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

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

pp: case inr
a b r : ℝ
h : -1 < r
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 inl
a b r : ℝ
h : -1 < r
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x ↦ x ^ r) volume 0 c
c : ℝ
hc : 0 ≤ c
⊢ IntervalIntegrable (fun x ↦ x ^ r) volume 0 c
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: case inr.inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hR : 0 < R
hw : 0 ∈ ball 0 R
⊢ Real.circleAverage (fun z ↦ ((z + 0) / (z - 0)).re • f z) 0 R = f 0

pp: case inr.inr
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
⊢ Real.circleAverage (fun z ↦ ((z + w) / (z - w)).re • f z) 0 R = f w

Tactic:
·

Goal state after:
case inr.inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hR : 0 < R
hw : 0 ∈ ball 0 R
⊢ Real.circleAverage (fun z ↦ ((z + 0) / (z - 0)).re • f z) 0 R = f 0
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case inr
a b r : ℝ
h : -1 < r
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
h : -1 < r
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.Trigonometric.Sinc

Goal state before:
pp: case pos
x : ℝ
hx : x = 0
⊢ ContinuousAt (dslope sin 0) x

pp: case neg
x : ℝ
hx : ¬x = 0
⊢ ContinuousAt (dslope sin 0) x

Tactic:
·

Goal state after:
case pos
x : ℝ
hx : x = 0
⊢ ContinuousAt (dslope sin 0) x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc

Goal state before:
pp: case neg
x : ℝ
hx : ¬x = 0
⊢ ContinuousAt (dslope sin 0) x

Tactic:
·

Goal state after:
case neg
x : ℝ
hx : ¬x = 0
⊢ ContinuousAt (dslope sin 0) x
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case pos
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)
h₀ : ∀ (x : ℝ), 0 ≤ x → f ↑x = 0
⊢ ∃ x, 0 ≤ x ∧ ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x‖

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)
h₀ : ∃ x, 0 ≤ x ∧ f ↑x ≠ 0
⊢ ∃ x, 0 ≤ x ∧ ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x‖

Tactic:
·

Goal state after:
case pos
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)
h₀ : ∀ (x : ℝ), 0 ≤ x → f ↑x = 0
⊢ ∃ x, 0 ≤ x ∧ ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x‖
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case refine_2
s t : ℝ
ht : 0 < t
h : -1 < s
⊢ IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume

pp: case refine_1
s t : ℝ
ht : 0 < t
h : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume
⊢ -1 < s

Tactic:
·

Goal state after:
case refine_2
s t : ℝ
ht : 0 < t
h : -1 < s
⊢ IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume
Module: Mathlib.Analysis.SpecialFunctions.Log.InvLog

Goal state before:
pp: case inl
x : ℝ
this✝¹ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) (-1)
this✝ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 1
this : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0
left✝¹ : x ≠ -1
left✝ : x ≠ 0
right✝ : x ≠ 1
⊢ deriv (fun x ↦ (log x)⁻¹) x = -x⁻¹ / log x ^ 2

pp: case inr.inl
this✝¹ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) (-1)
this✝ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 1
this : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0
⊢ deriv (fun x ↦ (log x)⁻¹) (-1) = -(-1)⁻¹ / log (-1) ^ 2

pp: case inr.inr.inl
this✝¹ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) (-1)
this✝ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 1
this : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0
⊢ deriv (fun x ↦ (log x)⁻¹) 0 = -0⁻¹ / log 0 ^ 2

pp: case inr.inr.inr
this✝¹ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) (-1)
this✝ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 1
this : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0
⊢ deriv (fun x ↦ (log x)⁻¹) 1 = -1⁻¹ / log 1 ^ 2

Tactic:
·

Goal state after:
case inl
x : ℝ
this✝¹ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) (-1)
this✝ : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 1
this : ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0
left✝¹ : x ≠ -1
left✝ : x ≠ 0
right✝ : x ≠ 1
⊢ deriv (fun x ↦ (log x)⁻¹) x = -x⁻¹ / log x ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

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

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

Tactic:
·

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

Goal state before:
pp: case inl
x : ℝ
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
left✝¹ : x ≠ -1
left✝ : x ≠ 0
right✝ : x ≠ 1
⊢ deriv (fun x ↦ log (log x)) x = x⁻¹ / log x

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
⊢ deriv (fun x ↦ log (log x)) (-1) = (-1)⁻¹ / log (-1)

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
⊢ deriv (fun x ↦ log (log x)) 0 = 0⁻¹ / log 0

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
⊢ deriv (fun x ↦ log (log x)) 1 = 1⁻¹ / log 1

Tactic:
·

Goal state after:
case inl
x : ℝ
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
left✝¹ : x ≠ -1
left✝ : x ≠ 0
right✝ : x ≠ 1
⊢ deriv (fun x ↦ log (log x)) x = x⁻¹ / log x
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

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

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

Tactic:
·

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

Goal state before:
pp: case neg.inr.refine_1
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
⊢ AEStronglyMeasurable (fun x ↦ ↑x ^ r) (μ.restrict (Ι a b))

pp: case neg.inr.refine_2
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
⊢ IntervalIntegrable (fun t ↦ ‖↑t ^ r‖) μ a b

Tactic:
·

Goal state after:
case neg.inr.refine_1
a b : ℝ
μ : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
⊢ AEStronglyMeasurable (fun x ↦ ↑x ^ r) (μ.restrict (Ι a b))
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
⊢ IntervalIntegrable (fun x ↦ ‖↑x ^ r‖) μ 0 c

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
⊢ IntervalIntegrable (fun x ↦ ‖↑x ^ r‖) μ 0 c

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
⊢ IntervalIntegrable (fun x ↦ ‖↑x ^ r‖) μ 0 c
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
⊢ IntervalIntegrable (fun x ↦ ‖↑x ^ r‖) μ 0 c

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
⊢ IntervalIntegrable (fun x ↦ ‖↑x ^ r‖) μ 0 c
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case neg.inr.refine_2.inr.left
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

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
⊢ IntegrableOn (fun x ↦ ‖↑x ^ r‖) (Ioo c 0) μ

Tactic:
·

Goal state after:
case neg.inr.refine_2.inr.left
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
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
⊢ IntegrableOn (fun x ↦ ‖↑x ^ r‖) (Ioo c 0) μ

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
⊢ IntegrableOn (fun x ↦ ‖↑x ^ r‖) (Ioo c 0) μ
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
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
⊢ IntegrableOn ?f (Ι 0 c) volume

pp: case hst
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ EqOn ?f (fun t ↦ ‖↑t ^ r‖) (Ι 0 c)

pp: case hs
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ MeasurableSet (Ι 0 c)

pp: case f
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
⊢ IntegrableOn ?f (Ι 0 c) volume
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
⊢ EqOn (fun x ↦ x ^ r.re) (fun t ↦ ‖↑t ^ r‖) (Ι 0 c)

pp: case hs
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ MeasurableSet (Ι 0 c)

Tactic:
·

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

Goal state before:
pp: case hs
a b : ℝ
r : ℂ
h : -1 < r.re
c : ℝ
hc : 0 ≤ c
⊢ MeasurableSet (Ι 0 c)

Tactic:
·

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

Tactic:
· refine ContinuousOn.aestronglyMeasurable ?_ measurableSet_uIoc
      refine continuousOn_of_forall_continuousAt fun x hx => ?_
      rw [uIoc_of_le hc] at hx
      refine (continuousAt_cpow_const (Or.inl ?_)).comp Complex.continuous_ofReal.continuousAt
      rw [Complex.ofReal_re]
      exact hx.1

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

Goal state before:
pp: case inl
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 : 0 ≤ c
⊢ IntervalIntegrable (fun x ↦ ↑x ^ r) volume 0 c

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 inl
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 : 0 ≤ c
⊢ 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
⊢ 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.Complex.PhragmenLindelof

Goal state before:
pp: case inl
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
hmax : ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x₀‖
h : ‖f ↑x₀‖ ≤ C
⊢ ∀ {z : ℂ}, 0 ≤ z.re → ‖f z‖ ≤ C

pp: case inr
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
hmax : ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x₀‖
h : C < ‖f ↑x₀‖
⊢ ∀ {z : ℂ}, 0 ≤ z.re → ‖f z‖ ≤ C

Tactic:
·

Goal state after:
case inl
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
hmax : ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x₀‖
h : ‖f ↑x₀‖ ≤ C
⊢ ∀ {z : ℂ}, 0 ≤ z.re → ‖f z‖ ≤ C
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
c : ℂ
hf : DiffContOnCl ℂ f (ball c R)
hw : w ∈ ball c R
hR : R ≤ 0
⊢ Real.circleAverage (re ∘ herglotzRieszKernel c w • f) c R = f w

pp: case inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
c : ℂ
hf : DiffContOnCl ℂ f (ball c R)
hw : w ∈ ball c R
hR : 0 < R
⊢ Real.circleAverage (re ∘ herglotzRieszKernel c w • f) c R = f w

Tactic:
·

Goal state after:
case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
c : ℂ
hf : DiffContOnCl ℂ f (ball c R)
hw : w ∈ ball c R
hR : R ≤ 0
⊢ Real.circleAverage (re ∘ herglotzRieszKernel c w • f) c R = f w
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: case inr
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
hmax : ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x₀‖
h : C < ‖f ↑x₀‖
⊢ ∀ {z : ℂ}, 0 ≤ z.re → ‖f z‖ ≤ C

Tactic:
·

Goal state after:
case inr
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
hmax : ∀ (y : ℝ), 0 ≤ y → ‖f ↑y‖ ≤ ‖f ↑x₀‖
h : C < ‖f ↑x₀‖
⊢ ∀ {z : ℂ}, 0 ≤ z.re → ‖f z‖ ≤ C
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

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_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 refine_2
s : ℂ
t : ℝ
ht : 0 < t
h : -1 < s.re
⊢ IntegrableOn (fun x ↦ ↑x ^ s) (Ioo 0 t) volume

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

Tactic:
·

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

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

pp: case hbc
a b x : ℝ
hx : 0 < x
⊢ IntervalIntegrable log volume 1 x

Tactic:
·

Goal state after:
case hab
a b x : ℝ
hx : 0 < x
⊢ IntervalIntegrable log volume 0 1
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case hab.hcont
a b x : ℝ
hx : 0 < x
⊢ ContinuousOn (fun x ↦ -(x * log x - x)) [[0, 1]]

pp: case hab.hderiv
a b x : ℝ
hx : 0 < x
⊢ ∀ x ∈ Ioo (min 0 1) (max 0 1), HasDerivAt (fun x ↦ -(x * log x - x)) ((-log) x) x

pp: case hab.hpos
a b x : ℝ
hx : 0 < x
⊢ ∀ x ∈ Ioo (min 0 1) (max 0 1), 0 ≤ (-log) x

Tactic:
·

Goal state after:
case hab.hcont
a b x : ℝ
hx : 0 < x
⊢ ContinuousOn (fun x ↦ -(x * log x - x)) [[0, 1]]
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case hab.hderiv
a b x : ℝ
hx : 0 < x
⊢ ∀ x ∈ Ioo (min 0 1) (max 0 1), HasDerivAt (fun x ↦ -(x * log x - x)) ((-log) x) x

pp: case hab.hpos
a b x : ℝ
hx : 0 < x
⊢ ∀ x ∈ Ioo (min 0 1) (max 0 1), 0 ≤ (-log) x

Tactic:
·

Goal state after:
case hab.hderiv
a b x : ℝ
hx : 0 < x
⊢ ∀ x ∈ Ioo (min 0 1) (max 0 1), HasDerivAt (fun x ↦ -(x * log x - x)) ((-log) x) x
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case hab.hpos
a b x : ℝ
hx : 0 < x
⊢ ∀ x ∈ Ioo (min 0 1) (max 0 1), 0 ≤ (-log) x

Tactic:
·

Goal state after:
case hab.hpos
a b x : ℝ
hx : 0 < x
⊢ ∀ x ∈ Ioo (min 0 1) (max 0 1), 0 ≤ (-log) x
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: case hbc
a b x : ℝ
hx : 0 < x
⊢ IntervalIntegrable log volume 1 x

Tactic:
·

Goal state after:
case hbc
a b x : ℝ
hx : 0 < x
⊢ IntervalIntegrable log volume 1 x
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: case pos
x : ℝ
h : 0 ≤ log x
⊢ max 0 (log x) - max 0 (-log x) = log x

pp: case neg
x : ℝ
h : log x < 0
⊢ max 0 (log x) - max 0 (-log x) = log x

Tactic:
·

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

Goal state before:
pp: case neg
x : ℝ
h : log x < 0
⊢ max 0 (log x) - max 0 (-log x) = log x

Tactic:
·

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

Goal state before:
pp: case pos
x : ℝ
hr : 0 ≤ log x
⊢ 2⁻¹ * (log x + |log x|) = log⁺ x

pp: case neg
x : ℝ
hr : log x < 0
⊢ 2⁻¹ * (log x + |log x|) = log⁺ x

Tactic:
·

Goal state after:
case pos
x : ℝ
hr : 0 ≤ log x
⊢ 2⁻¹ * (log x + |log x|) = log⁺ x
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: case neg
x : ℝ
hr : log x < 0
⊢ 2⁻¹ * (log x + |log x|) = log⁺ x

Tactic:
·

Goal state after:
case neg
x : ℝ
hr : log x < 0
⊢ 2⁻¹ * (log x + |log x|) = log⁺ x
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: case pos
n : ℕ
hn : n = 0
⊢ log⁺ ↑n = log ↑n

pp: case neg
n : ℕ
hn : ¬n = 0
⊢ log⁺ ↑n = log ↑n

Tactic:
·

Goal state after:
case pos
n : ℕ
hn : n = 0
⊢ log⁺ ↑n = log ↑n
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: case neg
n : ℕ
hn : ¬n = 0
⊢ log⁺ ↑n = log ↑n

Tactic:
·

Goal state after:
case neg
n : ℕ
hn : ¬n = 0
⊢ log⁺ ↑n = log ↑n
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: case pos
x : ℝ
hx : x ∈ Set.Ici 0
y : ℝ
hy : y ∈ Set.Ici 0
hxy : x ≤ y
h : log x ≤ 0
⊢ log x ≤ 0 ∨ 0 ≤ log y ∧ log x ≤ log y

pp: case neg
x : ℝ
hx : x ∈ Set.Ici 0
y : ℝ
hy : y ∈ Set.Ici 0
hxy : x ≤ y
h : 0 < log x
⊢ log x ≤ 0 ∨ 0 ≤ log y ∧ log x ≤ log y

Tactic:
·

Goal state after:
case pos
x : ℝ
hx : x ∈ Set.Ici 0
y : ℝ
hy : y ∈ Set.Ici 0
hxy : x ≤ y
h : log x ≤ 0
⊢ log x ≤ 0 ∨ 0 ≤ log y ∧ log x ≤ log y
Module: Mathlib.Analysis.SpecialFunctions.Log.PosLog

Goal state before:
pp: case neg
x : ℝ
hx : x ∈ Set.Ici 0
y : ℝ
hy : y ∈ Set.Ici 0
hxy : x ≤ y
h : 0 < log x
⊢ log x ≤ 0 ∨ 0 ≤ log y ∧ log x ≤ log y

Tactic:
·

Goal state after:
case neg
x : ℝ
hx : x ∈ Set.Ici 0
y : ℝ
hy : y ∈ Set.Ici 0
hxy : x ≤ y
h : 0 < log x
⊢ log x ≤ 0 ∨ 0 ≤ log y ∧ log x ≤ log y