mathlib-initiative/mathlib-tactics huggingface dataset:
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) = ∑ t ∈ insert a s, log⁺ (f t)

Tactic:
by

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

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

Tactic:
by

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:
by

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|
⊢ |∑ t ∈ s, f t| ∈ Set.Ici 0

Tactic:
by

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|
⊢ ∑ t ∈ s, |f t| ∈ Set.Ici 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
hg : CircleIntegrable f 0 R
hw : w ∈ ball 0 R
hR : 0 < R
d : ℝ
hd : d > 0
hsub : ball w d ⊆ ball 0 R
hdist : ∀ x ∈ ball w d, ∀ (θ : ℝ), d ≤ ‖circleMap 0 R θ - x‖
hgm : AEStronglyMeasurable (fun θ ↦ f (circleMap 0 R θ)) (volume.restrict (uIoc 0 (2 * π)))
θ : ℝ
a✝ : θ ∈ uIoc 0 (2 * π)
x : ℂ
hx : x ∈ ball w d
⊢ ‖2‖ = 2

Tactic:
by

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:
by

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|
⊢ ∑ t ∈ s, |f t_max| ∈ Set.Ici 0

Tactic:
by

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_max|)

Tactic:
by

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_max|) = log⁺ (↑s.card * |f t_max|)

Tactic:
by

Goal state after:
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
z : ℂ
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 : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f ↑x‖
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hz : 0 ≤ z.re
this : ∀ᶠ (ε : ℝ) in 𝓝[<] 0, ‖cexp (↑ε * z) • f z‖ ≤ C
⊢ Continuous fun c ↦ ‖cexp (↑c * z) • f z‖

Tactic:
by

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 ↑s.card + log⁺ |f t_max| ≤ log ↑s.card + ∑ t ∈ s, log⁺ (f t)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
s : Finset α
f : α → ℝ
⊢ log⁺ (∑ t ∈ s, f t) ≤ log ↑s.card + ∑ t ∈ s, log⁺ (f t)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝ : SeminormedAddCommGroup E
α : Type u_2
s : Finset α
f : α → E
⊢ log⁺ ‖∑ t ∈ s, f t‖ ≤ log ↑s.card + ∑ t ∈ s, log⁺ ‖f t‖

Tactic:
by

Goal state after:
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
z : ℂ
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 : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f ↑x‖
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hz : 0 ≤ z.re
this : ∀ᶠ (ε : ℝ) in 𝓝[<] 0, ‖cexp (↑ε * z) • f z‖ ≤ C
⊢ ‖cexp (↑0 * z) • f z‖ = ‖f z‖

Tactic:
by

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

Goal state before:
pp: x y : ℝ
⊢ log⁺ (x + y) ≤ log 2 + log⁺ x + log⁺ y

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝ : SeminormedAddCommGroup E
a b : E
⊢ log⁺ ‖a + b‖ ≤ log⁺ ‖a‖ + log⁺ ‖b‖ + log 2

Tactic:
by

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

Goal state before:
pp: case inr
a b : ℝ
r : ℂ
h : r ≠ -1 ∧ 0 ∉ [[a, b]]
⊢ r + 1 ≠ 0

Tactic:
· rw [Ne, ← add_eq_zero_iff_eq_neg] at h; exact h.1

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

Goal state before:
pp: a b : ℝ
r : ℂ
h : -1 < r.re ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]
⊢ r + 1 ≠ 0

Tactic:
by

Goal state after:
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
z : ℂ
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 : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f ↑x‖
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hz : 0 ≤ z.re
this : ∀ᶠ (ε : ℝ) in 𝓝[<] 0, ‖cexp (↑ε * z) • f z‖ ≤ C
⊢ ‖f z‖ ≤ C

Tactic:
by

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

Goal state before:
pp: a b : ℝ
r : ℂ
h : -1 < r.re ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]
hr : r + 1 ≠ 0
hab : ¬0 ∉ [[a, b]]
⊢ -1 < r.re

Tactic:
by

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

Goal state before:
pp: a b : ℝ
r : ℂ
hr : r + 1 ≠ 0
hab : ¬0 ∉ [[a, b]]
h : -1 < r.re
this : ∀ (c : ℝ), ∫ (x : ℝ) in 0..c, ↑x ^ r = ↑c ^ (r + 1) / (r + 1) - 0 ^ (r + 1) / (r + 1)
⊢ ∫ (x : ℝ) in a..b, ↑x ^ r = ↑b ^ (r + 1) / (r + 1) - ↑a ^ (r + 1) / (r + 1)

Tactic:
by

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

Goal state before:
pp: a b : ℝ
r : ℂ
h : -1 < r.re ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]
⊢ ∫ (x : ℝ) in a..b, ↑x ^ r = (↑b ^ (r + 1) - ↑a ^ (r + 1)) / (r + 1)

Tactic:
by

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

Goal state before:
pp: a b r : ℝ
h : -1 < r ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]
⊢ -1 < (↑r).re ∨ ↑r ≠ -1 ∧ 0 ∉ [[a, b]]

Tactic:
by

Goal state after:
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
z✝ : ℂ
hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
hre : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f ↑x‖
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hz : 0 ≤ z✝.re
ε : ℝ
ε₀ : ε < 0
g : ℂ → E := fun z ↦ cexp (↑ε * z) • f z
hd : DiffContOnCl ℂ g {z | 0 < z.re}
z : ℂ
⊢ ‖g z‖ = expR (ε * z.re) * ‖f z‖

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
hg : CircleIntegrable f 0 R
hw : w ∈ ball 0 R
hR : 0 < R
d : ℝ
hd : d > 0
hsub : ball w d ⊆ ball 0 R
hdist : ∀ x ∈ ball w d, ∀ (θ : ℝ), d ≤ ‖circleMap 0 R θ - x‖
hgm : AEStronglyMeasurable (fun θ ↦ f (circleMap 0 R θ)) (volume.restrict (uIoc 0 (2 * π)))
θ : ℝ
a✝ : θ ∈ uIoc 0 (2 * π)
x : ℂ
hx : x ∈ ball w d
h₁ : circleMap 0 R θ - x ≠ 0
⊢ HasDerivAt (fun x ↦ herglotzRieszKernel 0 x (circleMap 0 R θ)) (2 * circleMap 0 R θ / (circleMap 0 R θ - x) ^ 2) x

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hf : MeromorphicOn f [[a, b]]
t₀ : ∀ (u : ↑[[a, b]]), meromorphicOrderAt f ↑u ≠ ⊤
g : ℝ → E
h₁g : AnalyticOnNhd ℝ g [[a, b]]
h₂g : ∀ (u : ↑[[a, b]]), g ↑u ≠ 0
h₃g : f =ᶠ[codiscreteWithin [[a, b]]] (∏ᶠ (u : ℝ), (fun x ↦ x - u) ^ (divisor f [[a, b]]) u) • g
h₄g :
  (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin [[a, b]]]
    (∑ᶠ (u : ℝ), fun x ↦ ↑((divisor f [[a, b]]) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖
i : ℝ
⊢ a = a - i + i

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
hg : CircleIntegrable f 0 R
hw : w ∈ ball 0 R
⊢ HasDerivAt (fun w ↦ circleAverage (fun ζ ↦ herglotzRieszKernel 0 w ζ • f ζ) 0 R)
    (circleAverage (fun ζ ↦ (2 * ζ / (ζ - w) ^ 2) • f ζ) 0 R) w

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hf : MeromorphicOn f [[a, b]]
t₀ : ∀ (u : ↑[[a, b]]), meromorphicOrderAt f ↑u ≠ ⊤
g : ℝ → E
h₁g : AnalyticOnNhd ℝ g [[a, b]]
h₂g : ∀ (u : ↑[[a, b]]), g ↑u ≠ 0
h₃g : f =ᶠ[codiscreteWithin [[a, b]]] (∏ᶠ (u : ℝ), (fun x ↦ x - u) ^ (divisor f [[a, b]]) u) • g
h₄g :
  (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin [[a, b]]]
    (∑ᶠ (u : ℝ), fun x ↦ ↑((divisor f [[a, b]]) u) * log ‖x - u‖) + fun x ↦ log ‖g x‖
i : ℝ
⊢ b = b - i + i

Tactic:
by

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

Goal state before:
pp: ⊢ IntervalIntegrable (fun x ↦ log (sin x)) MeasureTheory.volume 0 (π / 2)

Tactic:
by

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

Goal state before:
pp: ⊢ IntervalIntegrable (fun x ↦ log (sin x)) MeasureTheory.volume (π / 2) π

Tactic:
by

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

Goal state before:
pp: ⊢ π - π / 2 = π / 2

Tactic:
by

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

Goal state before:
pp: a b r : ℝ
h : -1 < r ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]
h' : -1 < (↑r).re ∨ ↑r ≠ -1 ∧ 0 ∉ [[a, b]]
this : (∫ (x : ℝ) in a..b, ↑x ^ ↑r).re = ((↑b ^ (↑r + 1) - ↑a ^ (↑r + 1)) / (↑r + 1)).re
⊢ ↑r + 1 = ↑(r + 1)

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: ⊢ Ι 0 (π / 2) ⊆ Set.univ

Tactic:
by

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

Goal state before:
pp: ⊢ sin ⁻¹' {0}ᶜ ∈ codiscrete ℝ

Tactic:
by

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

Goal state before:
pp: t₀ : sin ⁻¹' {0}ᶜ ∈ codiscrete ℝ
⊢ cos ⁻¹' {0}ᶜ ∈ codiscrete ℝ

