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Update cauchyRiemann.lean

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Stefan Kebekus 2024-04-29 14:19:58 +02:00
parent 77d1fd2ae4
commit 38eaf677f9
1 changed files with 47 additions and 44 deletions
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nevanlinna/cauchyRiemann.lean
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@ -1,15 +1,15 @@
import Mathlib.Analysis.Calculus.Conformal.NormedSpace --import Mathlib.Analysis.Calculus.Conformal.NormedSpace
import Mathlib.Analysis.Calculus.ContDiff.Basic --import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs --import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.Deriv.Linear --import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Basic --import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp --import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.Linear --import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars --import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
import Mathlib.Analysis.Calculus.LineDeriv.Basic import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Complex.Basic --import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.CauchyIntegral --import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.Complex.Conformal --import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Complex.RealDeriv import Mathlib.Analysis.Complex.RealDeriv
variable {z : ℂ} {f : ℂ → ℂ} variable {z : ℂ} {f : ℂ → ℂ}
@ -26,12 +26,9 @@ theorem CauchyRiemann₁ : (DifferentiableAt ℂ f z)
have t₂a : Complex.I * (AR 1) = Complex.I • (AR 1) := by have t₂a : Complex.I * (AR 1) = Complex.I • (AR 1) := by
exact rfl exact rfl
rw [← t₂a] at t₂ rw [← t₂a] at t₂
have t₂b : Complex.I • 1 = Complex.I := by
simp
have t₂c : AR Complex.I = Complex.I • (AR 1) := by have t₂c : AR Complex.I = Complex.I • (AR 1) := by
simp at t₂ simp at t₂
exact t₂ exact t₂
let C : HasFDerivAt f (ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z)) z := h.hasFDerivAt.restrictScalars ℝ
have t₃ : fderiv ℝ f z = ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z) := by have t₃ : fderiv ℝ f z = ContinuousLinearMap.restrictScalars ℝ (fderiv ℂ f z) := by
exact DifferentiableAt.fderiv_restrictScalars ℝ h exact DifferentiableAt.fderiv_restrictScalars ℝ h
rw [t₃] rw [t₃]
@ -54,72 +51,78 @@ theorem CauchyRiemann₂ : (DifferentiableAt ℂ f z)
exact CauchyRiemann₁ h exact CauchyRiemann₁ h
theorem CauchyRiemann₃ : (DifferentiableAt ℂ f z) theorem CauchyRiemann₃ : (DifferentiableAt ℂ f z)
→ lineDeriv ℝ (Complex.reCLM ∘ f) z 1 = lineDeriv ℝ (Complex.imCLM ∘ f) z Complex.I := by → (lineDeriv ℝ (Complex.reCLM ∘ f) z 1 = lineDeriv ℝ (Complex.imCLM ∘ f) z Complex.I)
∧ (lineDeriv ℝ (Complex.reCLM ∘ f) z Complex.I = -lineDeriv ℝ (Complex.imCLM ∘ f) z 1)
:= by
-- Proof starts here -- Proof starts here
intro h intro h
have t₀₀ : DifferentiableAt ℝ f z := by have t₀₀ : DifferentiableAt ℝ f z := by
exact h.restrictScalars ℝ exact h.restrictScalars ℝ
have t₀₁ : DifferentiableAt ℝ Complex.reCLM (f z) := by have t₀₁ : DifferentiableAt ℝ Complex.reCLM (f z) := by
have : ∀ w, DifferentiableAt ℝ Complex.reCLM w := by have : ∀ w, DifferentiableAt ℝ Complex.reCLM w := by
intro w intro w
