w
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apply Differentiable.differentiableAt
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exact ContinuousLinearMap.differentiable (Complex.imCLM : ℂ →L[ℝ] ℝ)
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exact this (f z)
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have s₁ : DifferentiableAt ℝ (Complex.im ∘ f) z := by
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apply s₀₁.comp
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exact s₀₀
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have s₂ : lineDeriv ℝ (Complex.imCLM ∘ f) z Complex.I = (fderiv ℝ (Complex.imCLM ∘ f) z) Complex.I := by
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apply DifferentiableAt.lineDeriv_eq_fderiv
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exact s₁
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rw [s₂]
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rw [fderiv.comp, fderiv.comp]
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simp
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rw [CauchyRiemann₁ h]
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constructor
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· rw [t₂, s₂]
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rw [fderiv.comp, fderiv.comp]
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simp
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rw [CauchyRiemann₁ h]
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rw [Complex.I_mul]
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-- DifferentiableAt ℝ Complex.im (f z)
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exact Complex.imCLM.differentiableAt
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have u₁ : ∀ w, fderiv ℝ Complex.reCLM w = Complex.reCLM := by
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exact fun w => ContinuousLinearMap.fderiv Complex.reCLM
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-- DifferentiableAt ℝ f z
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exact t₀₀
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--rw [u₁ (f z)]
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have u₂ : ∀ w, fderiv ℝ Complex.imCLM w = Complex.imCLM := by
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exact fun w => ContinuousLinearMap.fderiv Complex.imCLM
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-- rw [u₂ (f z)]
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rw [Complex.I_mul]
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-- DifferentiableAt ℝ Complex.re (f z)
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exact Complex.reCLM.differentiableAt
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-- DifferentiableAt ℝ f z
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exact t₀₀
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· have r₂ : lineDeriv ℝ (Complex.reCLM ∘ f) z Complex.I = (fderiv ℝ (Complex.reCLM ∘ f) z) Complex.I := by
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apply DifferentiableAt.lineDeriv_eq_fderiv
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exact t₁
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have p₂ : lineDeriv ℝ (Complex.imCLM ∘ f) z 1 = (fderiv ℝ (Complex.imCLM ∘ f) z) 1 := by
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apply DifferentiableAt.lineDeriv_eq_fderiv
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exact s₁
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rw [r₂, p₂]
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rw [fderiv.comp, fderiv.comp]
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simp
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rw [CauchyRiemann₁ h]
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rw [Complex.I_mul]
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-- DifferentiableAt ℝ Complex.im (f z)
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exact Complex.imCLM.differentiableAt
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-- DifferentiableAt ℝ Complex.im (f z)
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exact Complex.imCLM.differentiableAt
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-- DifferentiableAt ℝ f z
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exact t₀₀
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-- DifferentiableAt ℝ Complex.re (f z)
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exact Complex.reCLM.differentiableAt
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-- DifferentiableAt ℝ f z
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exact t₀₀
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-- DifferentiableAt ℝ Complex.re (f z)
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exact Complex.reCLM.differentiableAt
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-- DifferentiableAt ℝ f z
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exact t₀₀
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-- DifferentiableAt ℝ f z
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exact t₀₀
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