working
This commit is contained in:
Stefan Kebekus
parent
6a12258093
commit
75ce3b31ef
File diff suppressed because it is too large
Load Diff
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@ -1,846 +0,0 @@
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import Mathlib.Analysis.Complex.TaylorSeries
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import Mathlib.Data.ENNReal.Basic
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noncomputable def primitive
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] :
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ℂ → (ℂ → E) → (ℂ → E) := by
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intro z₀
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intro f
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exact fun z ↦ (∫ (x : ℝ) in z₀.re..z.re, f ⟨x, z₀.im⟩) + Complex.I • ∫ (x : ℝ) in z₀.im..z.im, f ⟨z.re, x⟩
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theorem primitive_zeroAtBasepoint
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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(f : ℂ → E)
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(z₀ : ℂ) :
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(primitive z₀ f) z₀ = 0 := by
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unfold primitive
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simp
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theorem primitive_fderivAtBasepointZero
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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{f : ℂ → E}
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{R : ℝ}
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(hR : 0 < R)
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(hf : ContinuousOn f (Metric.ball 0 R)) :
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HasDerivAt (primitive 0 f) (f 0) 0 := by
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unfold primitive
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simp
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apply hasDerivAt_iff_isLittleO.2
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simp
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rw [Asymptotics.isLittleO_iff]
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intro c hc
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have {z : ℂ} {e : E} : z • e = (∫ (_ : ℝ) in (0)..(z.re), e) + Complex.I • ∫ (_ : ℝ) in (0)..(z.im), e:= by
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simp
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rw [smul_comm]
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rw [← smul_assoc]
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simp
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have : z.re • e = (z.re : ℂ) • e := by exact rfl
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rw [this, ← add_smul]
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simp
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conv =>
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left
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intro x
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left
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arg 1
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arg 2
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rw [this]
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obtain ⟨s, h₁s, h₂s⟩ : ∃ s ⊆ f⁻¹' Metric.ball (f 0) (c / (4 : ℝ)), IsOpen s ∧ 0 ∈ s := by
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apply eventually_nhds_iff.mp
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apply continuousAt_def.1
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apply Continuous.continuousAt
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fun_prop
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apply continuousAt_def.1
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apply hf.continuousAt
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exact Metric.ball_mem_nhds 0 hR
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apply Metric.ball_mem_nhds (f 0)
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simpa
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obtain ⟨ε, h₁ε, h₂ε⟩ : ∃ ε > 0, (Metric.ball 0 ε) ×ℂ (Metric.ball 0 ε) ⊆ s ∧ (Metric.ball 0 ε) ×ℂ (Metric.ball 0 ε) ⊆ Metric.ball 0 R := by
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obtain ⟨ε', h₁ε', h₂ε'⟩ : ∃ ε' > 0, Metric.ball 0 ε' ⊆ s ∩ Metric.ball 0 R := by
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apply Metric.mem_nhds_iff.mp
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apply IsOpen.mem_nhds
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apply IsOpen.inter
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exact h₂s.1
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exact Metric.isOpen_ball
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constructor
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· exact h₂s.2
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· simpa
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use (2 : ℝ)⁻¹ * ε'
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constructor
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· simpa
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· constructor
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· intro x hx
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apply (h₂ε' _).1
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simp
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calc Complex.abs x
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_ ≤ |x.re| + |x.im| := Complex.abs_le_abs_re_add_abs_im x
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_ < (2 : ℝ)⁻¹ * ε' + |x.im| := by
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apply (add_lt_add_iff_right |x.im|).mpr
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have : x.re ∈ Metric.ball 0 (2⁻¹ * ε') := (Complex.mem_reProdIm.1 hx).1
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simp at this
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exact this
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_ < (2 : ℝ)⁻¹ * ε' + (2 : ℝ)⁻¹ * ε' := by
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apply (add_lt_add_iff_left ((2 : ℝ)⁻¹ * ε')).mpr
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have : x.im ∈ Metric.ball 0 (2⁻¹ * ε') := (Complex.mem_reProdIm.1 hx).2
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simp at this
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exact this
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_ = ε' := by
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rw [← add_mul]
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abel_nf
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simp
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· intro x hx
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apply (h₂ε' _).2
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simp
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calc Complex.abs x
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_ ≤ |x.re| + |x.im| := Complex.abs_le_abs_re_add_abs_im x
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_ < (2 : ℝ)⁻¹ * ε' + |x.im| := by
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apply (add_lt_add_iff_right |x.im|).mpr
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have : x.re ∈ Metric.ball 0 (2⁻¹ * ε') := (Complex.mem_reProdIm.1 hx).1
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simp at this
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exact this
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_ < (2 : ℝ)⁻¹ * ε' + (2 : ℝ)⁻¹ * ε' := by
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apply (add_lt_add_iff_left ((2 : ℝ)⁻¹ * ε')).mpr
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have : x.im ∈ Metric.ball 0 (2⁻¹ * ε') := (Complex.mem_reProdIm.1 hx).2
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simp at this
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exact this
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_ = ε' := by
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rw [← add_mul]
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abel_nf
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simp
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have h₃ε : ∀ y ∈ (Metric.ball 0 ε) ×ℂ (Metric.ball 0 ε), ‖(f y) - (f 0)‖ < (c / (4 : ℝ)) := by
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intro y hy
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apply mem_ball_iff_norm.mp
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apply h₁s
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exact h₂ε.1 hy
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have intervalComputation_uIcc {x' y' : ℝ} (h : x' ∈ Set.uIcc 0 y') : |x'| ≤ |y'| := by
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let A := h.1
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let B := h.2
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rcases le_total 0 y' with hy | hy
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· simp [hy] at A
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simp [hy] at B
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rwa [abs_of_nonneg A, abs_of_nonneg hy]
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· simp [hy] at A
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simp [hy] at B
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rw [abs_of_nonpos hy]
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rw [abs_of_nonpos]
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linarith [h.1]
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exact B
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rw [Filter.eventually_iff_exists_mem]
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use Metric.ball 0 (ε / (4 : ℝ))
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constructor
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· apply Metric.ball_mem_nhds 0
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linarith
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· intro y hy
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have {A B C D :E} : (A + B) - (C + D) = (A - C) + (B - D) := by
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abel
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rw [this]
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rw [← smul_sub]
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have t₀ : IntervalIntegrable (fun x => f { re := x, im := 0 }) MeasureTheory.volume 0 y.re := by
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apply ContinuousOn.intervalIntegrable
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apply ContinuousOn.comp
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exact hf
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have : (fun x => ({ re := x, im := 0 } : ℂ)) = Complex.ofRealLI := by rfl
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rw [this]
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apply Continuous.continuousOn
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continuity
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intro x hx
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apply h₂ε.2
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simp
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constructor
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· simp
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calc |x|
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_ ≤ |y.re| := by apply intervalComputation_uIcc hx
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_ ≤ Complex.abs y := by exact Complex.abs_re_le_abs y
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_ < ε / 4 := by simp at hy; assumption
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_ < ε := by linarith
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· simpa
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have t₁ : IntervalIntegrable (fun _ => f 0) MeasureTheory.volume 0 y.re := by
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apply ContinuousOn.intervalIntegrable
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apply ContinuousOn.comp
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apply hf
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fun_prop
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intro x _
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simpa
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rw [← intervalIntegral.integral_sub t₀ t₁]
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have t₂ : IntervalIntegrable (fun x_1 => f { re := y.re, im := x_1 }) MeasureTheory.volume 0 y.im := by
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apply ContinuousOn.intervalIntegrable
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apply ContinuousOn.comp
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exact hf
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have : (Complex.mk y.re) = (fun x => Complex.I • Complex.ofRealCLM x + { re := y.re, 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 ContinuousOn.add
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apply Continuous.continuousOn
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continuity
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fun_prop
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intro x hx
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apply h₂ε.2
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constructor
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· simp
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calc |y.re|
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_ ≤ Complex.abs y := by exact Complex.abs_re_le_abs y
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_ < ε / 4 := by simp at hy; assumption
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_ < ε := by linarith
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· simp
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calc |x|
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_ ≤ |y.im| := by apply intervalComputation_uIcc hx
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_ ≤ Complex.abs y := by exact Complex.abs_im_le_abs y
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_ < ε / 4 := by simp at hy; assumption
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_ < ε := by linarith
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have t₃ : IntervalIntegrable (fun _ => f 0) MeasureTheory.volume 0 y.im := by
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apply ContinuousOn.intervalIntegrable
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apply ContinuousOn.comp
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exact hf
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fun_prop
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intro x _
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apply h₂ε.2
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simp
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constructor
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· simpa
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· simpa
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rw [← intervalIntegral.integral_sub t₂ t₃]
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have h₁y : |y.re| < ε / 4 := by
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calc |y.re|
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_ ≤ Complex.abs y := by apply Complex.abs_re_le_abs
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_ < ε / 4 := by
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let A := mem_ball_iff_norm.1 hy
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simp at A
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linarith
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have h₂y : |y.im| < ε / 4 := by
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calc |y.im|
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_ ≤ Complex.abs y := by apply Complex.abs_im_le_abs
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_ < ε / 4 := by
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let A := mem_ball_iff_norm.1 hy
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simp at A
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linarith
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have intervalComputation {x' y' : ℝ} (h : x' ∈ Ι 0 y') : |x'| ≤ |y'| := by
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let A := h.1
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let B := h.2
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rcases le_total 0 y' with hy | hy
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· simp [hy] at A
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simp [hy] at B
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rw [abs_of_nonneg hy]
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rw [abs_of_nonneg (le_of_lt A)]
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exact B
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· simp [hy] at A
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simp [hy] at B
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rw [abs_of_nonpos hy]
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rw [abs_of_nonpos]
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linarith [h.1]
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exact B
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have t₁ : ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ ≤ (c / (4 : ℝ)) * |y.re - 0| := by
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apply intervalIntegral.norm_integral_le_of_norm_le_const
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intro x hx
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have h₁x : |x| < ε / 4 := by
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calc |x|
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_ ≤ |y.re| := intervalComputation hx
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_ < ε / 4 := h₁y
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apply le_of_lt
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apply h₃ε { re := x, im := 0 }
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constructor
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· simp
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linarith
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· simp
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exact h₁ε
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have t₂ : ‖∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ ≤ (c / (4 : ℝ)) * |y.im - 0| := by
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apply intervalIntegral.norm_integral_le_of_norm_le_const
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intro x hx
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have h₁x : |x| < ε / 4 := by
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calc |x|
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_ ≤ |y.im| := intervalComputation hx
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_ < ε / 4 := h₂y
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apply le_of_lt
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apply h₃ε { re := y.re, im := x }
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constructor
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· simp
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linarith
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· simp
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linarith
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calc ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0) + Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖
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_ ≤ ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by
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apply norm_add_le
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_ ≤ ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by
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simp
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rw [norm_smul]
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simp
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_ ≤ (c / (4 : ℝ)) * |y.re - 0| + (c / (4 : ℝ)) * |y.im - 0| := by
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apply add_le_add
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exact t₁
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exact t₂
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_ ≤ (c / (4 : ℝ)) * (|y.re| + |y.im|) := by
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simp
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rw [mul_add]
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_ ≤ (c / (4 : ℝ)) * (4 * ‖y‖) := by
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have : |y.re| + |y.im| ≤ 4 * ‖y‖ := by
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calc |y.re| + |y.im|
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_ ≤ ‖y‖ + ‖y‖ := by
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apply add_le_add
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apply Complex.abs_re_le_abs
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apply Complex.abs_im_le_abs
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_ ≤ 4 * ‖y‖ := by
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rw [← two_mul]
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apply mul_le_mul
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linarith
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rfl
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exact norm_nonneg y
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linarith
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apply mul_le_mul
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rfl
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exact this
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apply add_nonneg
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apply abs_nonneg
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apply abs_nonneg
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linarith
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_ ≤ c * ‖y‖ := by
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linarith
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theorem primitive_translation
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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(f : ℂ → E)
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(z₀ t : ℂ) :
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primitive z₀ (f ∘ fun z ↦ (z - t)) = ((primitive (z₀ - t) f) ∘ fun z ↦ (z - t)) := by
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funext z
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unfold primitive
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simp
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let g : ℝ → E := fun x ↦ f ( {re := x, im := z₀.im - t.im} )
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have {x : ℝ} : f ({ re := x, im := z₀.im } - t) = g (1*x - t.re) := by
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congr 1
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apply Complex.ext <;> simp
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conv =>
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left
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left
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arg 1
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intro x
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rw [this]
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rw [intervalIntegral.integral_comp_mul_sub g one_ne_zero (t.re)]
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simp
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congr 1
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let g : ℝ → E := fun x ↦ f ( {re := z.re - t.re, im := x} )
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have {x : ℝ} : f ({ re := z.re, im := x} - t) = g (1*x - t.im) := by
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congr 1
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apply Complex.ext <;> simp
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conv =>
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left
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arg 1
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intro x
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rw [this]
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rw [intervalIntegral.integral_comp_mul_sub g one_ne_zero (t.im)]
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simp
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theorem primitive_hasDerivAtBasepoint
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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{f : ℂ → E}
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{R : ℝ}
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(z₀ : ℂ)
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(hR : 0 < R)
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(hf : ContinuousOn f (Metric.ball z₀ R)) :
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HasDerivAt (primitive z₀ f) (f z₀) z₀ := by
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let g := f ∘ fun z ↦ z + z₀
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have hg : ContinuousOn g (Metric.ball 0 R) := by
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apply ContinuousOn.comp
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fun_prop
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fun_prop
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intro x hx
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simp
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simp at hx
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assumption
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let B := primitive_translation g z₀ z₀
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simp at B
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have : (g ∘ fun z ↦ (z - z₀)) = f := by
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funext z
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dsimp [g]
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simp
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rw [this] at B
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rw [B]
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have : f z₀ = (1 : ℂ) • (f z₀) := (MulAction.one_smul (f z₀)).symm
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conv =>
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arg 2
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rw [this]
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apply HasDerivAt.scomp
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simp
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have : g 0 = f z₀ := by simp [g]
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rw [← this]
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exact primitive_fderivAtBasepointZero hR hg
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apply HasDerivAt.sub_const
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have : (fun (x : ℂ) ↦ x) = id := by
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funext x
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simp
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rw [this]
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exact hasDerivAt_id z₀
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theorem primitive_additivity
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{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
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{f : ℂ → E}
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{z₀ : ℂ}
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{rx ry : ℝ}
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(hf : DifferentiableOn ℂ f (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry))
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(hry : 0 < ry)
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{z₁ : ℂ}
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(hz₁ : z₁ ∈ (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry))
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:
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∃ εx > 0, ∃ εy > 0, ∀ z ∈ (Metric.ball z₁.re εx ×ℂ Metric.ball z₁.im εy), (primitive z₀ f z) - (primitive z₁ f z) - (primitive z₀ f z₁) = 0 := by
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let εx := rx - dist z₀.re z₁.re
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have hεx : εx > 0 := by
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let A := hz₁.1
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simp at A
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dsimp [εx]
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rw [dist_comm]
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||||
simpa
|
||||
let εy := ry - dist z₀.im z₁.im
|
||||
have hεy : εy > 0 := by
|
||||
let A := hz₁.2
|
||||
simp at A
|
||||
dsimp [εy]
|
||||
rw [dist_comm]
|
||||
simpa
|
||||
|
||||
use εx
|
||||
use hεx
|
||||
use εy
|
||||
use hεy
|
||||
|
||||
intro z hz
|
||||
|
||||
unfold primitive
|
||||
|
||||
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
|
||||
|
||||
rw [intervalIntegral.integral_add_adjacent_intervals]
|
||||
|
||||
-- IntervalIntegrable (fun x => f { re := x, im := z₀.im }) MeasureTheory.volume z₀.re z₁.re
|
||||
apply ContinuousOn.intervalIntegrable
|
||||
apply ContinuousOn.comp
|
||||
exact hf.continuousOn
|
||||
have {b : ℝ} : ((fun x => { re := x, im := b }) : ℝ → ℂ) = (fun x => Complex.ofRealCLM x + { re := 0, im := b }) := by
|
||||
funext x
|
||||
apply Complex.ext
|
||||
rw [Complex.add_re]
|
||||
simp
|
||||
rw [Complex.add_im]
|
||||
simp
|
||||
apply Continuous.continuousOn
|
||||
rw [this]
|
||||
continuity
|
||||
-- Remains: Set.MapsTo (fun x => { re := x, im := z₀.im }) (Set.uIcc z₀.re z₁.re) (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry)
|
||||
intro w hw
|
||||
simp
|
||||
apply Complex.mem_reProdIm.mpr
|
||||
constructor
|
||||
· simp
|
||||
calc dist w z₀.re
|
||||
_ ≤ dist z₁.re z₀.re := by apply Real.dist_right_le_of_mem_uIcc; rwa [Set.uIcc_comm] at hw
|
||||
_ < rx := by apply Metric.mem_ball.mp (Complex.mem_reProdIm.1 hz₁).1
|
||||
· simpa
|
||||
|
||||
-- IntervalIntegrable (fun x => f { re := x, im := z₀.im }) MeasureTheory.volume z₁.re z.re
|
||||
apply ContinuousOn.intervalIntegrable
|
||||
apply ContinuousOn.comp
|
||||
exact hf.continuousOn
|
||||
have {b : ℝ} : ((fun x => { re := x, im := b }) : ℝ → ℂ) = (fun x => Complex.ofRealCLM x + { re := 0, im := b }) := by
|
||||
funext x
|
||||
apply Complex.ext
|
||||
rw [Complex.add_re]
|
||||
simp
|
||||
rw [Complex.add_im]
|
||||
simp
|
||||
apply Continuous.continuousOn
|
||||
rw [this]
|
||||
continuity
|
||||
-- Remains: Set.MapsTo (fun x => { re := x, im := z₀.im }) (Set.uIcc z₁.re z.re) (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry)
|
||||
intro w hw
|
||||
simp
|
||||
constructor
|
||||
· simp
|
||||
calc dist w z₀.re
|
||||
_ ≤ dist w z₁.re + dist z₁.re z₀.re := by exact dist_triangle w z₁.re z₀.re
|
||||
_ ≤ dist z.re z₁.re + dist z₁.re z₀.re := by
|
||||
apply (add_le_add_iff_right (dist z₁.re z₀.re)).mpr
|
||||
rw [Set.uIcc_comm] at hw
|
||||
exact Real.dist_right_le_of_mem_uIcc hw
|
||||
_ < (rx - dist z₀.re z₁.re) + dist z₁.re z₀.re := by
|
||||
apply (add_lt_add_iff_right (dist z₁.re z₀.re)).mpr
|
||||
apply Metric.mem_ball.1 (Complex.mem_reProdIm.1 hz).1
|
||||
_ = rx := by
|
||||
rw [dist_comm]
|
||||
simp
|
||||
· simpa
|
||||
rw [this]
|
||||
|
||||
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
|
||||
rw [intervalIntegral.integral_add_adjacent_intervals]
|
||||
|
||||
-- IntervalIntegrable (fun x => f { re := z.re, im := x }) MeasureTheory.volume z₀.im z₁.im
|
||||
apply ContinuousOn.intervalIntegrable
|
||||
apply ContinuousOn.comp
|
||||
exact hf.continuousOn
|
||||
apply Continuous.continuousOn
|
||||
have {b : ℝ}: (Complex.mk b) = (fun x => Complex.I • Complex.ofRealCLM x + { re := b, im := 0 }) := by
|
||||
funext x
|
||||
apply Complex.ext
|
||||
rw [Complex.add_re]
|
||||
simp
|
||||
simp
|
||||
rw [this]
|
||||
apply Continuous.add
|
||||
fun_prop
|
||||
fun_prop
|
||||
-- Set.MapsTo (Complex.mk z.re) (Set.uIcc z₀.im z₁.im) (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry)
|
||||
intro w hw
|
||||
constructor
|
||||
· simp
|
||||
calc dist z.re z₀.re
|
||||
_ ≤ dist z.re z₁.re + dist z₁.re z₀.re := by exact dist_triangle z.re z₁.re z₀.re
|
||||
_ < (rx - dist z₀.re z₁.re) + dist z₁.re z₀.re := by
|
||||
apply (add_lt_add_iff_right (dist z₁.re z₀.re)).mpr
|
||||
apply Metric.mem_ball.1 (Complex.mem_reProdIm.1 hz).1
|
||||
_ = rx := by
|
||||
rw [dist_comm]
|
||||
simp
|
||||
· simp
|
||||
calc dist w z₀.im
|
||||
_ ≤ dist z₁.im z₀.im := by rw [Set.uIcc_comm] at hw; exact Real.dist_right_le_of_mem_uIcc hw
|
||||
_ < ry := by
|
||||
rw [← Metric.mem_ball]
|
||||
exact hz₁.2
|
||||
|
||||
-- IntervalIntegrable (fun x => f { re := z.re, im := x }) MeasureTheory.volume z₁.im z.im
|
||||
apply ContinuousOn.intervalIntegrable
|
||||
apply ContinuousOn.comp
|
||||
exact hf.continuousOn
|
||||
apply Continuous.continuousOn
|
||||
have {b : ℝ}: (Complex.mk b) = (fun x => Complex.I • Complex.ofRealCLM x + { re := b, im := 0 }) := by
|
||||
funext x
|
||||
apply Complex.ext
|
||||
rw [Complex.add_re]
|
||||
simp
|
||||
simp
|
||||
rw [this]
|
||||
apply Continuous.add
|
||||
fun_prop
|
||||
fun_prop
|
||||
-- Set.MapsTo (Complex.mk z.re) (Set.uIcc z₁.im z.im) (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry)
|
||||
intro w hw
|
||||
constructor
|
||||
· simp
|
||||
calc dist z.re z₀.re
|
||||
_ ≤ dist z.re z₁.re + dist z₁.re z₀.re := by exact dist_triangle z.re z₁.re z₀.re
|
||||
_ < (rx - dist z₀.re z₁.re) + dist z₁.re z₀.re := by
|
||||
apply (add_lt_add_iff_right (dist z₁.re z₀.re)).mpr
|
||||
apply Metric.mem_ball.1 (Complex.mem_reProdIm.1 hz).1
|
||||
_ = rx := by
|
||||
rw [dist_comm]
|
||||
simp
|
||||
· simp
|
||||
calc dist w z₀.im
|
||||
_ ≤ dist w z₁.im + dist z₁.im z₀.im := by exact dist_triangle w z₁.im z₀.im
|
||||
_ ≤ dist z.im z₁.im + dist z₁.im z₀.im := by
|
||||
apply (add_le_add_iff_right (dist z₁.im z₀.im)).mpr
|
||||
rw [Set.uIcc_comm] at hw
|
||||
exact Real.dist_right_le_of_mem_uIcc hw
|
||||
_ < (ry - dist z₀.im z₁.im) + dist z₁.im z₀.im := by
|
||||
apply (add_lt_add_iff_right (dist z₁.im z₀.im)).mpr
|
||||
apply Metric.mem_ball.1 (Complex.mem_reProdIm.1 hz).2
|
||||
_ = ry := by
|
||||
rw [dist_comm]
|
||||
simp
|
||||
rw [this]
|
||||
|
||||
simp
|
||||
|
||||
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
|
||||
abel
|
||||
rw [this]
|
||||
|
||||
|
||||
have H' : DifferentiableOn ℂ f (Set.uIcc z₁.re z.re ×ℂ Set.uIcc z₀.im z₁.im) := by
|
||||
apply DifferentiableOn.mono hf
|
||||
intro x hx
|
||||
constructor
|
||||
· simp
|
||||
calc dist x.re z₀.re
|
||||
_ ≤ dist x.re z₁.re + dist z₁.re z₀.re := by exact dist_triangle x.re z₁.re z₀.re
|
||||
_ ≤ dist z.re z₁.re + dist z₁.re z₀.re := by
|
||||
apply (add_le_add_iff_right (dist z₁.re z₀.re)).mpr
|
||||
rw [Set.uIcc_comm] at hx
|
||||
apply Real.dist_right_le_of_mem_uIcc (Complex.mem_reProdIm.1 hx).1
|
||||
_ < (rx - dist z₀.re z₁.re) + dist z₁.re z₀.re := by
|
||||
apply (add_lt_add_iff_right (dist z₁.re z₀.re)).mpr
|
||||
apply Metric.mem_ball.1 (Complex.mem_reProdIm.1 hz).1
|
||||
_ = rx := by
|
||||
rw [dist_comm]
|
||||
simp
|
||||
· simp
|
||||
calc dist x.im z₀.im
|
||||
_ ≤ dist z₀.im z₁.im := by rw [dist_comm]; exact Real.dist_left_le_of_mem_uIcc (Complex.mem_reProdIm.1 hx).2
|
||||
_ < ry := by
|
||||
rw [dist_comm]
|
||||
exact Metric.mem_ball.1 (Complex.mem_reProdIm.1 hz₁).2
|
||||
|
||||
let A := Complex.integral_boundary_rect_eq_zero_of_differentiableOn f ⟨z₁.re, z₀.im⟩ ⟨z.re, z₁.im⟩ H'
|
||||
have {x : ℝ} {w : ℂ} : ↑x + w.im * Complex.I = { re := x, im := w.im } := by
|
||||
apply Complex.ext
|
||||
· simp
|
||||
· simp
|
||||
simp_rw [this] at A
|
||||
have {x : ℝ} {w : ℂ} : w.re + x * Complex.I = { re := w.re, im := x } := by
|
||||
apply Complex.ext
|
||||
· simp
|
||||
· simp
|
||||
simp_rw [this] at A
|
||||
rw [← A]
|
||||
abel
|
||||
|
||||
|
||||
theorem primitive_additivity'
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
{z₀ z₁ : ℂ}
|
||||
{R : ℝ}
|
||||
(hf : DifferentiableOn ℂ f (Metric.ball z₀ R))
|
||||
(hz₁ : z₁ ∈ Metric.ball z₀ R)
|
||||
:
|
||||
primitive z₀ f =ᶠ[nhds z₁] fun z ↦ (primitive z₁ f z) + (primitive z₀ f z₁) := by
|
||||
|
||||
let d := fun ε ↦ √((dist z₁.re z₀.re + ε) ^ 2 + (dist z₁.im z₀.im + ε) ^ 2)
|
||||
have h₀d : Continuous d := by continuity
|
||||
have h₁d : ∀ ε, 0 ≤ d ε := fun ε ↦ Real.sqrt_nonneg ((dist z₁.re z₀.re + ε) ^ 2 + (dist z₁.im z₀.im + ε) ^ 2)
|
||||
|
||||
obtain ⟨ε, h₀ε, h₁ε⟩ : ∃ ε > 0, d ε < R := by
|
||||
let Omega := d⁻¹' Metric.ball 0 R
|
||||
|
||||
have lem₀Ω : IsOpen Omega := IsOpen.preimage h₀d Metric.isOpen_ball
|
||||
have lem₁Ω : 0 ∈ Omega := by
|
||||
dsimp [Omega, d]; simp
|
||||
have : dist z₁.re z₀.re = |z₁.re - z₀.re| := by exact rfl
|
||||
rw [this]
|
||||
have : dist z₁.im z₀.im = |z₁.im - z₀.im| := by exact rfl
|
||||
rw [this]
|
||||
simp
|
||||
rw [← Complex.dist_eq_re_im]; simp
|
||||
exact hz₁
|
||||
obtain ⟨ε, h₁ε, h₂ε⟩ := Metric.isOpen_iff.1 lem₀Ω 0 lem₁Ω
|
||||
|
||||
let ε' := (2 : ℝ)⁻¹ * ε
|
||||
|
||||
have h₀ε' : ε' ∈ Omega := by
|
||||
apply h₂ε
|
||||
dsimp [ε']; simp
|
||||
have : |ε| = ε := by apply abs_of_pos h₁ε
|
||||
rw [this]
|
||||
apply (inv_mul_lt_iff zero_lt_two).mpr
|
||||
linarith
|
||||
have h₁ε' : 0 < ε' := by
|
||||
apply Real.mul_pos _ h₁ε
|
||||
apply inv_pos.mpr
|
||||
exact zero_lt_two
|
||||
|
||||
use ε'
|
||||
|
||||
constructor
|
||||
· exact h₁ε'
|
||||
· dsimp [Omega] at h₀ε'; simp at h₀ε'
|
||||
rwa [abs_of_nonneg (h₁d ε')] at h₀ε'
|
||||
|
||||
let rx := dist z₁.re z₀.re + ε
|
||||
let ry := dist z₁.im z₀.im + ε
|
||||
|
||||
have h'ry : 0 < ry := by
|
||||
dsimp [ry]
|
||||
apply add_pos_of_nonneg_of_pos
|
||||
exact dist_nonneg
|
||||
simpa
|
||||
|
||||
have h'f : DifferentiableOn ℂ f (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry) := by
|
||||
apply hf.mono
|
||||
intro x hx
|
||||
simp
|
||||
rw [Complex.dist_eq_re_im]
|
||||
|
||||
have t₀ : dist x.re z₀.re < rx := Metric.mem_ball.mp hx.1
|
||||
have t₁ : dist x.im z₀.im < ry := Metric.mem_ball.mp hx.2
|
||||
have t₂ : √((x.re - z₀.re) ^ 2 + (x.im - z₀.im) ^ 2) < √( rx ^ 2 + ry ^ 2) := by
|
||||
rw [Real.sqrt_lt_sqrt_iff]
|
||||
apply add_lt_add
|
||||
· rw [sq_lt_sq]
|
||||
dsimp [dist] at t₀
|
||||
nth_rw 2 [abs_of_nonneg]
|
||||
assumption
|
||||
apply add_nonneg dist_nonneg (le_of_lt h₀ε)
|
||||
· rw [sq_lt_sq]
|
||||
dsimp [dist] at t₁
|
||||
nth_rw 2 [abs_of_nonneg]
|
||||
assumption
|
||||
apply add_nonneg dist_nonneg (le_of_lt h₀ε)
|
||||
apply add_nonneg
|
||||
exact sq_nonneg (x.re - z₀.re)
|
||||
exact sq_nonneg (x.im - z₀.im)
|
||||
|
||||
calc √((x.re - z₀.re) ^ 2 + (x.im - z₀.im) ^ 2)
|
||||
_ < √( rx ^ 2 + ry ^ 2) := by
|
||||
exact t₂
|
||||
_ = d ε := by dsimp [d, rx, ry]
|
||||
_ < R := by exact h₁ε
|
||||
|
||||
have h'z₁ : z₁ ∈ (Metric.ball z₀.re rx ×ℂ Metric.ball z₀.im ry) := by
|
||||
dsimp [rx, ry]
|
||||
constructor
|
||||
· simp; exact h₀ε
|
||||
· simp; exact h₀ε
|
||||
|
||||
obtain ⟨εx, hεx, εy, hεy, hε⟩ := primitive_additivity h'f h'ry h'z₁
|
||||
|
||||
apply Filter.eventuallyEq_iff_exists_mem.2
|
||||
use (Metric.ball z₁.re εx ×ℂ Metric.ball z₁.im εy)
|
||||
constructor
|
||||
· apply IsOpen.mem_nhds
|
||||
apply IsOpen.reProdIm
|
||||
exact Metric.isOpen_ball
|
||||
exact Metric.isOpen_ball
|
||||
constructor
|
||||
· simpa
|
||||
· simpa
|
||||
· intro x hx
|
||||
simp
|
||||
rw [← sub_zero (primitive z₀ f x), ← hε x hx]
|
||||
abel
|
||||
|
||||
|
||||
theorem primitive_hasDerivAt
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
{z₀ z : ℂ}
|
||||
{R : ℝ}
|
||||
(hf : DifferentiableOn ℂ f (Metric.ball z₀ R))
|
||||
(hz : z ∈ Metric.ball z₀ R) :
|
||||
HasDerivAt (primitive z₀ f) (f z) z := by
|
||||
|
||||
let A := primitive_additivity' hf hz
|
||||
rw [Filter.EventuallyEq.hasDerivAt_iff A]
|
||||
rw [← add_zero (f z)]
|
||||
apply HasDerivAt.add
|
||||
|
||||
let R' := R - dist z z₀
|
||||
have h₀R' : 0 < R' := by
|
||||
dsimp [R']
|
||||
simp
|
||||
exact hz
|
||||
have h₁R' : Metric.ball z R' ⊆ Metric.ball z₀ R := by
|
||||
intro x hx
|
||||
simp
|
||||
calc dist x z₀
|
||||
_ ≤ dist x z + dist z z₀ := dist_triangle x z z₀
|
||||
_ < R' + dist z z₀ := by
|
||||
refine add_lt_add_right ?bc (dist z z₀)
|
||||
exact hx
|
||||
_ = R := by
|
||||
dsimp [R']
|
||||
simp
|
||||
|
||||
apply primitive_hasDerivAtBasepoint
|
||||
exact h₀R'
|
||||
apply ContinuousOn.mono hf.continuousOn h₁R'
|
||||
apply hasDerivAt_const
|
||||
|
||||
|
||||
theorem primitive_differentiableOn
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
{z₀ : ℂ}
|
||||
{R : ℝ}
|
||||
(hf : DifferentiableOn ℂ f (Metric.ball z₀ R))
|
||||
:
|
||||
DifferentiableOn ℂ (primitive z₀ f) (Metric.ball z₀ R) := by
|
||||
intro z hz
|
||||
apply DifferentiableAt.differentiableWithinAt
|
||||
exact (primitive_hasDerivAt hf hz).differentiableAt
|
||||
|
||||
|
||||
theorem primitive_hasFderivAt
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
(z₀ : ℂ)
|
||||
(R : ℝ)
|
||||
(hf : DifferentiableOn ℂ f (Metric.ball z₀ R))
|
||||
:
|
||||
∀ z ∈ Metric.ball z₀ R, HasFDerivAt (primitive z₀ f) ((ContinuousLinearMap.lsmul ℂ ℂ).flip (f z)) z := by
|
||||
intro z hz
|
||||
rw [hasFDerivAt_iff_hasDerivAt]
|
||||
simp
|
||||
apply primitive_hasDerivAt hf hz
|
||||
|
||||
|
||||
theorem primitive_hasFderivAt'
|
||||
{f : ℂ → ℂ}
|
||||
{z₀ : ℂ}
|
||||
{R : ℝ}
|
||||
(hf : DifferentiableOn ℂ f (Metric.ball z₀ R))
|
||||
:
|
||||
∀ z ∈ Metric.ball z₀ R, HasFDerivAt (primitive z₀ f) (ContinuousLinearMap.lsmul ℂ ℂ (f z)) z := by
|
||||
intro z hz
|
||||
rw [hasFDerivAt_iff_hasDerivAt]
|
||||
simp
|
||||
exact primitive_hasDerivAt hf hz
|
||||
|
||||
|
||||
theorem primitive_fderiv
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
{z₀ : ℂ}
|
||||
{R : ℝ}
|
||||
(hf : DifferentiableOn ℂ f (Metric.ball z₀ R))
|
||||
:
|
||||
∀ z ∈ Metric.ball z₀ R, (fderiv ℂ (primitive z₀ f) z) = (ContinuousLinearMap.lsmul ℂ ℂ).flip (f z) := by
|
||||
intro z hz
|
||||
apply HasFDerivAt.fderiv
|
||||
exact primitive_hasFderivAt z₀ R hf z hz
|
||||
|
||||
|
||||
theorem primitive_fderiv'
|
||||
{f : ℂ → ℂ}
|
||||
{z₀ : ℂ}
|
||||
{R : ℝ}
|
||||
(hf : DifferentiableOn ℂ f (Metric.ball z₀ R))
|
||||
:
|
||||
∀ z ∈ Metric.ball z₀ R, (fderiv ℂ (primitive z₀ f) z) = ContinuousLinearMap.lsmul ℂ ℂ (f z) := by
|
||||
intro z hz
|
||||
apply HasFDerivAt.fderiv
|
||||
exact primitive_hasFderivAt' hf z hz
|
||||
|
|
@ -0,0 +1,666 @@
|
|||
import Mathlib.Analysis.Complex.TaylorSeries
|
||||
import Mathlib.Analysis.SpecialFunctions.Integrals
|
||||
import Mathlib.MeasureTheory.Integral.DivergenceTheorem
|
||||
import Mathlib.MeasureTheory.Integral.IntervalIntegral
|
||||
import Mathlib.MeasureTheory.Function.LocallyIntegrable
|
||||
import Nevanlinna.partialDeriv
|
||||
import Nevanlinna.cauchyRiemann
|
||||
|
||||
/-
|
||||
noncomputable def partialDeriv
|
||||
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
|
||||
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] : E → (E → F) → (E → F) :=
|
||||
fun v ↦ (fun f ↦ (fun w ↦ fderiv ℝ f w v))
|
||||
|
||||
|
||||
theorem partialDeriv_compContLinAt
|
||||
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
|
||||
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
|
||||
{G : Type*} [NormedAddCommGroup G] [NormedSpace ℂ G]
|
||||
{f : E → F}
|
||||
{l : F →L[ℝ] G}
|
||||
{v : E}
|
||||
{x : E}
|
||||
(h : DifferentiableAt ℝ f x) :
|
||||
(partialDeriv v (l ∘ f)) x = (l ∘ partialDeriv v f) x:= by
|
||||
unfold partialDeriv
|
||||
rw [fderiv.comp x (ContinuousLinearMap.differentiableAt l) h]
|
||||
simp
|
||||
|
||||
|
||||
theorem partialDeriv_compCLE
|
||||
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
|
||||
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
|
||||
{G : Type*} [NormedAddCommGroup G] [NormedSpace ℂ G]
|
||||
{f : E → F}
|
||||
{l : F ≃L[ℝ] G} {v : E} : partialDeriv v (l ∘ f) = l ∘ partialDeriv v f := by
|
||||
funext x
|
||||
by_cases hyp : DifferentiableAt ℝ f x
|
||||
· let lCLM : F →L[ℝ] G := l
|
||||
suffices shyp : partialDeriv v (lCLM ∘ f) x = (lCLM ∘ partialDeriv v f) x from by tauto
|
||||
apply partialDeriv_compContLinAt
|
||||
exact hyp
|
||||
· unfold partialDeriv
|
||||
rw [fderiv_zero_of_not_differentiableAt]
|
||||
simp
|
||||
rw [fderiv_zero_of_not_differentiableAt]
|
||||
simp
|
||||
exact hyp
|
||||
rw [ContinuousLinearEquiv.comp_differentiableAt_iff]
|
||||
exact hyp
|
||||
|
||||
theorem partialDeriv_smul'₂
|
||||
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
|
||||
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
|
||||
{f : E → F} {a : ℂ} {v : E} :
|
||||
partialDeriv v (a • f) = a • partialDeriv v f := by
|
||||
|
||||
funext w
|
||||
by_cases ha : a = 0
|
||||
· unfold partialDeriv
|
||||
have : a • f = fun y ↦ a • f y := by rfl
|
||||
rw [this, ha]
|
||||
simp
|
||||
· -- Now a is not zero. We present scalar multiplication with a as a continuous linear equivalence.
|
||||
let smulCLM : F ≃L[ℝ] F :=
|
||||
{
|
||||
toFun := fun x ↦ a • x
|
||||
map_add' := fun x y => DistribSMul.smul_add a x y
|
||||
map_smul' := fun m x => (smul_comm ((RingHom.id ℝ) m) a x).symm
|
||||
invFun := fun x ↦ a⁻¹ • x
|
||||
left_inv := by
|
||||
intro x
|
||||
simp
|
||||
rw [← smul_assoc, smul_eq_mul, mul_comm, mul_inv_cancel ha, one_smul]
|
||||
right_inv := by
|
||||
intro x
|
||||
simp
|
||||
rw [← smul_assoc, smul_eq_mul, mul_inv_cancel ha, one_smul]
|
||||
continuous_toFun := continuous_const_smul a
|
||||
continuous_invFun := continuous_const_smul a⁻¹
|
||||
}
|
||||
|
||||
have : a • f = smulCLM ∘ f := by tauto
|
||||
rw [this]
|
||||
rw [partialDeriv_compCLE]
|
||||
tauto
|
||||
|
||||
theorem CauchyRiemann₄
|
||||
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
|
||||
{f : ℂ → F} :
|
||||
(Differentiable ℂ f) → partialDeriv ℝ Complex.I f = Complex.I • partialDeriv ℝ 1 f := by
|
||||
intro h
|
||||
unfold partialDeriv
|
||||
|
||||
conv =>
|
||||
left
|
||||
intro w
|
||||
rw [DifferentiableAt.fderiv_restrictScalars ℝ (h w)]
|
||||
simp
|
||||
rw [← mul_one Complex.I]
|
||||
rw [← smul_eq_mul]
|
||||
conv =>
|
||||
right
|
||||
right
|
||||
intro w
|
||||
rw [DifferentiableAt.fderiv_restrictScalars ℝ (h w)]
|
||||
funext w
|
||||
simp
|
||||
-/
|
||||
|
||||
|
||||
theorem MeasureTheory.integral2_divergence₃
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
|
||||
(f g : ℝ × ℝ → E)
|
||||
(h₁f : ContDiff ℝ 1 f)
|
||||
(h₁g : ContDiff ℝ 1 g)
|
||||
(a₁ : ℝ)
|
||||
(a₂ : ℝ)
|
||||
(b₁ : ℝ)
|
||||
(b₂ : ℝ) :
|
||||
∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, ((fderiv ℝ f) (x, y)) (1, 0) + ((fderiv ℝ g) (x, y)) (0, 1) = (((∫ (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
|
||||
|
||||
apply integral2_divergence_prod_of_hasFDerivWithinAt_off_countable f g (fderiv ℝ f) (fderiv ℝ g) a₁ a₂ b₁ b₂ ∅
|
||||
exact Set.countable_empty
|
||||
-- ContinuousOn f (Set.uIcc a₁ b₁ ×ˢ Set.uIcc a₂ b₂)
|
||||
exact h₁f.continuous.continuousOn
|
||||
--
|
||||
exact h₁g.continuous.continuousOn
|
||||
--
|
||||
rw [Set.diff_empty]
|
||||
intro x _
|
||||
exact DifferentiableAt.hasFDerivAt ((h₁f.differentiable le_rfl) x)
|
||||
--
|
||||
rw [Set.diff_empty]
|
||||
intro y _
|
||||
exact DifferentiableAt.hasFDerivAt ((h₁g.differentiable le_rfl) y)
|
||||
--
|
||||
apply ContinuousOn.integrableOn_compact
|
||||
apply IsCompact.prod
|
||||
exact isCompact_uIcc
|
||||
exact isCompact_uIcc
|
||||
apply ContinuousOn.add
|
||||
apply Continuous.continuousOn
|
||||
exact Continuous.clm_apply (ContDiff.continuous_fderiv h₁f le_rfl) continuous_const
|
||||
apply Continuous.continuousOn
|
||||
exact Continuous.clm_apply (ContDiff.continuous_fderiv h₁g le_rfl) continuous_const
|
||||
|
||||
|
||||
theorem integral_divergence₄
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
|
||||
(f g : ℂ → E)
|
||||
(h₁f : ContDiff ℝ 1 f)
|
||||
(h₁g : ContDiff ℝ 1 g)
|
||||
(a₁ : ℝ)
|
||||
(a₂ : ℝ)
|
||||
(b₁ : ℝ)
|
||||
(b₂ : ℝ) :
|
||||
∫ (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
|
||||
|
||||
let fr : ℝ × ℝ → E := f ∘ Complex.equivRealProdCLM.symm
|
||||
let gr : ℝ × ℝ → E := g ∘ Complex.equivRealProdCLM.symm
|
||||
|
||||
have sfr {x y : ℝ} : f { re := x, im := y } = fr (x, y) := by exact rfl
|
||||
have sgr {x y : ℝ} : g { re := x, im := y } = gr (x, y) := by exact rfl
|
||||
repeat (conv in f { re := _, im := _ } => rw [sfr])
|
||||
repeat (conv in g { re := _, im := _ } => rw [sgr])
|
||||
|
||||
have sfr' {x y : ℝ} {z : ℂ} : (fderiv ℝ f { re := x, im := y }) z = fderiv ℝ fr (x, y) (Complex.equivRealProdCLM z) := by
|
||||
rw [fderiv.comp]
|
||||
rw [Complex.equivRealProdCLM.symm.fderiv]
|
||||
tauto
|
||||
apply Differentiable.differentiableAt
|
||||
exact h₁f.differentiable le_rfl
|
||||
exact Complex.equivRealProdCLM.symm.differentiableAt
|
||||
conv in ⇑(fderiv ℝ f { re := _, im := _ }) _ => rw [sfr']
|
||||
|
||||
have sgr' {x y : ℝ} {z : ℂ} : (fderiv ℝ g { re := x, im := y }) z = fderiv ℝ gr (x, y) (Complex.equivRealProdCLM z) := by
|
||||
rw [fderiv.comp]
|
||||
rw [Complex.equivRealProdCLM.symm.fderiv]
|
||||
tauto
|
||||
apply Differentiable.differentiableAt
|
||||
exact h₁g.differentiable le_rfl
|
||||
exact Complex.equivRealProdCLM.symm.differentiableAt
|
||||
conv in ⇑(fderiv ℝ g { re := _, im := _ }) _ => rw [sgr']
|
||||
|
||||
apply MeasureTheory.integral2_divergence₃ fr gr _ _ a₁ a₂ b₁ b₂
|
||||
-- ContDiff ℝ 1 fr
|
||||
exact (ContinuousLinearEquiv.contDiff_comp_iff (ContinuousLinearEquiv.symm Complex.equivRealProdCLM)).mpr h₁f
|
||||
-- ContDiff ℝ 1 gr
|
||||
exact (ContinuousLinearEquiv.contDiff_comp_iff (ContinuousLinearEquiv.symm Complex.equivRealProdCLM)).mpr h₁g
|
||||
|
||||
|
||||
theorem integral_divergence₅
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
(F : ℂ → E)
|
||||
(hF : Differentiable ℂ F)
|
||||
(lowerLeft upperRight : ℂ) :
|
||||
(∫ (x : ℝ) in lowerLeft.re..upperRight.re, F ⟨x, lowerLeft.im⟩) + Complex.I • ∫ (x : ℝ) in lowerLeft.im..upperRight.im, F ⟨upperRight.re, x⟩ =
|
||||
(∫ (x : ℝ) in lowerLeft.re..upperRight.re, F ⟨x, upperRight.im⟩) + Complex.I • ∫ (x : ℝ) in lowerLeft.im..upperRight.im, F ⟨lowerLeft.re, x⟩ := by
|
||||
|
||||
let h₁f : ContDiff ℝ 1 F := (hF.contDiff : ContDiff ℂ 1 F).restrict_scalars ℝ
|
||||
|
||||
let h₁g : ContDiff ℝ 1 (-Complex.I • F) := by
|
||||
have : -Complex.I • F = fun x ↦ -Complex.I • F x := by rfl
|
||||
rw [this]
|
||||
apply ContDiff.comp
|
||||
exact contDiff_const_smul _
|
||||
exact h₁f
|
||||
|
||||
let A := integral_divergence₄ (-Complex.I • F) F h₁g h₁f lowerLeft.re upperRight.im upperRight.re lowerLeft.im
|
||||
have {z : ℂ} : fderiv ℝ F z Complex.I = partialDeriv ℝ Complex.I F z := by rfl
|
||||
conv at A in (fderiv ℝ F _) _ => rw [this]
|
||||
have {z : ℂ} : fderiv ℝ (-Complex.I • F) z 1 = partialDeriv ℝ 1 (-Complex.I • F) z := by rfl
|
||||
conv at A in (fderiv ℝ (-Complex.I • F) _) _ => rw [this]
|
||||
conv at A =>
|
||||
left
|
||||
arg 1
|
||||
intro x
|
||||
arg 1
|
||||
intro y
|
||||
rw [CauchyRiemann₄ hF]
|
||||
rw [partialDeriv_smul'₂]
|
||||
simp
|
||||
simp at A
|
||||
|
||||
have {t₁ t₂ t₃ t₄ : E} : 0 = (t₁ - t₂) + t₃ + t₄ → t₁ + t₃ = t₂ - t₄ := by
|
||||
intro hyp
|
||||
calc
|
||||
t₁ + t₃ = t₁ + t₃ - 0 := by rw [sub_zero (t₁ + t₃)]
|
||||
_ = t₁ + t₃ - (t₁ - t₂ + t₃ + t₄) := by rw [hyp]
|
||||
_ = t₂ - t₄ := by abel
|
||||
let B := this A
|
||||
repeat
|
||||
rw [intervalIntegral.integral_symm lowerLeft.im upperRight.im] at B
|
||||
simp at B
|
||||
exact B
|
||||
|
||||
|
||||
noncomputable def primitive
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] :
|
||||
ℂ → (ℂ → E) → (ℂ → E) := by
|
||||
intro z₀
|
||||
intro f
|
||||
exact fun z ↦ (∫ (x : ℝ) in z₀.re..z.re, f ⟨x, z₀.im⟩) + Complex.I • ∫ (x : ℝ) in z₀.im..z.im, f ⟨z.re, x⟩
|
||||
|
||||
|
||||
theorem primitive_zeroAtBasepoint
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
(f : ℂ → E)
|
||||
(z₀ : ℂ) :
|
||||
(primitive z₀ f) z₀ = 0 := by
|
||||
unfold primitive
|
||||
simp
|
||||
|
||||
|
||||
theorem primitive_fderivAtBasepointZero
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
(f : ℂ → E)
|
||||
(hf : Continuous f) :
|
||||
HasDerivAt (primitive 0 f) (f 0) 0 := by
|
||||
unfold primitive
|
||||
simp
|
||||
apply hasDerivAt_iff_isLittleO.2
|
||||
simp
|
||||
rw [Asymptotics.isLittleO_iff]
|
||||
intro c hc
|
||||
|
||||
have {z : ℂ} {e : E} : z • e = (∫ (_ : ℝ) in (0)..(z.re), e) + Complex.I • ∫ (_ : ℝ) in (0)..(z.im), e:= by
|
||||
simp
|
||||
rw [smul_comm]
|
||||
rw [← smul_assoc]
|
||||
simp
|
||||
have : z.re • e = (z.re : ℂ) • e := by exact rfl
|
||||
rw [this, ← add_smul]
|
||||
simp
|
||||
conv =>
|
||||
left
|
||||
intro x
|
||||
left
|
||||
arg 1
|
||||
arg 2
|
||||
rw [this]
|
||||
have {A B C D :E} : (A + B) - (C + D) = (A - C) + (B - D) := by
|
||||
abel
|
||||
have t₀ {r : ℝ} : IntervalIntegrable (fun x => f { re := x, im := 0 }) MeasureTheory.volume 0 r := by
|
||||
apply Continuous.intervalIntegrable
|
||||
apply Continuous.comp
|
||||
exact hf
|
||||
have : (fun x => ({ re := x, im := 0 } : ℂ)) = Complex.ofRealLI := by rfl
|
||||
rw [this]
|
||||
continuity
|
||||
have t₁ {r : ℝ} : IntervalIntegrable (fun _ => f 0) MeasureTheory.volume 0 r := by
|
||||
apply Continuous.intervalIntegrable
|
||||
apply Continuous.comp
|
||||
exact hf
|
||||
fun_prop
|
||||
have t₂ {a b : ℝ} : IntervalIntegrable (fun x_1 => f { re := a, im := x_1 }) MeasureTheory.volume 0 b := by
|
||||
apply Continuous.intervalIntegrable
|
||||
apply Continuous.comp hf
|
||||
have : (Complex.mk a) = (fun x => Complex.I • Complex.ofRealCLM x + { re := a, im := 0 }) := by
|
||||
funext x
|
||||
apply Complex.ext
|
||||
rw [Complex.add_re]
|
||||
simp
|
||||
simp
|
||||
rw [this]
|
||||
apply Continuous.add
|
||||
continuity
|
||||
fun_prop
|
||||
|
||||
have t₃ {a : ℝ} : IntervalIntegrable (fun _ => f 0) MeasureTheory.volume 0 a := by
|
||||
apply Continuous.intervalIntegrable
|
||||
apply Continuous.comp
|
||||
exact hf
|
||||
fun_prop
|
||||
conv =>
|
||||
left
|
||||
intro x
|
||||
left
|
||||
arg 1
|
||||
rw [this]
|
||||
rw [← smul_sub]
|
||||
rw [← intervalIntegral.integral_sub t₀ t₁]
|
||||
rw [← intervalIntegral.integral_sub t₂ t₃]
|
||||
rw [Filter.eventually_iff_exists_mem]
|
||||
|
||||
let s := f⁻¹' Metric.ball (f 0) (c / (4 : ℝ))
|
||||
have h₁s : IsOpen s := IsOpen.preimage hf Metric.isOpen_ball
|
||||
have h₂s : 0 ∈ s := by
|
||||
apply Set.mem_preimage.mpr
|
||||
apply Metric.mem_ball_self
|
||||
linarith
|
||||
|
||||
obtain ⟨ε, h₁ε, h₂ε⟩ := Metric.isOpen_iff.1 h₁s 0 h₂s
|
||||
|
||||
have h₃ε : ∀ y ∈ Metric.ball 0 ε, ‖(f y) - (f 0)‖ < (c / (4 : ℝ)) := by
|
||||
intro y hy
|
||||
apply mem_ball_iff_norm.mp (h₂ε hy)
|
||||
|
||||
use Metric.ball 0 (ε / (4 : ℝ))
|
||||
constructor
|
||||
· apply Metric.ball_mem_nhds 0
|
||||
linarith
|
||||
· intro y hy
|
||||
have h₁y : |y.re| < ε / 4 := by
|
||||
calc |y.re|
|
||||
_ ≤ Complex.abs y := by apply Complex.abs_re_le_abs
|
||||
_ < ε / 4 := by
|
||||
let A := mem_ball_iff_norm.1 hy
|
||||
simp at A
|
||||
linarith
|
||||
have h₂y : |y.im| < ε / 4 := by
|
||||
calc |y.im|
|
||||
_ ≤ Complex.abs y := by apply Complex.abs_im_le_abs
|
||||
_ < ε / 4 := by
|
||||
let A := mem_ball_iff_norm.1 hy
|
||||
simp at A
|
||||
linarith
|
||||
|
||||
have intervalComputation {x' y' : ℝ} (h : x' ∈ Ι 0 y') : |x'| ≤ |y'| := by
|
||||
let A := h.1
|
||||
let B := h.2
|
||||
rcases le_total 0 y' with hy | hy
|
||||
· simp [hy] at A
|
||||
simp [hy] at B
|
||||
rw [abs_of_nonneg hy]
|
||||
rw [abs_of_nonneg (le_of_lt A)]
|
||||
exact B
|
||||
· simp [hy] at A
|
||||
simp [hy] at B
|
||||
rw [abs_of_nonpos hy]
|
||||
rw [abs_of_nonpos]
|
||||
linarith [h.1]
|
||||
exact B
|
||||
|
||||
have t₁ : ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ ≤ (c / (4 : ℝ)) * |y.re - 0| := by
|
||||
apply intervalIntegral.norm_integral_le_of_norm_le_const
|
||||
intro x hx
|
||||
|
||||
have h₁x : |x| < ε / 4 := by
|
||||
calc |x|
|
||||
_ ≤ |y.re| := intervalComputation hx
|
||||
_ < ε / 4 := h₁y
|
||||
apply le_of_lt
|
||||
apply h₃ε { re := x, im := 0 }
|
||||
rw [mem_ball_iff_norm]
|
||||
simp
|
||||
have : { re := x, im := 0 } = (x : ℂ) := by rfl
|
||||
rw [this]
|
||||
rw [Complex.abs_ofReal]
|
||||
linarith
|
||||
|
||||
have t₂ : ‖∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ ≤ (c / (4 : ℝ)) * |y.im - 0| := by
|
||||
apply intervalIntegral.norm_integral_le_of_norm_le_const
|
||||
intro x hx
|
||||
|
||||
have h₁x : |x| < ε / 4 := by
|
||||
calc |x|
|
||||
_ ≤ |y.im| := intervalComputation hx
|
||||
_ < ε / 4 := h₂y
|
||||
|
||||
apply le_of_lt
|
||||
apply h₃ε { re := y.re, im := x }
|
||||
simp
|
||||
|
||||
calc Complex.abs { re := y.re, im := x }
|
||||
_ ≤ |y.re| + |x| := by
|
||||
apply Complex.abs_le_abs_re_add_abs_im { re := y.re, im := x }
|
||||
_ < ε := by
|
||||
linarith
|
||||
|
||||
calc ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0) + Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖
|
||||
_ ≤ ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖Complex.I • ∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by
|
||||
apply norm_add_le
|
||||
_ ≤ ‖(∫ (x : ℝ) in (0)..(y.re), f { re := x, im := 0 } - f 0)‖ + ‖∫ (x : ℝ) in (0)..(y.im), f { re := y.re, im := x } - f 0‖ := by
|
||||
simp
|
||||
rw [norm_smul]
|
||||
simp
|
||||
_ ≤ (c / (4 : ℝ)) * |y.re - 0| + (c / (4 : ℝ)) * |y.im - 0| := by
|
||||
apply add_le_add
|
||||
exact t₁
|
||||
exact t₂
|
||||
_ ≤ (c / (4 : ℝ)) * (|y.re| + |y.im|) := by
|
||||
simp
|
||||
rw [mul_add]
|
||||
_ ≤ (c / (4 : ℝ)) * (4 * ‖y‖) := by
|
||||
have : |y.re| + |y.im| ≤ 4 * ‖y‖ := by
|
||||
calc |y.re| + |y.im|
|
||||
_ ≤ ‖y‖ + ‖y‖ := by
|
||||
apply add_le_add
|
||||
apply Complex.abs_re_le_abs
|
||||
apply Complex.abs_im_le_abs
|
||||
_ ≤ 4 * ‖y‖ := by
|
||||
rw [← two_mul]
|
||||
apply mul_le_mul
|
||||
linarith
|
||||
rfl
|
||||
exact norm_nonneg y
|
||||
linarith
|
||||
|
||||
apply mul_le_mul
|
||||
rfl
|
||||
exact this
|
||||
apply add_nonneg
|
||||
apply abs_nonneg
|
||||
apply abs_nonneg
|
||||
linarith
|
||||
_ ≤ c * ‖y‖ := by
|
||||
linarith
|
||||
|
||||
|
||||
theorem primitive_translation
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
(f : ℂ → E)
|
||||
(z₀ t : ℂ) :
|
||||
primitive z₀ (f ∘ fun z ↦ (z - t)) = ((primitive (z₀ - t) f) ∘ fun z ↦ (z - t)) := by
|
||||
funext z
|
||||
unfold primitive
|
||||
simp
|
||||
|
||||
let g : ℝ → E := fun x ↦ f ( {re := x, im := z₀.im - t.im} )
|
||||
have {x : ℝ} : f ({ re := x, im := z₀.im } - t) = g (1*x - t.re) := by
|
||||
congr 1
|
||||
apply Complex.ext <;> simp
|
||||
conv =>
|
||||
left
|
||||
left
|
||||
arg 1
|
||||
intro x
|
||||
rw [this]
|
||||
rw [intervalIntegral.integral_comp_mul_sub g one_ne_zero (t.re)]
|
||||
simp
|
||||
|
||||
congr 1
|
||||
let g : ℝ → E := fun x ↦ f ( {re := z.re - t.re, im := x} )
|
||||
have {x : ℝ} : f ({ re := z.re, im := x} - t) = g (1*x - t.im) := by
|
||||
congr 1
|
||||
apply Complex.ext <;> simp
|
||||
conv =>
|
||||
left
|
||||
arg 1
|
||||
intro x
|
||||
rw [this]
|
||||
rw [intervalIntegral.integral_comp_mul_sub g one_ne_zero (t.im)]
|
||||
simp
|
||||
|
||||
|
||||
theorem primitive_hasDerivAtBasepoint
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
(hf : Continuous f)
|
||||
(z₀ : ℂ) :
|
||||
HasDerivAt (primitive z₀ f) (f z₀) z₀ := by
|
||||
|
||||
let g := f ∘ fun z ↦ z + z₀
|
||||
have : Continuous g := by continuity
|
||||
let A := primitive_fderivAtBasepointZero g this
|
||||
simp at A
|
||||
|
||||
let B := primitive_translation g z₀ z₀
|
||||
simp at B
|
||||
have : (g ∘ fun z ↦ (z - z₀)) = f := by
|
||||
funext z
|
||||
dsimp [g]
|
||||
simp
|
||||
rw [this] at B
|
||||
rw [B]
|
||||
have : f z₀ = (1 : ℂ) • (f z₀) := by
|
||||
exact (MulAction.one_smul (f z₀)).symm
|
||||
conv =>
|
||||
arg 2
|
||||
rw [this]
|
||||
|
||||
apply HasDerivAt.scomp
|
||||
simp
|
||||
have : g 0 = f z₀ := by simp [g]
|
||||
rw [← this]
|
||||
exact A
|
||||
apply HasDerivAt.sub_const
|
||||
have : (fun (x : ℂ) ↦ x) = id := by
|
||||
funext x
|
||||
simp
|
||||
rw [this]
|
||||
exact hasDerivAt_id z₀
|
||||
|
||||
|
||||
lemma integrability₁
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
(f : ℂ → E)
|
||||
(hf : Differentiable ℂ f)
|
||||
(a₁ a₂ b : ℝ) :
|
||||
IntervalIntegrable (fun x => f { re := x, im := b }) MeasureTheory.volume a₁ a₂ := by
|
||||
apply Continuous.intervalIntegrable
|
||||
apply Continuous.comp
|
||||
exact Differentiable.continuous hf
|
||||
have : ((fun x => { re := x, im := b }) : ℝ → ℂ) = (fun x => Complex.ofRealCLM x + { re := 0, im := b }) := by
|
||||
funext x
|
||||
apply Complex.ext
|
||||
rw [Complex.add_re]
|
||||
simp
|
||||
rw [Complex.add_im]
|
||||
simp
|
||||
rw [this]
|
||||
continuity
|
||||
|
||||
|
||||
lemma integrability₂
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
(f : ℂ → E)
|
||||
(hf : Differentiable ℂ f)
|
||||
(a₁ a₂ b : ℝ) :
|
||||
IntervalIntegrable (fun x => f { re := b, im := x }) MeasureTheory.volume a₁ a₂ := by
|
||||
apply Continuous.intervalIntegrable
|
||||
apply Continuous.comp
|
||||
exact Differentiable.continuous hf
|
||||
have : (Complex.mk b) = (fun x => Complex.I • Complex.ofRealCLM x + { re := b, im := 0 }) := by
|
||||
funext x
|
||||
apply Complex.ext
|
||||
rw [Complex.add_re]
|
||||
simp
|
||||
simp
|
||||
rw [this]
|
||||
apply Continuous.add
|
||||
continuity
|
||||
fun_prop
|
||||
|
||||
|
||||
theorem primitive_additivity
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
(f : ℂ → E)
|
||||
(hf : Differentiable ℂ f)
|
||||
(z₀ z₁ : ℂ) :
|
||||
primitive z₀ f = fun z ↦ (primitive z₁ f) z + (primitive z₀ f z₁) := by
|
||||
funext z
|
||||
unfold primitive
|
||||
|
||||
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
|
||||
rw [intervalIntegral.integral_add_adjacent_intervals]
|
||||
apply integrability₁ f hf
|
||||
apply integrability₁ f hf
|
||||
rw [this]
|
||||
|
||||
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
|
||||
rw [intervalIntegral.integral_add_adjacent_intervals]
|
||||
apply integrability₂ f hf
|
||||
apply integrability₂ f hf
|
||||
rw [this]
|
||||
simp
|
||||
|
||||
let A := integral_divergence₅ f hf ⟨z₁.re, z₀.im⟩ ⟨z.re, z₁.im⟩
|
||||
simp at A
|
||||
|
||||
have {a b c d : E} : (b + a) + (c + d) = (a + c) + (b + d) := by
|
||||
abel
|
||||
rw [this]
|
||||
rw [A]
|
||||
abel
|
||||
|
||||
|
||||
theorem primitive_hasDerivAt
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
(hf : Differentiable ℂ f)
|
||||
(z₀ z : ℂ) :
|
||||
HasDerivAt (primitive z₀ f) (f z) z := by
|
||||
rw [primitive_additivity f hf z₀ z]
|
||||
rw [← add_zero (f z)]
|
||||
apply HasDerivAt.add
|
||||
apply primitive_hasDerivAtBasepoint
|
||||
exact hf.continuous
|
||||
apply hasDerivAt_const
|
||||
|
||||
|
||||
theorem primitive_differentiable
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
(hf : Differentiable ℂ f)
|
||||
(z₀ : ℂ) :
|
||||
Differentiable ℂ (primitive z₀ f) := by
|
||||
intro z
|
||||
exact (primitive_hasDerivAt hf z₀ z).differentiableAt
|
||||
|
||||
|
||||
theorem primitive_hasFderivAt
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
(hf : Differentiable ℂ f)
|
||||
(z₀ : ℂ) :
|
||||
∀ z, HasFDerivAt (primitive z₀ f) ((ContinuousLinearMap.lsmul ℂ ℂ).flip (f z)) z := by
|
||||
intro z
|
||||
rw [hasFDerivAt_iff_hasDerivAt]
|
||||
simp
|
||||
exact primitive_hasDerivAt hf z₀ z
|
||||
|
||||
|
||||
theorem primitive_hasFderivAt'
|
||||
{f : ℂ → ℂ}
|
||||
(hf : Differentiable ℂ f)
|
||||
(z₀ : ℂ) :
|
||||
∀ z, HasFDerivAt (primitive z₀ f) (ContinuousLinearMap.lsmul ℂ ℂ (f z)) z := by
|
||||
intro z
|
||||
rw [hasFDerivAt_iff_hasDerivAt]
|
||||
simp
|
||||
exact primitive_hasDerivAt hf z₀ z
|
||||
|
||||
|
||||
theorem primitive_fderiv
|
||||
{E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E]
|
||||
{f : ℂ → E}
|
||||
(hf : Differentiable ℂ f)
|
||||
(z₀ : ℂ) :
|
||||
∀ z, (fderiv ℂ (primitive z₀ f) z) = (ContinuousLinearMap.lsmul ℂ ℂ).flip (f z) := by
|
||||
intro z
|
||||
apply HasFDerivAt.fderiv
|
||||
exact primitive_hasFderivAt hf z₀ z
|
||||
|
||||
|
||||
theorem primitive_fderiv'
|
||||
{f : ℂ → ℂ}
|
||||
(hf : Differentiable ℂ f)
|
||||
(z₀ : ℂ) :
|
||||
∀ z, (fderiv ℂ (primitive z₀ f) z) = ContinuousLinearMap.lsmul ℂ ℂ (f z) := by
|
||||
intro z
|
||||
apply HasFDerivAt.fderiv
|
||||
exact primitive_hasFderivAt' hf z₀ z
|
||||
|
||||
Loading…
Reference in New Issue