refine Differentiable.differentiableAt ?h refine Differentiable.differentiableAt ?h
exact ContinuousLinearMap.differentiable (Complex.reCLM : ℂ →L[ℝ] ℝ) exact ContinuousLinearMap.differentiable (Complex.reCLM : ℂ →L[ℝ] ℝ)
exact this (f z) exact this (f z)
have t₁ : DifferentiableAt ℝ (Complex.reCLM ∘ f) z := by have t₁ : DifferentiableAt ℝ (Complex.reCLM ∘ f) z := by
apply t₀₁.comp apply t₀₁.comp
exact t₀₀ exact t₀₀
have t₂ : lineDeriv ℝ (Complex.reCLM ∘ f) z 1 = (fderiv ℝ (Complex.reCLM ∘ f) z) 1 := by have t₂ : lineDeriv ℝ (Complex.reCLM ∘ f) z 1 = (fderiv ℝ (Complex.reCLM ∘ f) z) 1 := by
apply DifferentiableAt.lineDeriv_eq_fderiv apply DifferentiableAt.lineDeriv_eq_fderiv
exact t₁ exact t₁
rw [t₂]
have s₀₀ : DifferentiableAt ℝ f z := by have s₀₀ : DifferentiableAt ℝ f z := by
exact h.restrictScalars ℝ exact h.restrictScalars ℝ
have s₀₁ : DifferentiableAt ℝ Complex.imCLM (f z) := by have s₀₁ : DifferentiableAt ℝ Complex.imCLM (f z) := by
have : ∀ w, DifferentiableAt ℝ Complex.imCLM w := by have : ∀ w, DifferentiableAt ℝ Complex.imCLM w := by
intro w intro w
apply Differentiable.differentiableAt apply Differentiable.differentiableAt
exact ContinuousLinearMap.differentiable (Complex.imCLM : ℂ →L[ℝ] ℝ) exact ContinuousLinearMap.differentiable (Complex.imCLM : ℂ →L[ℝ] ℝ)
exact this (f z) exact this (f z)
have s₁ : DifferentiableAt ℝ (Complex.im ∘ f) z := by have s₁ : DifferentiableAt ℝ (Complex.im ∘ f) z := by
apply s₀₁.comp apply s₀₁.comp
exact s₀₀ exact s₀₀
have s₂ : lineDeriv ℝ (Complex.imCLM ∘ f) z Complex.I = (fderiv ℝ (Complex.imCLM ∘ f) z) Complex.I := by have s₂ : lineDeriv ℝ (Complex.imCLM ∘ f) z Complex.I = (fderiv ℝ (Complex.imCLM ∘ f) z) Complex.I := by
apply DifferentiableAt.lineDeriv_eq_fderiv apply DifferentiableAt.lineDeriv_eq_fderiv
exact s₁ exact s₁
rw [s₂]
rw [fderiv.comp, fderiv.comp] constructor
simp · rw [t₂, s₂]
rw [CauchyRiemann₁ h] rw [fderiv.comp, fderiv.comp]
simp
rw [CauchyRiemann₁ h]
rw [Complex.I_mul]
-- DifferentiableAt ℝ Complex.im (f z)
exact Complex.imCLM.differentiableAt
have u₁ : ∀ w, fderiv ℝ Complex.reCLM w = Complex.reCLM := by -- DifferentiableAt ℝ f z
exact fun w => ContinuousLinearMap.fderiv Complex.reCLM exact t₀₀
--rw [u₁ (f z)] -- DifferentiableAt ℝ Complex.re (f z)
have u₂ : ∀ w, fderiv ℝ Complex.imCLM w = Complex.imCLM := by exact Complex.reCLM.differentiableAt
exact fun w => ContinuousLinearMap.fderiv Complex.imCLM
-- rw [u₂ (f z)]
rw [Complex.I_mul]
-- DifferentiableAt ℝ f z
exact t₀₀
· have r₂ : lineDeriv ℝ (Complex.reCLM ∘ f) z Complex.I = (fderiv ℝ (Complex.reCLM ∘ f) z) Complex.I := by
apply DifferentiableAt.lineDeriv_eq_fderiv
exact t₁
have p₂ : lineDeriv ℝ (Complex.imCLM ∘ f) z 1 = (fderiv ℝ (Complex.imCLM ∘ f) z) 1 := by
apply DifferentiableAt.lineDeriv_eq_fderiv
exact s₁
rw [r₂, p₂]
rw [fderiv.comp, fderiv.comp]
simp
rw [CauchyRiemann₁ h]
rw [Complex.I_mul]
-- DifferentiableAt ℝ Complex.im (f z)
exact Complex.imCLM.differentiableAt
-- DifferentiableAt ℝ Complex.im (f z) -- DifferentiableAt ℝ f z
exact Complex.imCLM.differentiableAt exact t₀₀
-- DifferentiableAt ℝ Complex.re (f z)
exact Complex.reCLM.differentiableAt
-- DifferentiableAt ℝ f z -- DifferentiableAt ℝ f z
exact t₀₀ exact t₀₀
-- DifferentiableAt ℝ Complex.re (f z)
exact Complex.reCLM.differentiableAt
-- DifferentiableAt ℝ f z
exact t₀₀