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: a b r : ℝ
h : -1 < r ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]
⊢ ∫ (x : ℝ) in a..b, x ^ r = (b ^ (r + 1) - a ^ (r + 1)) / (r + 1)

Tactic:
by

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

Goal state before:
pp: a b : ℝ
n : ℤ
h : 0 ≤ n ∨ n ≠ -1 ∧ 0 ∉ [[a, b]]
⊢ ∫ (x : ℝ) in a..b, x ^ n = (b ^ (n + 1) - a ^ (n + 1)) / (↑n + 1)

Tactic:
by

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

Goal state before:
pp: a b : ℝ
n : ℕ
⊢ ∫ (x : ℝ) in a..b, x ^ n = (b ^ (n + 1) - a ^ (n + 1)) / (↑n + 1)

Tactic:
by

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

Goal state before:
pp: a b : ℝ
n : ℕ
hab : a ≤ b
⊢ ∫ (x : ℝ) in Ι a b, |x - a| ^ n = ∫ (x : ℝ) in a..b, |x - a| ^ n

Tactic:
by

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

Goal state before:
pp: a b : ℝ
n : ℕ
hab : a ≤ b
⊢ ∫ (x : ℝ) in a..b, |x - a| ^ n = ∫ (x : ℝ) in 0..b - a, x ^ n

Tactic:
by

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

Goal state before:
pp: a b : ℝ
n : ℕ
hab : a ≤ b
⊢ ∫ (x : ℝ) in 0..b - a, x ^ n = |b - a| ^ (n + 1) / (↑n + 1)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hf : MeromorphicOn f [[a, b]]
t₀ : ∀ (u : ↑[[a, b]]), meromorphicOrderAt f ↑u = ⊤
⊢ (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin (Ι a b)] 0

Tactic:
by

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

Goal state before:
pp: a b : ℝ
n : ℕ
hab : b < a
⊢ ∫ (x : ℝ) in Ι a b, |x - a| ^ n = ∫ (x : ℝ) in b..a, |x - a| ^ n

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: a b : ℝ
n : ℕ
hab : b < a
⊢ ∫ (x : ℝ) in b..a, |x - a| ^ n = ∫ (x : ℝ) in b - a..0, (-x) ^ n

Tactic:
by

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

Goal state before:
pp: ⊢ 2 * (π / 2) = π

Tactic:
by

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

Goal state before:
pp: a b : ℝ
n : ℕ
hab : b < a
⊢ ∫ (x : ℝ) in b - a..0, (-x) ^ n = |b - a| ^ (n + 1) / (↑n + 1)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hf : MeromorphicOn f [[a, b]]
⊢ IntervalIntegrable (fun x ↦ log ‖f x‖) volume a b

Tactic:
by

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

Goal state before:
pp: a b : ℝ
n : ℕ
⊢ ∫ (x : ℝ) in Ι a b, |x - a| ^ n = |b - a| ^ (n + 1) / (↑n + 1)

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: a b : ℝ
⊢ ∫ (x : ℝ) in a..b, x = (b ^ 2 - a ^ 2) / 2

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: a b : ℝ
⊢ ∫ (x : ℝ) in a..b, 1 = b - a

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
hf : MeromorphicOn f [[a, b]]
⊢ IntervalIntegrable (fun x ↦ log⁺ ‖f x‖) volume a b

Tactic:
by

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

Goal state before:
pp: a b : ℝ
f : ℝ → ℝ
hf : MeromorphicOn f [[a, b]]
⊢ log ∘ f = fun x ↦ log ‖f x‖

Tactic:
by

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

Goal state before:
pp: a b : ℝ
⊢ ∫ (x : ℝ) in a..a + 1, b = b

Tactic:
by

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

Goal state before:
pp: a b : ℝ
f : ℝ → ℝ
hf : MeromorphicOn f [[a, b]]
⊢ IntervalIntegrable (log ∘ f) volume a b

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: a b : ℝ
h : 0 ∉ [[a, b]]
⊢ ∫ (x : ℝ) in a..b, x⁻¹ = log (b / a)

Tactic:
by

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

Goal state before:
pp: a b : ℝ
h : 0 ∉ [[a, b]]
⊢ ∫ (x : ℝ) in a..b, 1 / x = log (b / a)

Tactic:
by

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

Goal state before:
pp: a c : ℂ
r : ℝ
z : ℂ
hz : z ∈ sphere c |r|
⊢ MeromorphicAt (fun x ↦ x - a) z

Tactic:
by

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

Goal state before:
pp: a : ℂ
h✝ : ‖a‖ < 1
z : ℂ
hz : ‖z‖ = 1
h : z = 0
⊢ False

Tactic:
by

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

Goal state before:
pp: a : ℂ
h : ‖a‖ < 1
z : ℂ
hz : ‖z‖ = 1
this : z ≠ 0
⊢ ‖z - a‖ = ‖z⁻¹ * (z - a)‖

Tactic:
by

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

Goal state before:
pp: a : ℂ
h : ‖a‖ < 1
z : ℂ
hz : ‖z‖ = 1
this : z ≠ 0
⊢ ‖z⁻¹ * (z - a)‖ = ‖1 - z⁻¹ * a‖

Tactic:
by

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

Goal state before:
pp: a : ℂ
h : ‖a‖ < 1
⊢ circleAverage (fun x ↦ log ‖x - a‖) 0 1 = circleAverage (fun x ↦ log ‖1 - x⁻¹ * a‖) 0 1

Tactic:
by

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

Goal state before:
pp: a b : ℝ
ha : 0 < a
hb : 0 < b
⊢ ∫ (x : ℝ) in a..b, 1 / x = log (b / a)

Tactic:
by

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

Goal state before:
pp: a b : ℝ
ha : a < 0
hb : b < 0
⊢ ∫ (x : ℝ) in a..b, 1 / x = log (b / a)

Tactic:
by

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

Goal state before:
pp: a b : ℝ
⊢ ∫ (x : ℝ) in a..b, rexp x = rexp b - rexp a

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: a : ℂ
h : ‖a‖ < 1
x : ℂ
hx : x ∈ closedBall 0 |1|
this : ‖x * a‖ < 1
⊢ AnalyticAt ℂ (fun x ↦ 1 - x * a) x

Tactic:
by

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

Goal state before:
pp: a : ℂ
h : ‖a‖ < 1
⊢ circleAverage (fun x ↦ log ‖1 - x⁻¹ * a‖) 0 1 = 0

Tactic:
by

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

Goal state before:
pp: a b : ℝ
c : ℂ
hc : c ≠ 0
⊢ ∀ (x : ℝ), HasDerivAt (fun y ↦ Complex.exp (c * ↑y) / c) (Complex.exp (c * ↑x)) x

Tactic:
by

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

Goal state before:
pp: a : ℂ
h : ‖a‖ < 1
⊢ circleAverage (fun x ↦ log ‖x - a‖) 0 1 = 0

Tactic:
by

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

Goal state before:
pp: x : ℝ
⊢ x / 2 = 2⁻¹ * x

Tactic:
by

Goal state after:
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
z : ℂ
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 : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f ↑x‖
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hz : 0 ≤ z.re
⊢ ‖f z‖ ≤ C

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: a b : ℝ
c : ℂ
hc : c ≠ 0
⊢ ∫ (x : ℝ) in a..b, Complex.exp (c * ↑x) = (Complex.exp (c * ↑b) - Complex.exp (c * ↑a)) / c

Tactic:
by

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

Goal state before:
pp: ⊢ Ι 0 π ⊆ Set.univ

Tactic:
by

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

Goal state before:
pp: r : ℝ
⊢ ∫ (t : ℝ) in -r..r, Complex.exp (↑t * Complex.I) =
    (Complex.exp (Complex.I * ↑r) - Complex.exp (Complex.I * -↑r)) / Complex.I

Tactic:
by

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

Goal state before:
pp: x✝¹ : ℝ
x✝ : x✝¹ ∈ Set.univ
⊢ AnalyticAt ℝ (fun x ↦ 4 * sin x ^ 2) x✝¹

Tactic:
by

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

Goal state before:
pp: r : ℝ
⊢ (Complex.exp (Complex.I * ↑r) - Complex.exp (Complex.I * -↑r)) / Complex.I = 2 * ↑(sin r)

Tactic:
by

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

Goal state before:
pp: r : ℝ
⊢ ∫ (t : ℝ) in -r..r, Complex.exp (↑t * Complex.I) = 2 * ↑r * ↑(sinc r)

Tactic:
by

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

Goal state before:
pp: 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
this : ∀ (z : ℂ), 0 < z.re → f z = 0
⊢ EqOn f 0 {z | 0 ≤ z.re}

Tactic:
by

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

Goal state before:
pp: b : ℝ
ht : 0 < b
⊢ ContinuousAt (fun x ↦ x * log x - x) b

Tactic:
by

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

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

Tactic:
by

Goal state after:
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
⊢ 4 ≠ 0

Tactic:
by

Goal state after:
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
⊢ sin a ^ 2 ≠ 0

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: ⊢ IntervalIntegrable (fun x ↦ 2 * log (sin x)) volume 0 π

Tactic:
by

Goal state after: