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nevanlinna/cauchyRiemann.lean

@@ -1,19 +1,22 @@
-import Mathlib.Analysis.Complex.CauchyIntegral
-import Mathlib.Analysis.Complex.Basic
-import Mathlib.Analysis.Calculus.LineDeriv.Basic
-import Mathlib.Analysis.Calculus.ContDiff.Defs
-import Mathlib.Analysis.Calculus.FDeriv.Basic
-import Mathlib.Analysis.Complex.RealDeriv
-import Mathlib.Analysis.Calculus.ContDiff.Basic
-import Mathlib.Analysis.Calculus.Deriv.Linear
-import Mathlib.Analysis.Complex.Conformal
 import Mathlib.Analysis.Calculus.Conformal.NormedSpace
+import Mathlib.Analysis.Calculus.ContDiff.Basic
+import Mathlib.Analysis.Calculus.ContDiff.Defs
+import Mathlib.Analysis.Calculus.Deriv.Linear
 import Mathlib.Analysis.Calculus.FDeriv.Basic
+import Mathlib.Analysis.Calculus.FDeriv.Comp
+import Mathlib.Analysis.Calculus.FDeriv.Linear
 import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
+import Mathlib.Analysis.Calculus.LineDeriv.Basic
+import Mathlib.Analysis.Complex.Basic
+import Mathlib.Analysis.Complex.CauchyIntegral
+import Mathlib.Analysis.Complex.Conformal
+import Mathlib.Analysis.Complex.RealDeriv
 
-variable {z : ℂ} {f : ℂ → ℂ} {h : DifferentiableAt ℂ f z}
+variable {z : ℂ} {f : ℂ → ℂ}
 
-theorem CauchyRiemann₁ : fderiv ℝ f z Complex.I = Complex.I * fderiv ℝ f z 1 := by
+theorem CauchyRiemann₁ : (DifferentiableAt ℂ f z)
+  → (fderiv ℝ f z) Complex.I = Complex.I * (fderiv ℝ f z) 1 := by
+  intro h
   let A := fderiv ℂ f z
   have t₁ : A (Complex.I • 1) = Complex.I • (A 1) := by
     exact ContinuousLinearMap.map_smul_of_tower A Complex.I 1
@@ -33,3 +36,90 @@ theorem CauchyRiemann₁ : fderiv ℝ f z Complex.I = Complex.I * fderiv ℝ f z
     exact DifferentiableAt.fderiv_restrictScalars ℝ h
   rw [t₃]
   exact t₂c
+
+theorem CauchyRiemann₂ : (DifferentiableAt ℂ f z)
+  → lineDeriv ℝ f z Complex.I = Complex.I * lineDeriv ℝ f z 1 := by
+  intro h
+
+  have : lineDeriv ℝ f z Complex.I = (fderiv ℝ f z) Complex.I := by
+    apply DifferentiableAt.lineDeriv_eq_fderiv
+    apply h.restrictScalars ℝ
+  rw [this]
+
+  have : lineDeriv ℝ f z 1 = (fderiv ℝ f z) 1 := by
+    apply DifferentiableAt.lineDeriv_eq_fderiv
+    apply h.restrictScalars ℝ
+  rw [this]
+
+  exact CauchyRiemann₁ h
+
+theorem CauchyRiemann₃ : (DifferentiableAt ℂ f z)
+  → lineDeriv ℝ (Complex.reCLM ∘ f) z 1 = lineDeriv ℝ (Complex.imCLM ∘ f) z Complex.I := by
+
+  -- Proof starts here
+  intro h
+
+  have t₀₀ : DifferentiableAt ℝ f z := by
+    exact h.restrictScalars ℝ
+
+  have t₀₁ : DifferentiableAt ℝ Complex.reCLM (f z) := by
+    have : ∀ w, DifferentiableAt ℝ Complex.reCLM w := by
+      intro w
+      refine Differentiable.differentiableAt ?h
+      exact ContinuousLinearMap.differentiable (Complex.reCLM : ℂ →L[ℝ] ℝ)
+    exact this (f z)
+
+  have t₁ : DifferentiableAt ℝ (Complex.reCLM ∘ f) z := by
+    apply t₀₁.comp
+    exact t₀₀
+
+  have t₂ : lineDeriv ℝ (Complex.reCLM ∘ f) z 1 = (fderiv ℝ (Complex.reCLM ∘ f) z) 1 := by
+    apply DifferentiableAt.lineDeriv_eq_fderiv
+    exact t₁
+  rw [t₂]
+
+  have s₀₀ : DifferentiableAt ℝ f z := by
+    exact h.restrictScalars ℝ
+
+  have s₀₁ : DifferentiableAt ℝ Complex.imCLM (f z) := by
+    have : ∀ w, DifferentiableAt ℝ Complex.imCLM w := by
+      intro w
+      apply Differentiable.differentiableAt
+      exact ContinuousLinearMap.differentiable (Complex.imCLM : ℂ →L[ℝ] ℝ)
+    exact this (f z)
+
+  have s₁ : DifferentiableAt ℝ (Complex.im ∘ f) z := by
+    apply s₀₁.comp
+    exact s₀₀
+
+  have s₂ : lineDeriv ℝ (Complex.imCLM ∘ f) z Complex.I = (fderiv ℝ (Complex.imCLM ∘ f) z) Complex.I := by
+    apply DifferentiableAt.lineDeriv_eq_fderiv
+    exact s₁
+  rw [s₂]
+
+  rw [fderiv.comp, fderiv.comp]
+  simp
+  rw [CauchyRiemann₁ h]
+
+  have u₁ : ∀ w, fderiv ℝ Complex.reCLM w = Complex.reCLM := by
+    exact fun w => ContinuousLinearMap.fderiv Complex.reCLM
+
+  --rw [u₁ (f z)]
+  have u₂ : ∀ w, fderiv ℝ Complex.imCLM w = Complex.imCLM := by
+    exact fun w => ContinuousLinearMap.fderiv Complex.imCLM
+  -- rw [u₂ (f z)]
+  rw [Complex.I_mul]
+
+
+  -- DifferentiableAt ℝ Complex.im (f z)
+  exact Complex.imCLM.differentiableAt
+
+
+  -- DifferentiableAt ℝ f z
+  exact t₀₀
+
+  -- DifferentiableAt ℝ Complex.re (f z)
+  exact Complex.reCLM.differentiableAt
+
+  -- DifferentiableAt ℝ f z
+  exact t₀₀