Update holomorphic_primitive2.lean
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Stefan Kebekus
parent
103fd5fb0f
commit
c376bd3c46
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@ -393,31 +393,10 @@ theorem primitive_additivity
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unfold primitive
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have integrability₁
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(a₁ a₂ b : ℝ) :
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IntervalIntegrable (fun x => f { re := x, im := b }) MeasureTheory.volume a₁ a₂ := by
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apply ContinuousOn.intervalIntegrable
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apply ContinuousOn.comp
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exact hf.continuousOn
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have : ((fun x => { re := x, im := b }) : ℝ → ℂ) = (fun x => Complex.ofRealCLM x + { re := 0, im := b }) := by
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funext x
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apply Complex.ext
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rw [Complex.add_re]
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simp
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rw [Complex.add_im]
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simp
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apply Continuous.continuousOn
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rw [this]
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continuity
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--
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intro w hw
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simp
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rw [Complex.dist_eq_re_im]
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have : (∫ (x : ℝ) in z₀.re..z.re, f { re := x, im := z₀.im }) = (∫ (x : ℝ) in z₀.re..z₁.re, f { re := x, im := z₀.im }) + (∫ (x : ℝ) in z₁.re..z.re, f { re := x, im := z₀.im }) := by
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rw [intervalIntegral.integral_add_adjacent_intervals]
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-- IntervalIntegrable (fun x => f { re := x, im := z₀.im }) MeasureTheory.volume z₀.re z₁.re
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apply ContinuousOn.intervalIntegrable
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apply ContinuousOn.comp
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@ -432,18 +411,71 @@ theorem primitive_additivity
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apply Continuous.continuousOn
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rw [this]
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continuity
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-- Remains: Set.MapsTo (fun x => { re := x, im := z₀.im }) (Set.uIcc z₀.re z₁.re) (Metric.ball z₀ R)
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-- -- Remains: Set.MapsTo (fun x => { re := x, im := z₀.im }) (Set.uIcc z₀.re z₁.re) (Metric.ball z₀ R)
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intro w hw
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simp
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simp_rw [Complex.dist_of_im_eq]
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calc dist w z₀.re
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_ ≤ dist z₁.re z₀.re := by apply Real.dist_right_le_of_mem_uIcc; rwa [Set.uIcc_comm] at hw
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_ ≤ dist z₁ z₀ := by sorry
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_ < R := by simp at hz₁; assumption
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-- IntervalIntegrable (fun x => f { re := x, im := z₀.im }) MeasureTheory.volume z₁.re z.re
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apply ContinuousOn.intervalIntegrable
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apply ContinuousOn.comp
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exact hf.continuousOn
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have {b : ℝ} : ((fun x => { re := x, im := b }) : ℝ → ℂ) = (fun x => Complex.ofRealCLM x + { re := 0, im := b }) := by
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funext x
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apply Complex.ext
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rw [Complex.add_re]
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simp
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rw [Complex.add_im]
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simp
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apply Continuous.continuousOn
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rw [this]
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continuity
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-- -- Remains: Set.MapsTo (fun x => { re := x, im := z₀.im }) (Set.uIcc z₁.re z.re) (Metric.ball z₀ R)
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intro w hw
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simp
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simp_rw [Complex.dist_of_im_eq]
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calc dist w z₀.re
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_ ≤ dist w z₁.re + dist z₁.re z₀.re := by exact dist_triangle w z₁.re z₀.re
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_ ≤ dist z₁.re z.re + dist z₁.re z₀.re := by
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apply add_le_add_right
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apply Real.dist_le_of_mem_uIcc hw
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simp
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_ ≤ dist z₁ z + dist z₁ z₀ := by
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sorry
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_ < (R - dist z₁ z₀) + dist z₁ z₀ := by
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apply add_lt_add_right
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simp at hz
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rwa [dist_comm]
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_ = R := by simp
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rw [this]
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have : (∫ (x : ℝ) in z₀.im..z.im, f { re := z.re, im := x }) = (∫ (x : ℝ) in z₀.im..z₁.im, f { re := z.re, im := x }) + (∫ (x : ℝ) in z₁.im..z.im, f { re := z.re, im := x }) := by
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rw [intervalIntegral.integral_add_adjacent_intervals]
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-- IntervalIntegrable (fun x => f { re := z.re, im := x }) MeasureTheory.volume z₀.im z₁.im
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apply ContinuousOn.intervalIntegrable
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apply ContinuousOn.comp
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exact hf.continuousOn
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apply Continuous.continuousOn
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have {b : ℝ}: (Complex.mk b) = (fun x => Complex.I • Complex.ofRealCLM x + { re := b, im := 0 }) := by
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funext x
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apply Complex.ext
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rw [Complex.add_re]
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simp
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simp
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rw [this]
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apply Continuous.add
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fun_prop
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fun_prop
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-- Set.MapsTo (Complex.mk z.re) (Set.uIcc z₀.im z₁.im) (Metric.ball z₀ R)
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intro w hw
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simp
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sorry --apply integrability₁ f hf
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sorry --apply integrability₁ f hf
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rw [this]
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have : (∫ (x : ℝ) in z₀.im..z.im, f { re := z.re, im := x }) = (∫ (x : ℝ) in z₀.im..z₁.im, f { re := z.re, im := x }) + (∫ (x : ℝ) in z₁.im..z.im, f { re := z.re, im := x }) := by
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rw [intervalIntegral.integral_add_adjacent_intervals]
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sorry --apply integrability₂ f hf
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sorry --apply integrability₂ f hf
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rw [this]
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simp
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@ -451,22 +483,16 @@ theorem primitive_additivity
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have {a b c d e f g h : E} : (a + b) + (c + d) - (e + f) - (g + h) = b + (a - g) - e - f + d - h + (c) := by
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abel
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rw [this]
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simp
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have H : DifferentiableOn ℂ f (Set.uIcc z₁.re z.re ×ℂ Set.uIcc z₀.im z₁.im) := by
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have H' : DifferentiableOn ℂ f (Set.uIcc z₁.re z.re ×ℂ Set.uIcc z₀.im z₁.im) := by
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apply DifferentiableOn.mono hf
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intro x hx
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simp
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let A₀ : dist x.re z₀.re ≤ dist x.re z₁.re + dist z₁.re z₀.re := by apply dist_triangle
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let A₁ : dist x.im z₀.im ≤ dist z₁.im z₀.im := by sorry
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exact H hx
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sorry
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let A := Complex.integral_boundary_rect_eq_zero_of_differentiableOn f ⟨z₁.re, z₀.im⟩ ⟨z.re, z₁.im⟩ H
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let A := Complex.integral_boundary_rect_eq_zero_of_differentiableOn f ⟨z₁.re, z₀.im⟩ ⟨z.re, z₁.im⟩ H'
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have {x : ℝ} {w : ℂ} : ↑x + w.im * Complex.I = { re := x, im := w.im } := by
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apply Complex.ext
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· simp
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