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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
hfc : ContinuousOn (fun x ↦ f ↑x) (Ici 0)
x₀ : ℝ
hx₀ : 0 ≤ x₀
hne : f ↑x₀ ≠ 0
⊢ f ↑x₀ ≠ 0

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
hfc : ContinuousOn (fun x ↦ f ↑x) (Ici 0)
x₀ : ℝ
hx₀ : 0 ≤ x₀
hne : f ↑x₀ ≠ 0
⊢ f ↑x₀ ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case neg
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
⊢ IntervalIntegrable (fun x ↦ ↑x ^ r) μ a b

Tactic:
]

Goal state after:
case neg
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
⊢ IntervalIntegrable (fun x ↦ ↑x ^ r) μ a b
Module: Mathlib.Analysis.SpecialFunctions.Log.InvLog

Goal state before:
pp: case inr.inl
this✝¹ : ¬ContinuousAt (fun x ↦ log (log x)) (-1)
this✝ : ¬ContinuousAt (fun x ↦ log (log x)) 0
this : ¬ContinuousAt (fun x ↦ log (log x)) 1
⊢ 0 = (-1)⁻¹ / log (-1)

Tactic:
]

Goal state after:
case inr.inl
this✝¹ : ¬ContinuousAt (fun x ↦ log (log x)) (-1)
this✝ : ¬ContinuousAt (fun x ↦ log (log x)) 0
this : ¬ContinuousAt (fun x ↦ log (log x)) 1
⊢ 0 = (-1)⁻¹ / log (-1)
Module: Mathlib.Analysis.SpecialFunctions.Log.InvLog

Goal state before:
pp: case inr.inr.inl
this✝¹ : ¬ContinuousAt (fun x ↦ log (log x)) (-1)
this✝ : ¬ContinuousAt (fun x ↦ log (log x)) 0
this : ¬ContinuousAt (fun x ↦ log (log x)) 1
⊢ 0 = 0⁻¹ / log 0

Tactic:
]

Goal state after:
case inr.inr.inl
this✝¹ : ¬ContinuousAt (fun x ↦ log (log x)) (-1)
this✝ : ¬ContinuousAt (fun x ↦ log (log x)) 0
this : ¬ContinuousAt (fun x ↦ log (log x)) 1
⊢ 0 = 0⁻¹ / log 0
Module: Mathlib.Analysis.SpecialFunctions.Log.InvLog

Goal state before:
pp: case inr.inr.inr
this✝¹ : ¬ContinuousAt (fun x ↦ log (log x)) (-1)
this✝ : ¬ContinuousAt (fun x ↦ log (log x)) 0
this : ¬ContinuousAt (fun x ↦ log (log x)) 1
⊢ 0 = 1⁻¹ / log 1

Tactic:
]

Goal state after:
case inr.inr.inr
this✝¹ : ¬ContinuousAt (fun x ↦ log (log x)) (-1)
this✝ : ¬ContinuousAt (fun x ↦ log (log x)) 0
this : ¬ContinuousAt (fun x ↦ log (log x)) 1
⊢ 0 = 1⁻¹ / log 1
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case neg.inr.refine_2.inl
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : 0 ≤ c
this : IntegrableOn (fun x ↦ 1) (Ioc 0 c) μ
⊢ IntegrableOn (fun x ↦ ‖↑x ^ r‖) (Ioc 0 c) μ

Tactic:
]

Goal state after:
case neg.inr.refine_2.inl
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : 0 ≤ c
this : IntegrableOn (fun x ↦ 1) (Ioc 0 c) μ
⊢ IntegrableOn (fun x ↦ ‖↑x ^ r‖) (Ioc 0 c) μ
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
h0 : ∮ (z : ℂ) in C(0, R), z⁻¹ • (↑q * z / (↑q * z - W)) • f z = 0
z : ℂ
hz : z ∈ sphere 0 |R|
hzR : ‖z‖ = R
⊢ ↑R * cexp (↑z.arg * I) / (↑R * cexp (↑z.arg * I) - w) -
      ↑‖w‖ * cexp (↑z.arg * I) / (↑‖w‖ * cexp (↑z.arg * I) - ↑R * cexp (↑w.arg * I)) =
    ↑R * cexp (↑z.arg * I) / (↑R * cexp (↑z.arg * I) - w) -
      ↑‖w‖ / ↑R * (↑R * cexp (↑z.arg * I)) / (↑‖w‖ / ↑R * (↑R * cexp (↑z.arg * I)) - ↑R * cexp (↑w.arg * I))

Tactic:
]

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
h0 : ∮ (z : ℂ) in C(0, R), z⁻¹ • (↑q * z / (↑q * z - W)) • f z = 0
z : ℂ
hz : z ∈ sphere 0 |R|
hzR : ‖z‖ = R
⊢ ↑R * cexp (↑z.arg * I) / (↑R * cexp (↑z.arg * I) - w) -
      ↑‖w‖ * cexp (↑z.arg * I) / (↑‖w‖ * cexp (↑z.arg * I) - ↑R * cexp (↑w.arg * I)) =
    ↑R * cexp (↑z.arg * I) / (↑R * cexp (↑z.arg * I) - w) -
      ↑‖w‖ / ↑R * (↑R * cexp (↑z.arg * I)) / (↑‖w‖ / ↑R * (↑R * cexp (↑z.arg * I)) - ↑R * cexp (↑w.arg * I))
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case neg.inr.refine_2.inl
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : 0 ≤ c
this : IntegrableOn (fun x ↦ 1) (Ioc 0 c) μ
x : ℝ
hx : x ∈ Ioc 0 c
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case neg.inr.refine_2.inr
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : c < 0
⊢ IntegrableOn (fun x ↦ ‖↑x ^ r‖) (Ioc c 0) μ

Tactic:
]

Goal state after:
case neg.inr.refine_2.inr
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : c < 0
⊢ IntegrableOn (fun x ↦ ‖↑x ^ r‖) (Ioc c 0) μ
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case neg.inr.refine_2.inr
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : c < 0
⊢ IntegrableOn (fun x ↦ ‖↑x ^ r‖) {0} μ ∧ IntegrableOn (fun x ↦ ‖↑x ^ r‖) (Ioo c 0) μ

Tactic:
]

Goal state after:
case neg.inr.refine_2.inr
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : c < 0
⊢ IntegrableOn (fun x ↦ ‖↑x ^ r‖) {0} μ ∧ IntegrableOn (fun x ↦ ‖↑x ^ r‖) (Ioo c 0) μ
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : c < 0
x : ℝ
hx : x ∈ Ioo c 0
⊢ ‖Complex.exp (↑π * Complex.I * r)‖ = ‖Complex.exp (↑π * Complex.I * r)‖

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case neg.inr.refine_2.inr.right
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : c < 0
this : ∀ x ∈ Ioo c 0, ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖
⊢ ‖‖Complex.exp (↑π * Complex.I * r)‖‖ₑ = 0 ∨ μ (Ioo c 0) < ⊤

Tactic:
]

Goal state after:
case neg.inr.refine_2.inr.right
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
c : ℝ
hc : c < 0
this : ∀ x ∈ Ioo c 0, ‖Complex.exp (↑π * Complex.I * r)‖ = ‖↑x ^ r‖
⊢ ‖‖Complex.exp (↑π * Complex.I * r)‖‖ₑ = 0 ∨ μ (Ioo c 0) < ⊤
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
hfc : ContinuousOn (fun x ↦ f ↑x) (Ici 0)
x₀ : ℝ
hx₀ : 0 ≤ x₀
hne : f ↑x₀ ≠ 0
hlt : ‖0‖ < ‖f ↑x₀‖
⊢ ∀ᶠ (x : ℝ) in atTop ⊓ 𝓟 (Ici 0), ‖f ↑x‖ ≤ ‖f ↑x₀‖

Tactic:
]

Goal state after:
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
hfc : ContinuousOn (fun x ↦ f ↑x) (Ici 0)
x₀ : ℝ
hx₀ : 0 ≤ x₀
hne : f ↑x₀ ≠ 0
hlt : ‖0‖ < ‖f ↑x₀‖
⊢ ∀ᶠ (x : ℝ) in atTop ⊓ 𝓟 (Ici 0), ‖f ↑x‖ ≤ ‖f ↑x₀‖
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ IntervalIntegrable (fun t ↦ ‖↑t ^ r‖) volume 0 c

pp: a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ AEStronglyMeasurable (fun x ↦ ↑x ^ r) (volume.restrict (Ι 0 c))

Tactic:
]

Goal state after:
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ IntervalIntegrable (fun t ↦ ‖↑t ^ r‖) volume 0 c

a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ AEStronglyMeasurable (fun x ↦ ↑x ^ r) (volume.restrict (Ι 0 c))
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ IntegrableOn (fun t ↦ ‖↑t ^ r‖) (Ι 0 c) volume

Tactic:
]

Goal state after:
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ IntegrableOn (fun t ↦ ‖↑t ^ r‖) (Ι 0 c) volume
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case h
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ IntervalIntegrable ?m.108 volume 0 c

pp: a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ ℝ → ℝ

Tactic:
]

Goal state after:
case h
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ IntervalIntegrable ?m.108 volume 0 c

a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ ℝ → ℝ
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case hst
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
x : ℝ
hx : x ∈ Ioc 0 c
⊢ (fun x ↦ x ^ r.re) x = (fun t ↦ ‖↑t ^ r‖) x

Tactic:
]

Goal state after:
case hst
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
x : ℝ
hx : x ∈ Ioc 0 c
⊢ (fun x ↦ x ^ r.re) x = (fun t ↦ ‖↑t ^ r‖) x
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case hst
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
x : ℝ
hx : x ∈ Ioc 0 c
⊢ x ^ r.re = x ^ r.re

Tactic:
]

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
h0 : ∮ (z : ℂ) in C(0, R), z⁻¹ • (↑q * z / (↑q * z - W)) • f z = 0
h₁ : ∀ z ∈ sphere 0 R, z - w ≠ 0
h₃ : ContinuousOn f (sphere 0 R)
⊢ Real.circleAverage (fun z ↦ (z / (z - w)) • f z) 0 R - Real.circleAverage (fun z ↦ (q • z / (q • z - W)) • f z) 0 R =
    Real.circleAverage (fun z ↦ (z / (z - w)) • f z) 0 R - Real.circleAverage (fun z ↦ (q • z / (q • z - W)) • f z) 0 R

pp: case hf₁
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
h0 : ∮ (z : ℂ) in C(0, R), z⁻¹ • (↑q * z / (↑q * z - W)) • f z = 0
h₁ : ∀ z ∈ sphere 0 R, z - w ≠ 0
h₃ : ContinuousOn f (sphere 0 R)
⊢ CircleIntegrable (fun z ↦ (z / (z - w)) • f z) 0 R

pp: case hf₂
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
h0 : ∮ (z : ℂ) in C(0, R), z⁻¹ • (↑q * z / (↑q * z - W)) • f z = 0
h₁ : ∀ z ∈ sphere 0 R, z - w ≠ 0
h₃ : ContinuousOn f (sphere 0 R)
⊢ CircleIntegrable (fun z ↦ (q • z / (q • z - W)) • f z) 0 R

Tactic:
]

Goal state after:
case hf₁
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
h0 : ∮ (z : ℂ) in C(0, R), z⁻¹ • (↑q * z / (↑q * z - W)) • f z = 0
h₁ : ∀ z ∈ sphere 0 R, z - w ≠ 0
h₃ : ContinuousOn f (sphere 0 R)
⊢ CircleIntegrable (fun z ↦ (z / (z - w)) • f z) 0 R

case hf₂
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
h0 : ∮ (z : ℂ) in C(0, R), z⁻¹ • (↑q * z / (↑q * z - W)) • f z = 0
h₁ : ∀ z ∈ sphere 0 R, z - w ≠ 0
h₃ : ContinuousOn f (sphere 0 R)
⊢ CircleIntegrable (fun z ↦ (q • z / (q • z - W)) • f z) 0 R
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
x : ℝ
hx : x ∈ Ioc 0 c
⊢ ContinuousAt (fun x ↦ ↑x ^ r) x

Tactic:
]

Goal state after:
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
x : ℝ
hx : x ∈ Ioc 0 c
⊢ ContinuousAt (fun x ↦ ↑x ^ r) x
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
x : ℝ
hx : x ∈ Ioc 0 c
⊢ 0 < x

Tactic:
]

Goal state after:
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
x : ℝ
hx : x ∈ Ioc 0 c
⊢ 0 < x
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 |R|)
hw : w ∈ ball 0 |R|
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
h0 : ∮ (z : ℂ) in C(0, R), z⁻¹ • (↑q * z / (↑q * z - W)) • f z = 0
⊢ Real.circleAverage (fun z ↦ (z / (z - w)) • f z) 0 R - Real.circleAverage (fun z ↦ (q • z / (q • z - W)) • f z) 0 R =
    f w

Tactic:
]

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 |R|)
hw : w ∈ ball 0 |R|
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
η₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑q * x - W ≠ 0
h0 : ∮ (z : ℂ) in C(0, R), z⁻¹ • (↑q * z / (↑q * z - W)) • f z = 0
⊢ Real.circleAverage (fun z ↦ (z / (z - w)) • f z) 0 R - Real.circleAverage (fun z ↦ (q • z / (q • z - W)) • f z) 0 R =
    f w
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case inr
a b : ℝ
r : ℂ
h : -1 < r.re
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x ↦ ↑x ^ r) volume 0 c
c : ℝ
hc : c ≤ 0
⊢ IntervalIntegrable (fun x ↦ ↑(-x) ^ r) volume 0 (-c)

Tactic:
]

Goal state after:
case inr
a b : ℝ
r : ℂ
h : -1 < r.re
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x ↦ ↑x ^ r) volume 0 c
c : ℝ
hc : c ≤ 0
⊢ IntervalIntegrable (fun x ↦ ↑(-x) ^ r) volume 0 (-c)
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case inr
a b : ℝ
r : ℂ
h : -1 < r.re
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x ↦ ↑x ^ r) volume 0 c
c : ℝ
hc : c ≤ 0
m : IntegrableOn (fun x ↦ Complex.exp (↑π * Complex.I * r) * ↑x ^ r) (Ioc 0 (-c)) volume
⊢ IntegrableOn (fun x ↦ ↑(-x) ^ r) (Ioc 0 (-c)) volume

Tactic:
]

Goal state after:
case inr
a b : ℝ
r : ℂ
h : -1 < r.re
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x ↦ ↑x ^ r) volume 0 c
c : ℝ
hc : c ≤ 0
m : IntegrableOn (fun x ↦ Complex.exp (↑π * Complex.I * r) * ↑x ^ r) (Ioc 0 (-c)) volume
⊢ IntegrableOn (fun x ↦ ↑(-x) ^ r) (Ioc 0 (-c)) volume
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: case refine_1
R : ℝ
w c : ℂ
hw : w ∈ ball c R
this : Disjoint {w} (ball c R)ᶜ
d : ℝ
hd : 0 < d
h_disj : Disjoint (ball w d) (thickening d (ball c R)ᶜ)
⊢ ball w d ⊆ ball c R

Tactic:
]

Goal state after:
case refine_1
R : ℝ
w c : ℂ
hw : w ∈ ball c R
this : Disjoint {w} (ball c R)ᶜ
d : ℝ
hd : 0 < d
h_disj : Disjoint (ball w d) (thickening d (ball c R)ᶜ)
⊢ ball w d ⊆ ball c R
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: case refine_2
R : ℝ
w c : ℂ
hw : w ∈ ball c R
this : Disjoint {w} (ball c R)ᶜ
d : ℝ
hd : 0 < d
h_disj : Disjoint (ball w d) (thickening d (ball c R)ᶜ)
⊢ ∀ x ∈ ball w d, ∀ (θ : ℝ), d ≤ ‖circleMap c R θ - x‖

Tactic:
]

Goal state after:
case refine_2
R : ℝ
w c : ℂ
hw : w ∈ ball c R
this : Disjoint {w} (ball c R)ᶜ
d : ℝ
hd : 0 < d
h_disj : Disjoint (ball w d) (thickening d (ball c R)ᶜ)
⊢ ∀ x ∈ ball w d, ∀ (θ : ℝ), d ≤ ‖circleMap c R θ - x‖
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case inr
a b : ℝ
r : ℂ
h : -1 < r.re
this✝ : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x ↦ ↑x ^ r) volume 0 c
c : ℝ
hc : c ≤ 0
m : IntegrableOn (fun x ↦ Complex.exp (↑π * Complex.I * r) * ↑x ^ r) (Ioc 0 (-c)) volume
x : ℝ
hx : x ∈ Ioc 0 (-c)
this : -x ≤ 0
⊢ ↑x ^ r * Complex.exp (↑π * Complex.I * r) = (-↑(-x)) ^ r * Complex.exp (↑π * Complex.I * r)

Tactic:
]

Goal state after:
case inr
a b : ℝ
r : ℂ
h : -1 < r.re
this✝ : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x ↦ ↑x ^ r) volume 0 c
c : ℝ
hc : c ≤ 0
m : IntegrableOn (fun x ↦ Complex.exp (↑π * Complex.I * r) * ↑x ^ r) (Ioc 0 (-c)) volume
x : ℝ
hx : x ∈ Ioc 0 (-c)
this : -x ≤ 0
⊢ ↑x ^ r * Complex.exp (↑π * Complex.I * r) = (-↑(-x)) ^ r * Complex.exp (↑π * Complex.I * r)
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case refine_2
s : ℂ
t : ℝ
ht : 0 < t
h : -1 < s.re
⊢ IntervalIntegrable (fun x ↦ ↑x ^ s) volume 0 t

Tactic:
]

Goal state after:
case refine_2
s : ℂ
t : ℝ
ht : 0 < t
h : -1 < s.re
⊢ IntervalIntegrable (fun x ↦ ↑x ^ s) volume 0 t
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case refine_1
s : ℂ
t : ℝ
ht : 0 < t
h : IntegrableOn (fun x ↦ ↑x ^ s) (Ioo 0 t) volume
B : -1 < s.re
⊢ -1 < s.re

Tactic:
]

Goal state after:
case refine_1
s : ℂ
t : ℝ
ht : 0 < t
h : IntegrableOn (fun x ↦ ↑x ^ s) (Ioo 0 t) volume
B : -1 < s.re
⊢ -1 < s.re
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
x₀ : ℝ
hx₀ : 0 ≤ x₀
h : C < ‖f ↑x₀‖
hmax : IsMaxOn (norm ∘ f) {z | 0 < z.re} ↑x₀
z : ℂ
hz : ‖z - ↑x₀‖ < x₀
⊢ z ∈ {z | 0 < z.re}

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
x₀ : ℝ
hx₀ : 0 ≤ x₀
h : C < ‖f ↑x₀‖
hmax : IsMaxOn (norm ∘ f) {z | 0 < z.re} ↑x₀
z : ℂ
hz : ‖z - ↑x₀‖ < x₀
⊢ z ∈ {z | 0 < z.re}
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case hab
a b x : ℝ
hx : 0 < x
⊢ IntervalIntegrable (- -log) volume 0 1

Tactic:
]

Goal state after:
case hab
a b x : ℝ
hx : 0 < x
⊢ IntervalIntegrable (- -log) volume 0 1
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
x₀ : ℝ
hx₀ : 0 ≤ x₀
h : C < ‖f ↑x₀‖
hmax : IsMaxOn (norm ∘ f) {z | 0 < z.re} ↑x₀
z : ℂ
hz : ‖z - ↑x₀‖ < x₀
⊢ 0 < z.re

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
x₀ : ℝ
hx₀ : 0 ≤ x₀
h : C < ‖f ↑x₀‖
hmax : IsMaxOn (norm ∘ f) {z | 0 < z.re} ↑x₀
z : ℂ
hz : ‖z - ↑x₀‖ < x₀
⊢ 0 < z.re
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: x : ℝ
⊢ max 0 (log x) - max 0 (-log x) = log x

Tactic:
]

Goal state after:
x : ℝ
⊢ max 0 (log x) - max 0 (-log x) = log x
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: x : ℝ
⊢ log⁺ |x| = 0 ↔ log |x| ≤ 0

Tactic:
]

Goal state after:
x : ℝ
⊢ log⁺ |x| = 0 ↔ log |x| ≤ 0
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: x : ℝ
hx : 1 ≤ |x|
⊢ 0 ≤ log |x|

Tactic:
]

Goal state after:
x : ℝ
hx : 1 ≤ |x|
⊢ 0 ≤ log |x|
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hle : ∀ (C' : ℝ), (∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C') → ∀ (z : ℂ), 0 ≤ z.re → ‖f z‖ ≤ max C C'
x₀ : ℝ
hx₀ : 0 ≤ x₀
h : C < ‖f ↑x₀‖
hmax : IsMaxOn (norm ∘ f) {z | 0 < z.re} ↑x₀
z : ℂ
hz : z.re ≤ 0
⊢ |z.re - x₀| = |z.re - x₀|

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: case neg
n : ℕ
x : ℝ
hn : ¬n = 0
hx : 1 < |x|
⊢ log⁺ (x ^ n) = ↑n * log⁺ x

Tactic:
]

Goal state after:
case neg
n : ℕ
x : ℝ
hn : ¬n = 0
hx : 1 < |x|
⊢ log⁺ (x ^ n) = ↑n * log⁺ x
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: ⊢ ∀ (x : ℝ), ContinuousAt log⁺ x

Tactic:
]

Goal state after:
⊢ ∀ (x : ℝ), ContinuousAt log⁺ x
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: x : ℝ
hx : x = 0
y : ℝ
hy : y ∈ Metric.ball x 1
⊢ |y| ≤ 1

Tactic:
]

Goal state after:
x : ℝ
hx : x = 0
y : ℝ
hy : y ∈ Metric.ball x 1
⊢ |y| ≤ 1
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: case neg
x : ℝ
hx : ¬x = 0
⊢ ContinuousAt (0 ⊔ fun x ↦ log x) x

Tactic:
]

Goal state after:
case neg
x : ℝ
hx : ¬x = 0
⊢ ContinuousAt (0 ⊔ fun x ↦ log x) x
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: case neg
x y : ℝ
ha : ¬x = 0
hb : ¬y = 0
⊢ max (0 + 0) (log x + log y) ≤ max 0 (log x) + max 0 (log y)

Tactic:
]

Goal state after:
case neg
x y : ℝ
ha : ¬x = 0
hb : ¬y = 0
⊢ max (0 + 0) (log x + log y) ≤ max 0 (log x) + max 0 (log y)
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: x : ℝ
n : ℕ
⊢ log⁺ (↑n * x) ≤ log⁺ ↑n + log⁺ x

Tactic:
]

Goal state after:
x : ℝ
n : ℕ
⊢ log⁺ (↑n * x) ≤ log⁺ ↑n + log⁺ x
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: α : Type u_1
f : α → ℝ
a : α
s : Finset α
ha : a ∉ s
hs : log⁺ (∏ t ∈ s, f t) ≤ ∑ t ∈ s, log⁺ (f t)
⊢ log⁺ (f a * ∏ x ∈ s, f x) = log⁺ (f a * ∏ t ∈ s, f t)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: α : Type u_1
f : α → ℝ
a : α
s : Finset α
ha : a ∉ s
hs : log⁺ (∏ t ∈ s, f t) ≤ ∑ t ∈ s, log⁺ (f t)
⊢ log⁺ (f a) + ∑ t ∈ s, log⁺ (f t) = log⁺ (f a) + ∑ x ∈ s, log⁺ (f x)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: α : Type u_1
s : Finset α
f : α → ℝ
hs : s.Nonempty
t_max : α
ht_max : t_max ∈ s ∧ ∀ x' ∈ s, |f x'| ≤ |f t_max|
⊢ log⁺ (∑ t ∈ s, f t) = log⁺ (∑ t ∈ s, f t)

Tactic:
]

Goal state after: