mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: b : ℝ
ht : 0 < b
⊢ b * log b - b - 0 = b * log b - b

pp: case hab
b : ℝ
ht : 0 < b
⊢ 0 < b

pp: case hderiv
b : ℝ
ht : 0 < b
⊢ ∀ x ∈ Set.Ioo 0 b, HasDerivAt (fun x ↦ x * log x - x) (log x) x

pp: case ha
b : ℝ
ht : 0 < b
⊢ Filter.Tendsto (fun x ↦ x * log x - x) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0)

pp: case hb
b : ℝ
ht : 0 < b
⊢ Filter.Tendsto (fun x ↦ x * log x - x) (nhdsWithin b (Set.Iio b)) (nhds (b * log b - b))

Tactic:
]

Goal state after:
b : ℝ
ht : 0 < b
⊢ b * log b - b - 0 = b * log b - b

case hab
b : ℝ
ht : 0 < b
⊢ 0 < b

case hderiv
b : ℝ
ht : 0 < b
⊢ ∀ x ∈ Set.Ioo 0 b, HasDerivAt (fun x ↦ x * log x - x) (log x) x

case ha
b : ℝ
ht : 0 < b
⊢ Filter.Tendsto (fun x ↦ x * log x - x) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0)

case hb
b : ℝ
ht : 0 < b
⊢ Filter.Tendsto (fun x ↦ x * log x - x) (nhdsWithin b (Set.Iio b)) (nhds (b * log b - b))
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: case inl
b : ℝ
h : b < 0
⊢ -(-b * log b - -b) = b * log b - b

Tactic:
]

Goal state after:
case inl
b : ℝ
h : b < 0
⊢ -(-b * log b - -b) = b * log b - b
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: this✝ : AnalyticOnNhd ℝ (fun x ↦ 4 * sin x ^ 2) Set.univ
this : ((fun x ↦ 4 * sin x ^ 2) ⁻¹' {0})ᶜ ∈ codiscrete ℝ
a : ℝ
ha : ¬sin a = 0
⊢ log 4 + 2 * log (sin a) = log 4 + 2 * log (sin a)

Tactic:
]

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

Goal state before:
pp: case inr.inr
b : ℝ
h : 0 < b
⊢ ∫ (s : ℝ) in 0..b, log s = b * log b - b

Tactic:
exact integral_log_from_zero_of_pos h

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

Goal state before:
pp: case inr.inr
b : ℝ
h : 0 < b
⊢ ∫ (s : ℝ) in 0..b, log s = b * log b - b

Tactic:
· exact integral_log_from_zero_of_pos h

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

Goal state before:
pp: a b : ℝ
⊢ (∫ (x : ℝ) in a..0, log x) + ∫ (x : ℝ) in 0..b, log x = b * log b - a * log a - b + a

pp: case hab
a b : ℝ
⊢ IntervalIntegrable log MeasureTheory.volume a 0

pp: case hbc
a b : ℝ
⊢ IntervalIntegrable log MeasureTheory.volume 0 b

Tactic:
]

Goal state after:
a b : ℝ
⊢ (∫ (x : ℝ) in a..0, log x) + ∫ (x : ℝ) in 0..b, log x = b * log b - a * log a - b + a

case hab
a b : ℝ
⊢ IntervalIntegrable log MeasureTheory.volume a 0

case hbc
a b : ℝ
⊢ IntervalIntegrable log MeasureTheory.volume 0 b
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: ⊢ (∫ (x : ℝ) in 0..π, log 4) + 2 * ∫ (x : ℝ) in 0..π, log (sin x) =
    (∫ (x : ℝ) in 0..π, log 4) + 2 * ∫ (x : ℝ) in 0..π, log (sin x)

Tactic:
]

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

Goal state before:
pp: a b : ℝ
⊢ -(a * log a - a) + (b * log b - b) = b * log b - a * log a - b + a

Tactic:
]

Goal state after:
a b : ℝ
⊢ -(a * log a - a) + (b * log b - b) = b * log b - a * log a - b + a
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
⊢ -cos b - -cos a = cos a - cos b

pp: case hderiv
a b : ℝ
⊢ (deriv fun x ↦ -cos x) = sin

pp: case hdiff
a b : ℝ
⊢ ∀ x ∈ [[a, b]], DifferentiableAt ℝ (fun x ↦ -cos x) x

pp: case hcont
a b : ℝ
⊢ ContinuousOn sin [[a, b]]

Tactic:
]

Goal state after:
a b : ℝ
⊢ -cos b - -cos a = cos a - cos b

case hderiv
a b : ℝ
⊢ (deriv fun x ↦ -cos x) = sin

case hdiff
a b : ℝ
⊢ ∀ x ∈ [[a, b]], DifferentiableAt ℝ (fun x ↦ -cos x) x

case hcont
a b : ℝ
⊢ ContinuousOn sin [[a, b]]
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ = 1
⊢ ∃ ζ, a⁻¹ = circleMap 0 1 ζ

Tactic:
by simp [Set.exists_range_iff.1, h]

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

Goal state before:
pp: a b : ℝ
⊢ ?f b - ?f a = sin b - sin a

pp: case f
a b : ℝ
⊢ ℝ → ℝ

pp: case hderiv
a b : ℝ
⊢ deriv ?f = cos

pp: case hdiff
a b : ℝ
⊢ ∀ x ∈ [[a, b]], DifferentiableAt ℝ ?f x

pp: case hcont
a b : ℝ
⊢ ContinuousOn cos [[a, b]]

Tactic:
]

Goal state after:
case hderiv
a b : ℝ
⊢ deriv sin = cos

case hdiff
a b : ℝ
⊢ ∀ x ∈ [[a, b]], DifferentiableAt ℝ sin x

case hcont
a b : ℝ
⊢ ContinuousOn cos [[a, b]]
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ log (Complex.normSq (circleMap 0 1 x - 1)) / 2 = log (4 * sin (x / 2) ^ 2) / 2

Tactic:
]

Goal state after:
a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ log (Complex.normSq (circleMap 0 1 x - 1)) / 2 = log (4 * sin (x / 2) ^ 2) / 2
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ log ‖circleMap 0 1 x - 1‖ = log (4 * sin (x / 2) ^ 2) / 2

Tactic:
rw [Complex.norm_def, log_sqrt (circleMap 0 1 x - 1).normSq_nonneg]

Goal state after:
a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ log (Complex.normSq (circleMap 0 1 x - 1)) / 2 = log (4 * sin (x / 2) ^ 2) / 2
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ 1 + 1 - 2 * cos x = 2 - 2 * cos x

Tactic:
]

Goal state after:
a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ 1 + 1 - 2 * cos x = 2 - 2 * cos x
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ 2 - 2 * cos x = 2 - 2 * cos (2 * x / 2)

Tactic:
]

Goal state after:
a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ 2 - 2 * cos x = 2 - 2 * cos (2 * x / 2)
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ 2 - 2 * cos x = 2 - 2 * cos (2 * (x / 2))

Tactic:
by
      rw [← mul_div_assoc]
      simp

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ 2 - 2 * (2 * cos (x / 2) ^ 2 - 1) = 4 - 4 * cos (x / 2) ^ 2

Tactic:
]

Goal state after:
a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ 2 - 2 * (2 * cos (x / 2) ^ 2 - 1) = 4 - 4 * cos (x / 2) ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ 4 * (sin (x / 2) ^ 2 + cos (x / 2) ^ 2) - 4 * cos (x / 2) ^ 2 = 4 * sin (x / 2) ^ 2

Tactic:
]

Goal state after:
a : ℂ
h : ‖a‖ = 1
ζ : ℝ
hζ : a⁻¹ = circleMap 0 1 ζ
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ 4 * (sin (x / 2) ^ 2 + cos (x / 2) ^ 2) - 4 * cos (x / 2) ^ 2 = 4 * sin (x / 2) ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic

Goal state before:
pp: case h₀
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
R : ℝ
f : ℂ → E
hf : MeromorphicOn f (sphere c |R|)
t₀ : ∀ (u : ↑(sphere c |R|)), meromorphicOrderAt f ↑u ≠ ⊤
g : ℂ → E
h₁g : AnalyticOnNhd ℂ g (sphere c |R|)
h₂g : ∀ (u : ↑(sphere c |R|)), g ↑u ≠ 0
h₃g : f =ᶠ[codiscreteWithin (sphere c |R|)] (∏ᶠ (u : ℂ), (fun x ↦ x - u) ^ (divisor f (sphere c |R|)) u) • g
h₄g :
  (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin (sphere c |R|)]
    (∑ᶠ (u : ℂ), fun x ↦ ↑((divisor f (sphere c |R|)) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ ¬g (circleMap c R x) = 0

Tactic:
]

Goal state after:
case h₀
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
R : ℝ
f : ℂ → E
hf : MeromorphicOn f (sphere c |R|)
t₀ : ∀ (u : ↑(sphere c |R|)), meromorphicOrderAt f ↑u ≠ ⊤
g : ℂ → E
h₁g : AnalyticOnNhd ℂ g (sphere c |R|)
h₂g : ∀ (u : ↑(sphere c |R|)), g ↑u ≠ 0
h₃g : f =ᶠ[codiscreteWithin (sphere c |R|)] (∏ᶠ (u : ℂ), (fun x ↦ x - u) ^ (divisor f (sphere c |R|)) u) • g
h₄g :
  (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin (sphere c |R|)]
    (∑ᶠ (u : ℂ), fun x ↦ ↑((divisor f (sphere c |R|)) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖
x : ℝ
hx : x ∈ [[0, 2 * π]]
⊢ ¬g (circleMap c R x) = 0
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : 1 < ‖a‖
⊢ log ‖a‖ = log ‖a‖

pp: a : ℂ
h : 1 < ‖a‖
⊢ InnerProductSpace.HarmonicOnNhd (fun x ↦ log ‖x - a‖) (closedBall 0 |1|)

Tactic:
]

Goal state after:
a : ℂ
h : 1 < ‖a‖
⊢ InnerProductSpace.HarmonicOnNhd (fun x ↦ log ‖x - a‖) (closedBall 0 |1|)
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: case neg
a b c : ℝ
hc : ¬c = 0
⊢ c * (c⁻¹ * (arctan (b / c) - arctan (a / c))) = arctan (b / c) - arctan (a / c)

Tactic:
]

Goal state after:
case neg
a b c : ℝ
hc : ¬c = 0
⊢ c * (c⁻¹ * (arctan (b / c) - arctan (a / c))) = arctan (b / c) - arctan (a / c)
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a : ℂ
h : 1 < ‖a‖
x : ℂ
hx : x ∈ closedBall 0 |1|
⊢ x ≠ a

Tactic:
]

Goal state after:
a : ℂ
h : 1 < ‖a‖
x : ℂ
hx : x ∈ closedBall 0 |1|
⊢ x ≠ a
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
ha : 1 < a
hb : 1 < b
⊢ ∫ (x : ℝ) in a..b, deriv (fun x ↦ log (log x)) x = log (log b) - log (log a)

Tactic:
]

Goal state after:
a b : ℝ
ha : 1 < a
hb : 1 < b
⊢ ∫ (x : ℝ) in a..b, deriv (fun x ↦ log (log x)) x = log (log b) - log (log a)
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: case inl
a : ℂ
h : 1 < ‖a‖
⊢ log ‖a‖ = log⁺ ‖a‖

Tactic:
]

Goal state after:
case inl
a : ℂ
h : 1 < ‖a‖
⊢ log ‖a‖ = log⁺ ‖a‖
Module: Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic

Goal state before:
pp: case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
R : ℝ
f : ℂ → E
hf : MeromorphicOn f (sphere c |R|)
t₀ : ¬∃ u, meromorphicOrderAt f ↑u ≠ ⊤
⊢ CircleIntegrable (fun x ↦ log ‖f x‖) c R

Tactic:
]

Goal state after:
case neg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
R : ℝ
f : ℂ → E
hf : MeromorphicOn f (sphere c |R|)
t₀ : ¬∃ u, meromorphicOrderAt f ↑u ≠ ⊤
⊢ CircleIntegrable (fun x ↦ log ‖f x‖) c R
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: case inr.inl
a : ℂ
h : 1 = ‖a‖
⊢ |‖a‖| ≤ 1

Tactic:
]

Goal state after:
case inr.inl
a : ℂ
h : 1 = ‖a‖
⊢ |‖a‖| ≤ 1
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
t : ℂ
ht : t = -1
⊢ t = -1

Tactic:
]

Goal state after:
a b : ℝ
t : ℂ
ht : t = -1
⊢ t = -1
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: case inr.inr
a : ℂ
h : ‖a‖ < 1
⊢ |‖a‖| ≤ 1

Tactic:
]

Goal state after:
case inr.inr
a : ℂ
h : ‖a‖ < 1
⊢ |‖a‖| ≤ 1
Module: Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
R : ℝ
f : ℂ → E
hf : MeromorphicOn f (sphere c |R|)
hR : 0 ≤ R
⊢ CircleIntegrable (fun x ↦ log ‖f x‖) c R

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
R : ℝ
f : ℂ → E
hf : MeromorphicOn f (sphere c |R|)
hR : 0 ≤ R
⊢ CircleIntegrable (fun x ↦ log ‖f x‖) c R
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a c : ℂ
R : ℝ
hR : R ≠ 0
⊢ circleAverage (fun z ↦ log ‖↑R * z + c - a‖) 0 1 = circleAverage (fun z ↦ log ‖↑R * (z + ↑R⁻¹ * (c - a))‖) 0 1

Tactic:
]

Goal state after:
a c : ℂ
R : ℝ
hR : R ≠ 0
⊢ circleAverage (fun z ↦ log ‖↑R * z + c - a‖) 0 1 = circleAverage (fun z ↦ log ‖↑R * (z + ↑R⁻¹ * (c - a))‖) 0 1
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: case e_f
a c : ℂ
R : ℝ
hR : R ≠ 0
z : ℂ
⊢ ↑R * z + c - a = ↑R * (z + (↑R)⁻¹ * (c - a))

Tactic:
]

Goal state after:
case e_f
a c : ℂ
R : ℝ
hR : R ≠ 0
z : ℂ
⊢ ↑R * z + c - a = ↑R * (z + (↑R)⁻¹ * (c - a))
Module: Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
R : ℝ
f : ℂ → E
hf : MeromorphicOn f (sphere c |R|)
hR : 0 ≤ R
⊢ CircleIntegrable (fun x ↦ log⁺ ‖f x‖) c R

Tactic:
]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
c : ℂ
R : ℝ
f : ℂ → E
hf : MeromorphicOn f (sphere c |R|)
hR : 0 ≤ R
⊢ CircleIntegrable (fun x ↦ log⁺ ‖f x‖) c R
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b t : ℝ
ht : t ≠ -1
x s : ℝ
⊢ (1 + ↑x ^ 2) ^ ↑s = (1 + ↑x ^ 2) ^ ↑s

pp: case hx
a b t : ℝ
ht : t ≠ -1
x s : ℝ
⊢ 0 ≤ 1 + x ^ 2

Tactic:
]

Goal state after:
case hx
a b t : ℝ
ht : t ≠ -1
x s : ℝ
⊢ 0 ≤ 1 + x ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: case h
a c : ℂ
R : ℝ
z z₁ z₂ : ℂ
hR : ¬R = 0
hz : ‖z‖ = 1
hz₁ : ‖z₁‖ = 1 ∧ z₁ + (↑R)⁻¹ * (c - a) = 0
hz₂ : ‖z₂‖ = 1 ∧ z₂ + (↑R)⁻¹ * (c - a) = 0
⊢ -((↑R)⁻¹ * (c - a)) = -((↑R)⁻¹ * (c - a))

Tactic:
]

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

Goal state before:
pp: a b t : ℝ
ht : t ≠ -1
this : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s
⊢ ∫ (x : ℝ) in a..b, x * (1 + x ^ 2) ^ t = (1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1))

Tactic:
← ofReal_inj

Goal state after:
a b t : ℝ
ht : t ≠ -1
this : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s
⊢ ↑(∫ (x : ℝ) in a..b, x * (1 + x ^ 2) ^ t) =
    ↑((1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1)))
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b t : ℝ
ht : t ≠ -1
this : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s
⊢ ↑(∫ (x : ℝ) in a..b, x * (1 + x ^ 2) ^ t) =
    ↑((1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1)))

Tactic:
]

Goal state after:
a b t : ℝ
ht : t ≠ -1
this : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s
⊢ ↑(∫ (x : ℝ) in a..b, x * (1 + x ^ 2) ^ t) =
    ↑((1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1)))
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: case e'_2
a b t : ℝ
ht : t ≠ -1
this : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s
⊢ ∫ (x : ℝ) in a..b, ↑(x * (1 + x ^ 2) ^ t) = ∫ (x : ℝ) in ?convert_1..?convert_2, ↑x * (1 + ↑x ^ 2) ^ ↑t

Tactic:
]

Goal state after:
case e'_2
a b t : ℝ
ht : t ≠ -1
this : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s
⊢ ∫ (x : ℝ) in a..b, ↑(x * (1 + x ^ 2) ^ t) = ∫ (x : ℝ) in ?convert_1..?convert_2, ↑x * (1 + ↑x ^ 2) ^ ↑t
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: case e'_2
a b t : ℝ
ht : t ≠ -1
this : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s
⊢ ↑(∫ (x : ℝ) in a..b, x * (1 + x ^ 2) ^ t) = ∫ (x : ℝ) in ?convert_1..?convert_2, ↑x * (1 + ↑x ^ 2) ^ ↑t

Tactic:
rw [← intervalIntegral.integral_ofReal]

Goal state after:
case e'_2
a b t : ℝ
ht : t ≠ -1
this : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s
⊢ ∫ (x : ℝ) in a..b, ↑(x * (1 + x ^ 2) ^ t) = ∫ (x : ℝ) in ?convert_1..?convert_2, ↑x * (1 + ↑x ^ 2) ^ ↑t
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a c : ℂ
R : ℝ
hR : R ≠ 0
this : {z | ‖z + ↑R⁻¹ * (c - a)‖ ≠ 0} ∈ codiscreteWithin (sphere 0 |1|)
z : ℂ
hz : ‖z + ↑R⁻¹ * (c - a)‖ ≠ 0
⊢ log ‖↑R‖ + log ‖z + ↑R⁻¹ * (c - a)‖ = log ‖R‖ + log ‖z + ↑R⁻¹ * (c - a)‖

Tactic:
]

Goal state after:
a c : ℂ
R : ℝ
hR : R ≠ 0
this : {z | ‖z + ↑R⁻¹ * (c - a)‖ ≠ 0} ∈ codiscreteWithin (sphere 0 |1|)
z : ℂ
hz : ‖z + ↑R⁻¹ * (c - a)‖ ≠ 0
⊢ log ‖↑R‖ + log ‖z + ↑R⁻¹ * (c - a)‖ = log ‖R‖ + log ‖z + ↑R⁻¹ * (c - a)‖
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: case e'_2.e_f
a b t : ℝ
ht : t ≠ -1
this : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s
x : ℝ
⊢ ↑x * (1 + ↑x ^ 2) ^ ↑t = ↑x * (1 + ↑x ^ 2) ^ ↑t

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: a c : ℂ
R : ℝ
hR : R ≠ 0
⊢ log ‖R‖ + circleAverage (fun x ↦ log ‖x + ↑R⁻¹ * (c - a)‖) 0 1 = log R + log⁺ (|R|⁻¹ * ‖c - a‖)

Tactic:
]

Goal state after:
a c : ℂ
R : ℝ
hR : R ≠ 0
⊢ log ‖R‖ + circleAverage (fun x ↦ log ‖x + ↑R⁻¹ * (c - a)‖) 0 1 = log R + log⁺ (|R|⁻¹ * ‖c - a‖)
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
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 : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖
C : ℝ
hC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
g : ℕ → ℂ → E := fun n z ↦ cexp z ^ n • f z
hg : ∀ (n : ℕ) (z : ℂ), ‖g n z‖ = expR z.re ^ n * ‖f z‖
z : ℂ
hz : 0 < z.re
n : ℕ
⊢ Tendsto (fun x ↦ ‖g n ↑x‖) atTop (𝓝 0)

Tactic:
]

Goal state after:
case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
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 : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖
C : ℝ
hC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
g : ℕ → ℂ → E := fun n z ↦ cexp z ^ n • f z
hg : ∀ (n : ℕ) (z : ℂ), ‖g n z‖ = expR z.re ^ n * ‖f z‖
z : ℂ
hz : 0 < z.re
n : ℕ
⊢ Tendsto (fun x ↦ ‖g n ↑x‖) atTop (𝓝 0)
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: case e_a.inr
a c : ℂ
R : ℝ
hR : R ≠ 0
h : 0 < R
⊢ log⁺ (R⁻¹ * ‖c - a‖) = log⁺ (R⁻¹ * ‖c - a‖)

Tactic:
]

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

Goal state before:
pp: case convert_3
a b t : ℝ
ht : t ≠ -1
this : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s
⊢ ¬t = -1

Tactic:
]

Goal state after:
case convert_3
a b t : ℝ
ht : t ≠ -1
this : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s
⊢ ¬t = -1
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
n : ℕ
C : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b
α β γ : ℝ
⊢ β * α * γ * α = β * (α * α * γ)

Tactic:
by ring

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: case neg
a c : ℂ
R : ℝ
hu : a ∈ closedBall c |R|
hR : ¬R = 0
⊢ |R|⁻¹ * ‖c - a‖ ≤ 1

Tactic:
]

Goal state after:
case neg
a c : ℂ
R : ℝ
hu : a ∈ closedBall c |R|
hR : ¬R = 0
⊢ |R|⁻¹ * ‖c - a‖ ≤ 1
Module: Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage

Goal state before:
pp: case neg
a c : ℂ
R : ℝ
hu : ‖c - a‖ ≤ |R|
hR : ¬R = 0
⊢ |R|⁻¹ * ‖c - a‖ ≤ 1

Tactic:
]

Goal state after:
case neg
a c : ℂ
R : ℝ
hu : ‖c - a‖ ≤ |R|
hR : ¬R = 0
⊢ |R|⁻¹ * ‖c - a‖ ≤ 1
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
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 : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖
C : ℝ
hC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
g : ℕ → ℂ → E := fun n z ↦ cexp z ^ n • f z
hg : ∀ (n : ℕ) (z : ℂ), ‖g n z‖ = expR z.re ^ n * ‖f z‖
z : ℂ
hz : 0 < z.re
n : ℕ
y : ℝ
⊢ ‖f (↑y * I)‖ ≤ C

Tactic:
]

Goal state after:
case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
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 : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖
C : ℝ
hC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
g : ℕ → ℂ → E := fun n z ↦ cexp z ^ n • f z
hg : ∀ (n : ℕ) (z : ℂ), ‖g n z‖ = expR z.re ^ n * ‖f z‖
z : ℂ
hz : 0 < z.re
n : ℕ
y : ℝ
⊢ ‖f (↑y * I)‖ ≤ C
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
n : ℕ
C : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x
H :
  ∫ (x : ℝ) in a..b, sin x ^ (n + 1) * sin x =
    sin b ^ (n + 1) * (-cos) b - sin a ^ (n + 1) * (-cos) a - ∫ (x : ℝ) in a..b, ↑(n + 1) * cos x * sin x ^ n * (-cos) x
⊢ C + (((↑n + 1) * ∫ (x : ℝ) in a..b, sin x ^ n) - (↑n + 1) * ∫ (x : ℝ) in a..b, sin x ^ (n + 2)) =
    C + (((↑n + 1) * ∫ (x : ℝ) in a..b, sin x ^ n) - (↑n + 1) * ∫ (x : ℝ) in a..b, sin x ^ (n + 2))

pp: case hf
a b : ℝ
n : ℕ
C : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x
H :
  ∫ (x : ℝ) in a..b, sin x ^ (n + 1) * sin x =
    sin b ^ (n + 1) * (-cos) b - sin a ^ (n + 1) * (-cos) a - ∫ (x : ℝ) in a..b, ↑(n + 1) * cos x * sin x ^ n * (-cos) x
⊢ IntervalIntegrable (fun x ↦ sin x ^ n) MeasureTheory.volume a b

pp: case hg
a b : ℝ
n : ℕ
C : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x
H :
  ∫ (x : ℝ) in a..b, sin x ^ (n + 1) * sin x =
    sin b ^ (n + 1) * (-cos) b - sin a ^ (n + 1) * (-cos) a - ∫ (x : ℝ) in a..b, ↑(n + 1) * cos x * sin x ^ n * (-cos) x
⊢ IntervalIntegrable (fun x ↦ sin x ^ (n + 2)) MeasureTheory.volume a b

Tactic:
]

Goal state after:
case hf
a b : ℝ
n : ℕ
C : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x
H :
  ∫ (x : ℝ) in a..b, sin x ^ (n + 1) * sin x =
    sin b ^ (n + 1) * (-cos) b - sin a ^ (n + 1) * (-cos) a - ∫ (x : ℝ) in a..b, ↑(n + 1) * cos x * sin x ^ n * (-cos) x
⊢ IntervalIntegrable (fun x ↦ sin x ^ n) MeasureTheory.volume a b

case hg
a b : ℝ
n : ℕ
C : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x
H :
  ∫ (x : ℝ) in a..b, sin x ^ (n + 1) * sin x =
    sin b ^ (n + 1) * (-cos) b - sin a ^ (n + 1) * (-cos) a - ∫ (x : ℝ) in a..b, ↑(n + 1) * cos x * sin x ^ n * (-cos) x
⊢ IntervalIntegrable (fun x ↦ sin x ^ (n + 2)) MeasureTheory.volume a b
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ : ℂ
R r : ℝ
hρ : ‖ρ‖ = R
hr_lt : r < R
hwr : ‖w‖ < r
⊢ ∀ (x : ℝ), ContinuousAt (fun θ ↦ herglotzLogIntegrand w ρ (circleMap 0 r θ)) x

Tactic:
]

Goal state after:
w ρ : ℂ
R r : ℝ
hρ : ‖ρ‖ = R
hr_lt : r < R
hwr : ‖w‖ < r
⊢ ∀ (x : ℝ), ContinuousAt (fun θ ↦ herglotzLogIntegrand w ρ (circleMap 0 r θ)) x
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: r₀ r R : ℝ
ρ : ℂ
hρ : ‖ρ‖ = R
hr₀ : 0 < r₀
hR : 0 < R
hr₀r : r₀ ≤ r
hrR : r ≤ R
θ r₁ : ℝ
⊢ ↑(normSq (circleMap 0 r₁ θ) + normSq ρ - 2 * (circleMap 0 r₁ θ * (starRingEnd ℂ) ρ).re) =
    ↑(r₁ ^ 2 + R ^ 2 - 2 * r₁ * R * Real.cos (θ - ρ.arg))

Tactic:
]

Goal state after:
r₀ r R : ℝ
ρ : ℂ
hρ : ‖ρ‖ = R
hr₀ : 0 < r₀
hR : 0 < R
hr₀r : r₀ ≤ r
hrR : r ≤ R
θ r₁ : ℝ
⊢ ↑(normSq (circleMap 0 r₁ θ) + normSq ρ - 2 * (circleMap 0 r₁ θ * (starRingEnd ℂ) ρ).re) =
    ↑(r₁ ^ 2 + R ^ 2 - 2 * r₁ * R * Real.cos (θ - ρ.arg))
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: r₀ r R : ℝ
ρ : ℂ
hρ : ‖ρ‖ = R
hr₀ : 0 < r₀
hR : 0 < R
hr₀r : r₀ ≤ r
hrR : r ≤ R
θ r₁ : ℝ
| (circleMap 0 r₁ θ * circleMap 0 ‖ρ‖ (-ρ.arg)).re

Tactic:
circleMap_zero_mul,

Goal state after:
r₀ r R : ℝ
ρ : ℂ
hρ : ‖ρ‖ = R
hr₀ : 0 < r₀
hR : 0 < R
hr₀r : r₀ ≤ r
hrR : r ≤ R
θ r₁ : ℝ
| (circleMap 0 (r₁ * ‖ρ‖) (θ + -ρ.arg)).re