mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case inr.refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
ζ✝ : ℂ
hζ : ζ✝.im ∈ Icc 0 (π / 2)
hz_re : 0 ≤ (cexp ζ✝).re
hz_im : 0 ≤ (cexp ζ✝).im
hzne : cexp ζ✝ ≠ 0
H : MapsTo cexp (im ⁻¹' Ioo 0 (π / 2)) (Ioi 0 ×ℂ Ioi 0)
ζ : ℝ
⊢ ‖f ↑(expR ζ)‖ ≤ C

Tactic:
exact hre _ (Real.exp_pos _).le

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℂ → E
c : ℂ
R : ℝ
inst✝ : CompleteSpace E
a : E
hf : ∀ x ∈ sphere c |R|, f x = a
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ) = a

Tactic:
]

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℂ → E
c : ℂ
R : ℝ
inst✝ : CompleteSpace E
a : E
hf : ∀ x ∈ sphere c |R|, f x = a
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ) = a
Module: Mathlib.Analysis.Complex.Harmonic.MeanValue

Goal state before:
pp: F✝ : Type u_1
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
inst✝ : CompleteSpace F✝
f : ℂ → F✝
c : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F✝
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + |R|) ⊆ {x | HarmonicAt (⇑g ∘ f) x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + |R|))
h₂F : Set.EqOn (fun z ↦ (F z).re) (⇑g ∘ f) (ball c (e + |R|))
h₃F : DifferentiableOn ℂ F (closure (ball c |R|))
h₄F : Set.EqOn (⇑Complex.reCLM ∘ F) (⇑g ∘ f) (sphere c |R|)
⊢ Complex.reCLM (F c) = g (f c)

pp: F✝ : Type u_1
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
inst✝ : CompleteSpace F✝
f : ℂ → F✝
c : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F✝
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + |R|) ⊆ {x | HarmonicAt (⇑g ∘ f) x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + |R|))
h₂F : Set.EqOn (fun z ↦ (F z).re) (⇑g ∘ f) (ball c (e + |R|))
h₃F : DifferentiableOn ℂ F (closure (ball c |R|))
h₄F : Set.EqOn (⇑Complex.reCLM ∘ F) (⇑g ∘ f) (sphere c |R|)
⊢ CircleIntegrable F c R

Tactic:
]

Goal state after:
F✝ : Type u_1
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
inst✝ : CompleteSpace F✝
f : ℂ → F✝
c : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F✝
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + |R|) ⊆ {x | HarmonicAt (⇑g ∘ f) x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + |R|))
h₂F : Set.EqOn (fun z ↦ (F z).re) (⇑g ∘ f) (ball c (e + |R|))
h₃F : DifferentiableOn ℂ F (closure (ball c |R|))
h₄F : Set.EqOn (⇑Complex.reCLM ∘ F) (⇑g ∘ f) (sphere c |R|)
⊢ Complex.reCLM (F c) = g (f c)

F✝ : Type u_1
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
inst✝ : CompleteSpace F✝
f : ℂ → F✝
c : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F✝
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + |R|) ⊆ {x | HarmonicAt (⇑g ∘ f) x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + |R|))
h₂F : Set.EqOn (fun z ↦ (F z).re) (⇑g ∘ f) (ball c (e + |R|))
h₃F : DifferentiableOn ℂ F (closure (ball c |R|))
h₄F : Set.EqOn (⇑Complex.reCLM ∘ F) (⇑g ∘ f) (sphere c |R|)
⊢ CircleIntegrable F c R
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: c : ℂ
R : ℝ
f : ℂ → ℝ
a : ℝ
hf : CircleIntegrable f c R
h₂f : ∀ x ∈ sphere c |R|, f x ≤ a
⊢ ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ) ≤ ∫ (θ : ℝ) in 0..2 * π, a

Tactic:
]

Goal state after:
c : ℂ
R : ℝ
f : ℂ → ℝ
a : ℝ
hf : CircleIntegrable f c R
h₂f : ∀ x ∈ sphere c |R|, f x ≤ a
⊢ ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ) ≤ ∫ (θ : ℝ) in 0..2 * π, a
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: c : ℂ
R : ℝ
f : ℂ → ℝ
⊢ |∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)| ≤ ∫ (θ : ℝ) in 0..2 * π, |f| (circleMap c R θ)

Tactic:
]

Goal state after:
c : ℂ
R : ℝ
f : ℂ → ℝ
⊢ |∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)| ≤ ∫ (θ : ℝ) in 0..2 * π, |f| (circleMap c R θ)
Module: Mathlib.Analysis.Complex.Harmonic.MeanValue

Goal state before:
pp: F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℂ → F
c : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c |R|)
h : CircleIntegrable f c R
⊢ ∀ (g : StrongDual ℝ F), g (circleAverage f c R) = g (f c)

Tactic:
]

Goal state after:
F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℂ → F
c : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c |R|)
h : CircleIntegrable f c R
⊢ ∀ (g : StrongDual ℝ F), g (circleAverage f c R) = g (f c)
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: case pos
c : ℂ
R : ℝ
f : ℂ → ℝ
h₂f : ∀ x ∈ sphere c |R|, 0 ≤ f x
hf : CircleIntegrable f c R
⊢ ∫ (θ : ℝ) in 0..2 * π, 0 ≤ ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)

Tactic:
]

Goal state after:
case pos
c : ℂ
R : ℝ
f : ℂ → ℝ
h₂f : ∀ x ∈ sphere c |R|, 0 ≤ f x
hf : CircleIntegrable f c R
⊢ ∫ (θ : ℝ) in 0..2 * π, 0 ≤ ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: case neg
c : ℂ
R : ℝ
f : ℂ → ℝ
h₂f : ∀ x ∈ sphere c |R|, 0 ≤ f x
hf : ¬CircleIntegrable f c R
⊢ 0 ≤ 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.Harmonic.MeanValue

Goal state before:
pp: F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℂ → F
c : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F
⊢ circleAverage (⇑g ∘ f) c R = g (f c)

Tactic:
]

Goal state after:
F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℂ → F
c : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F
⊢ circleAverage (⇑g ∘ f) c R = g (f c)
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
c : ℂ
R : ℝ
inst✝ : CompleteSpace E
L : E →L[ℝ] F
f : ℂ → E
hf : CircleIntegrable f c R
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, (⇑L ∘ f) (circleMap c R θ) =
    (2 * π)⁻¹ • L (∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ))

Tactic:
]

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
c : ℂ
R : ℝ
inst✝ : CompleteSpace E
L : E →L[ℝ] F
f : ℂ → E
hf : CircleIntegrable f c R
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, (⇑L ∘ f) (circleMap c R θ) =
    (2 * π)⁻¹ • L (∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ))
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
𝕜 : Type u_3
inst✝³ : NormedDivisionRing 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : NormSMulClass 𝕜 E
inst✝ : SMulCommClass ℝ 𝕜 E
f : ℂ → E
c : ℂ
R : ℝ
a : 𝕜
this : SMulCommClass 𝕜 ℝ E
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, (a • f) (circleMap c R θ) =
    (2 * π)⁻¹ • a • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)

Tactic:
]

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
𝕜 : Type u_3
inst✝³ : NormedDivisionRing 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : NormSMulClass 𝕜 E
inst✝ : SMulCommClass ℝ 𝕜 E
f : ℂ → E
c : ℂ
R : ℝ
a : 𝕜
this : SMulCommClass 𝕜 ℝ E
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, (a • f) (circleMap c R θ) =
    (2 * π)⁻¹ • a • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ)
Module: Mathlib.Analysis.Complex.Harmonic.MeanValue

Goal state before:
pp: F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℂ → F
c : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F
hR : ¬R = 0
x : ℂ
hx : x ∈ {z | ‖z - c‖ ∈ Set.Ioc 0 |R|}
⊢ ‖x - c‖ ≤ |R|

Tactic:
]

Goal state after:
F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℂ → F
c : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F
hR : ¬R = 0
x : ℂ
hx : x ∈ {z | ‖z - c‖ ∈ Set.Ioc 0 |R|}
⊢ ‖x - c‖ ≤ |R|
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case inr.refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
ζ✝ : ℂ
hζ✝ : ζ✝.im ∈ Icc 0 (π / 2)
hz_re : 0 ≤ (cexp ζ✝).re
hz_im : 0 ≤ (cexp ζ✝).im
hzne : cexp ζ✝ ≠ 0
H : MapsTo cexp (im ⁻¹' Ioo 0 (π / 2)) (Ioi 0 ×ℂ Ioi 0)
ζ : ℂ
hζ : ζ.im = π / 2
⊢ ‖f (↑(expR ζ.re) * I)‖ ≤ C

Tactic:
]

Goal state after:
case inr.refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
ζ✝ : ℂ
hζ✝ : ζ✝.im ∈ Icc 0 (π / 2)
hz_re : 0 ≤ (cexp ζ✝).re
hz_im : 0 ≤ (cexp ζ✝).im
hzne : cexp ζ✝ ≠ 0
H : MapsTo cexp (im ⁻¹' Ioo 0 (π / 2)) (Ioi 0 ×ℂ Ioi 0)
ζ : ℂ
hζ : ζ.im = π / 2
⊢ ‖f (↑(expR ζ.re) * I)‖ ≤ C
Module: Mathlib.Analysis.Complex.Harmonic.MeanValue

Goal state before:
pp: case neg
F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℂ → F
c : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F
hR : ¬R = 0
H : ContinuousOn (circleAverage (⇑g ∘ f) c) (Set.Ioc 0 |R|)
⊢ circleAverage (⇑g ∘ f) c |R| = g (f c)

Tactic:
]

Goal state after:
case neg
F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℂ → F
c : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F
hR : ¬R = 0
H : ContinuousOn (circleAverage (⇑g ∘ f) c) (Set.Ioc 0 |R|)
⊢ circleAverage (⇑g ∘ f) c |R| = g (f c)
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f₁ f₂ : ℂ → E
c : ℂ
R : ℝ
hf₁ : CircleIntegrable f₁ c R
hf₂ : CircleIntegrable f₂ c R
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, (f₁ + f₂) (circleMap c R θ) =
    (2 * π)⁻¹ • ((∫ (θ : ℝ) in 0..2 * π, f₁ (circleMap c R θ)) + ∫ (θ : ℝ) in 0..2 * π, f₂ (circleMap c R θ))

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f₁ f₂ : ℂ → E
c : ℂ
R : ℝ
hf₁ : CircleIntegrable f₁ c R
hf₂ : CircleIntegrable f₂ c R
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, (f₁ + f₂) (circleMap c R θ) =
    (2 * π)⁻¹ • ((∫ (θ : ℝ) in 0..2 * π, f₁ (circleMap c R θ)) + ∫ (θ : ℝ) in 0..2 * π, f₂ (circleMap c R θ))
Module: Mathlib.Analysis.Complex.Harmonic.MeanValue

Goal state before:
pp: case neg
F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℂ → F
c : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F
hR : ¬R = 0
H : ContinuousOn (circleAverage (⇑g ∘ f) c) (Set.Ioc 0 |R|)
r : ℝ
hr : r ∈ Set.Ioo 0 |R|
⊢ closedBall c r ⊆ ball c |R|

Tactic:
]

Goal state after:
case neg
F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : ℂ → F
c : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c |R|)
h : CircleIntegrable f c R
g : StrongDual ℝ F
hR : ¬R = 0
H : ContinuousOn (circleAverage (⇑g ∘ f) c) (Set.Ioc 0 |R|)
r : ℝ
hr : r ∈ Set.Ioo 0 |R|
⊢ closedBall c r ⊆ ball c |R|
Module: Mathlib.MeasureTheory.Integral.CircleAverage

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f₁ f₂ : ℂ → E
c : ℂ
R : ℝ
hf₁ : CircleIntegrable f₁ c R
hf₂ : CircleIntegrable f₂ c R
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, (f₁ - f₂) (circleMap c R θ) =
    (2 * π)⁻¹ • ((∫ (θ : ℝ) in 0..2 * π, f₁ (circleMap c R θ)) - ∫ (θ : ℝ) in 0..2 * π, f₂ (circleMap c R θ))

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f₁ f₂ : ℂ → E
c : ℂ
R : ℝ
hf₁ : CircleIntegrable f₁ c R
hf₂ : CircleIntegrable f₂ c R
⊢ (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, (f₁ - f₂) (circleMap c R θ) =
    (2 * π)⁻¹ • ((∫ (θ : ℝ) in 0..2 * π, f₁ (circleMap c R θ)) - ∫ (θ : ℝ) in 0..2 * π, f₂ (circleMap c R θ))
Module: Mathlib.Analysis.Complex.Harmonic.Poisson

Goal state before:
pp: f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
⊢ circleAverage (re ∘ herglotzRieszKernel c w • f) c R = f w

Tactic:
]

Goal state after:
f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
⊢ circleAverage (re ∘ herglotzRieszKernel c w • f) c R = f w
Module: Mathlib.Analysis.Complex.Harmonic.Poisson

Goal state before:
pp: f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + R))
h₂F : Set.EqOn (fun z ↦ (F z).re) f (ball c (e + R))
h₃F : DifferentiableOn ℂ F (closure (ball c R))
h₄F :
  Set.EqOn (re ∘ herglotzRieszKernel c w • f) (⇑reCLM ∘ fun z ↦ ((z - c + (w - c)) / (z - c - (w - c))).re • F z)
    (sphere c |R|)
⊢ circleAverage (re ∘ herglotzRieszKernel c w • f) c R = f w

Tactic:
]

Goal state after:
f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + R))
h₂F : Set.EqOn (fun z ↦ (F z).re) f (ball c (e + R))
h₃F : DifferentiableOn ℂ F (closure (ball c R))
h₄F :
  Set.EqOn (re ∘ herglotzRieszKernel c w • f) (⇑reCLM ∘ fun z ↦ ((z - c + (w - c)) / (z - c - (w - c))).re • F z)
    (sphere c |R|)
⊢ circleAverage (re ∘ herglotzRieszKernel c w • f) c R = f w
Module: Mathlib.Analysis.Complex.Harmonic.Poisson

Goal state before:
pp: f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + R))
h₂F : Set.EqOn (fun z ↦ (F z).re) f (ball c (e + R))
h₃F : DifferentiableOn ℂ F (closure (ball c R))
h₄F :
  Set.EqOn (re ∘ herglotzRieszKernel c w • f) (⇑reCLM ∘ fun z ↦ ((z - c + (w - c)) / (z - c - (w - c))).re • F z)
    (sphere c |R|)
⊢ reCLM (F w) = f w

pp: f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + R))
h₂F : Set.EqOn (fun z ↦ (F z).re) f (ball c (e + R))
h₃F : DifferentiableOn ℂ F (closure (ball c R))
h₄F :
  Set.EqOn (re ∘ herglotzRieszKernel c w • f) (⇑reCLM ∘ fun z ↦ ((z - c + (w - c)) / (z - c - (w - c))).re • F z)
    (sphere c |R|)
⊢ CircleIntegrable (fun z ↦ ((z - c + (w - c)) / (z - c - (w - c))).re • F z) c R

Tactic:
]

Goal state after:
f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + R))
h₂F : Set.EqOn (fun z ↦ (F z).re) f (ball c (e + R))
h₃F : DifferentiableOn ℂ F (closure (ball c R))
h₄F :
  Set.EqOn (re ∘ herglotzRieszKernel c w • f) (⇑reCLM ∘ fun z ↦ ((z - c + (w - c)) / (z - c - (w - c))).re • F z)
    (sphere c |R|)
⊢ reCLM (F w) = f w

f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
e : ℝ
h₁e : 0 < e
h₂e : ball c (e + R) ⊆ {x | HarmonicAt f x}
F : ℂ → ℂ
h₁F : AnalyticOnNhd ℂ F (ball c (e + R))
h₂F : Set.EqOn (fun z ↦ (F z).re) f (ball c (e + R))
h₃F : DifferentiableOn ℂ F (closure (ball c R))
h₄F :
  Set.EqOn (re ∘ herglotzRieszKernel c w • f) (⇑reCLM ∘ fun z ↦ ((z - c + (w - c)) / (z - c - (w - c))).re • F z)
    (sphere c |R|)
⊢ CircleIntegrable (fun z ↦ ((z - c + (w - c)) / (z - c - (w - c))).re • F z) c R
Module: Mathlib.Analysis.Complex.UpperHalfPlane.Basic

Goal state before:
pp: z : ℍ
⊢ ↑z ≠ 0

Tactic:
]

Goal state after:
z : ℍ
⊢ ↑z ≠ 0
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.im
hz_im : 0 ≤ z.re
H : MapsTo (fun x ↦ x * I) (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Ioi 0)
c : ℝ
hc : c < 2
B : ℝ
hO : f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)
x : ℝ
hx : 0 ≤ x
⊢ ‖f ↑(-x)‖ ≤ C

Tactic:
]

Goal state after:
case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.im
hz_im : 0 ≤ z.re
H : MapsTo (fun x ↦ x * I) (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Ioi 0)
c : ℝ
hc : c < 2
B : ℝ
hO : f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)
x : ℝ
hx : 0 ≤ x
⊢ ‖f ↑(-x)‖ ≤ C
Module: Mathlib.Analysis.Complex.IsIntegral

Goal state before:
pp: ⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.IsIntegral

Goal state before:
pp: ⊢ eval₂ (algebraMap ℤ ℂ) I (X ^ 2 + C 1) = 0

Tactic:
rw [eval₂_add, eval₂_X_pow, eval₂_C, I_sq, eq_intCast, Int.cast_one, neg_add_cancel]

Goal state after:
Module: Mathlib.Analysis.Complex.Harmonic.Poisson

Goal state before:
pp: case h₁f.hf
f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c R)
hw : w ∈ ball c R
⊢ ContinuousOn ((re ∘ fun z ↦ (z - c + (w - c)) / (z - c - (w - c))) • f) {z | ‖z - c‖ ∈ Set.Ioc ‖w - c‖ R}

Tactic:
]

Goal state after:
case h₁f.hf
f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c R)
hw : w ∈ ball c R
⊢ ContinuousOn ((re ∘ fun z ↦ (z - c + (w - c)) / (z - c - (w - c))) • f) {z | ‖z - c‖ ∈ Set.Ioc ‖w - c‖ R}
Module: Mathlib.Analysis.Complex.Harmonic.Poisson

Goal state before:
pp: case h₂f
f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c R)
hw : w ∈ ball c R
r : ℝ
hr : r ∈ Set.Ioo ‖w - c‖ R
⊢ f w = (fun x ↦ f w) r

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.Harmonic.Poisson

Goal state before:
pp: f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
⊢ circleAverage (poissonKernel c w • f) c R = circleAverage (re ∘ herglotzRieszKernel c w • f) c R

Tactic:
]

Goal state after:
f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicOnNhd f (closedBall c R)
hw : w ∈ ball c R
⊢ circleAverage (poissonKernel c w • f) c R = circleAverage (re ∘ herglotzRieszKernel c w • f) c R
Module: Mathlib.Analysis.Complex.Harmonic.Poisson

Goal state before:
pp: f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c R)
hw : w ∈ ball c R
⊢ circleAverage (poissonKernel c w • f) c R = circleAverage (re ∘ herglotzRieszKernel c w • f) c R

Tactic:
]

Goal state after:
f : ℂ → ℝ
c w : ℂ
R : ℝ
hf : HarmonicContOnCl f (ball c R)
hw : w ∈ ball c R
⊢ circleAverage (poissonKernel c w • f) c R = circleAverage (re ∘ herglotzRieszKernel c w • f) c R
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)
x : ℝ
hx : 0 ≤ x
⊢ ‖f ↑(-x)‖ ≤ C

Tactic:
]

Goal state after:
case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)
x : ℝ
hx : 0 ≤ x
⊢ ‖f ↑(-x)‖ ≤ C
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)
x : ℝ
hx : 0 ≤ x
⊢ ‖f (↑(-x) * I)‖ ≤ C

Tactic:
]

Goal state after:
case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)
hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z ↦ expR (B * ‖z‖ ^ c)
hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C
him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C
z : ℂ
hz_re : 0 ≤ z.re
hz_im : 0 ≤ z.im
H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)
x : ℝ
hx : 0 ≤ x
⊢ ‖f (↑(-x) * I)‖ ≤ C
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: a b : ℂ
⊢ ((a.re + b.re) * (a.re - b.re) + (a.im + b.im) * (a.im - b.im)) / ‖a - b‖ ^ 2 = (‖a‖ ^ 2 - ‖b‖ ^ 2) / ‖a - b‖ ^ 2

Tactic:
]

Goal state after:
a b : ℂ
⊢ ((a.re + b.re) * (a.re - b.re) + (a.im + b.im) * (a.im - b.im)) / ‖a - b‖ ^ 2 = (‖a‖ ^ 2 - ‖b‖ ^ 2) / ‖a - b‖ ^ 2
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: c w z : ℂ
⊢ (‖z - c‖ ^ 2 - ‖w - c‖ ^ 2) / ‖z - c - (w - c)‖ ^ 2 = (‖z - c‖ ^ 2 - ‖w - c‖ ^ 2) / ‖z - c - (w - c)‖ ^ 2

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: φ θ r R : ℝ
h₁ : 0 < r
h₂ : r < R
⊢ ((↑R * cexp (↑θ * I) + ↑r * cexp (↑φ * I)) * (↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I))⁻¹).re ≤ (R + r) / (R - r)

Tactic:
]

Goal state after:
φ θ r R : ℝ
h₁ : 0 < r
h₂ : r < R
⊢ ((↑R * cexp (↑θ * I) + ↑r * cexp (↑φ * I)) * (↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I))⁻¹).re ≤ (R + r) / (R - r)
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: φ θ r R : ℝ
h₁ : 0 < r
h₂ : r < R
h_cos : R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ) ≥ (R - r) ^ 2
⊢ (R ^ 2 - r ^ 2) * (R - r) ≤ (R + r) * (R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ))

pp: case hb
φ θ r R : ℝ
h₁ : 0 < r
h₂ : r < R
h_cos : R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ) ≥ (R - r) ^ 2
⊢ 0 < R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ)

pp: case hd
φ θ r R : ℝ
h₁ : 0 < r
h₂ : r < R
h_cos : R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ) ≥ (R - r) ^ 2
⊢ 0 < R - r

Tactic:
]

Goal state after:
φ θ r R : ℝ
h₁ : 0 < r
h₂ : r < R
h_cos : R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ) ≥ (R - r) ^ 2
⊢ (R ^ 2 - r ^ 2) * (R - r) ≤ (R + r) * (R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ))

case hb
φ θ r R : ℝ
h₁ : 0 < r
h₂ : r < R
h_cos : R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ) ≥ (R - r) ^ 2
⊢ 0 < R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ)

case hd
φ θ r R : ℝ
h₁ : 0 < r
h₂ : r < R
h_cos : R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ) ≥ (R - r) ^ 2
⊢ 0 < R - r
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: φ θ r R : ℝ
h₁ : 0 < r
h₂ : r < R
h_cos : R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ) ≥ (R - r) ^ 2
h_subst : (R ^ 2 - r ^ 2) / (R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ)) ≤ (R + r) / (R - r)
⊢ (‖↑R * cexp (↑θ * I)‖ ^ 2 - ‖↑r * cexp (↑φ * I)‖ ^ 2) / ‖↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I)‖ ^ 2 =
    (R ^ 2 - r ^ 2) / (R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ))

Tactic:
]

Goal state after:
φ θ r R : ℝ
h₁ : 0 < r
h₂ : r < R
h_cos : R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ) ≥ (R - r) ^ 2
h_subst : (R ^ 2 - r ^ 2) / (R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ)) ≤ (R + r) / (R - r)
⊢ (‖↑R * cexp (↑θ * I)‖ ^ 2 - ‖↑r * cexp (↑φ * I)‖ ^ 2) / ‖↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I)‖ ^ 2 =
    (R ^ 2 - r ^ 2) / (R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ))