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

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

Tactic:
]

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

Goal state before:
pp: θ φ r R : ℝ
h₁ : 0 < r
h₂ : r < R
⊢ (R - r) / (R + r) ≤
    (‖↑R * cexp (↑θ * I)‖ ^ 2 - ‖↑r * cexp (↑φ * I)‖ ^ 2) / ‖↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I)‖ ^ 2

Tactic:
]

Goal state after:
θ φ r R : ℝ
h₁ : 0 < r
h₂ : r < R
⊢ (R - r) / (R + r) ≤
    (‖↑R * cexp (↑θ * I)‖ ^ 2 - ‖↑r * cexp (↑φ * I)‖ ^ 2) / ‖↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I)‖ ^ 2
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: θ φ r R : ℝ
h₁ : 0 < r
h₂ : r < R
⊢ 1 *
      (‖↑R * cexp (↑θ * I)‖ ^ 2 + ‖↑r * cexp (↑φ * I)‖ ^ 2 -
        2 * (↑R * cexp (↑θ * I) * (starRingEnd ℂ) (↑r * cexp (↑φ * I))).re) ≤
    (R + r) * (R + r)

pp: case hpos
θ φ r R : ℝ
h₁ : 0 < r
h₂ : r < R
⊢ 0 < ‖↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I)‖ ^ 2

Tactic:
]

Goal state after:
θ φ r R : ℝ
h₁ : 0 < r
h₂ : r < R
⊢ 1 *
      (‖↑R * cexp (↑θ * I)‖ ^ 2 + ‖↑r * cexp (↑φ * I)‖ ^ 2 -
        2 * (↑R * cexp (↑θ * I) * (starRingEnd ℂ) (↑r * cexp (↑φ * I))).re) ≤
    (R + r) * (R + r)

case hpos
θ φ r R : ℝ
h₁ : 0 < r
h₂ : r < R
⊢ 0 < ‖↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I)‖ ^ 2
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: θ φ r R : ℝ
h₁ : 0 < r
h₂ : r < R
key : (-(↑R * cexp (↑θ * I) * (starRingEnd ℂ) (↑r * cexp (↑φ * I)))).re ≤ R * r
⊢ R ^ 2 + r ^ 2 + 2 * (-(↑R * cexp (↑θ * I) * (starRingEnd ℂ) (↑r * cexp (↑φ * I)))).re ≤ R ^ 2 + r ^ 2 + 2 * (R * r)

Tactic:
]

Goal state after:
θ φ r R : ℝ
h₁ : 0 < r
h₂ : r < R
key : (-(↑R * cexp (↑θ * I) * (starRingEnd ℂ) (↑r * cexp (↑φ * I)))).re ≤ R * r
⊢ R ^ 2 + r ^ 2 + 2 * (-(↑R * cexp (↑θ * I) * (starRingEnd ℂ) (↑r * cexp (↑φ * I)))).re ≤ R ^ 2 + r ^ 2 + 2 * (R * r)
Module: Mathlib.Analysis.Complex.PhragmenLindelof

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

Tactic:
]

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

Goal state before:
pp: z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
⊢ HarmonicAt (⇑reCLM ∘ (⇑conjCLE ∘ log ∘ g + log ∘ g)) z

pp: z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
⊢ Real.log ∘ ⇑normSq ∘ g =ᶠ[𝓝 z] ⇑reCLM ∘ (⇑conjCLE ∘ log ∘ g + log ∘ g)

Tactic:
]

Goal state after:
z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
⊢ HarmonicAt (⇑reCLM ∘ (⇑conjCLE ∘ log ∘ g + log ∘ g)) z

z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
⊢ Real.log ∘ ⇑normSq ∘ g =ᶠ[𝓝 z] ⇑reCLM ∘ (⇑conjCLE ∘ log ∘ g + log ∘ g)
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: θ φ r R : ℝ
h₁ : 0 < r
h₂ : r < R
key : (-(↑R * cexp (↑θ * I) * (starRingEnd ℂ) (↑r * cexp (↑φ * I)))).re ≤ R * r
⊢ R ^ 2 + r ^ 2 + 2 * (R * r) = (R + r) * (R + r)

Tactic:
by ring

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions

Goal state before:
pp: z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
⊢ reCLM (log ((starRingEnd ℂ) (g x) * g x)) = reCLM (log ((starRingEnd ℂ) (g x) * g x))

pp: z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
⊢ 0 ≤ normSq (g x)

Tactic:
]

Goal state after:
z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
⊢ 0 ≤ normSq (g x)
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: ⊢ ∀ (x : ℝ), ContinuousAt (fun x ↦ x * log x) x

Tactic:
]

Goal state after:
⊢ ∀ (x : ℝ), ContinuousAt (fun x ↦ x * log x) x
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: case inr
⊢ Filter.Tendsto (fun x ↦ x * log x) (𝓝 0) (𝓝 0)

Tactic:
]

Goal state after:
case inr
⊢ Filter.Tendsto (fun x ↦ x * log x) (𝓝 0) (𝓝 0)
Module: Mathlib.Analysis.Complex.PhragmenLindelof

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

Tactic:
]

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

Goal state before:
pp: case inr
this : Set.univ = Set.Iio 0 ∪ Set.Ioi 0 ∪ {0}
⊢ Filter.Tendsto (fun x ↦ log x * x) (𝓝[<] 0 ⊔ 𝓝[>] 0 ⊔ 𝓝[{0}] 0) (𝓝 0)

Tactic:
]

Goal state after:
case inr
this : Set.univ = Set.Iio 0 ∪ Set.Ioi 0 ∪ {0}
⊢ Filter.Tendsto (fun x ↦ log x * x) (𝓝[<] 0 ⊔ 𝓝[>] 0 ⊔ 𝓝[{0}] 0) (𝓝 0)
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions

Goal state before:
pp: case e_6
z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
hx : x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0})
⊢ (if (g x).arg = Real.pi then Real.pi else -(g x).arg) + (g x).arg ∈ Set.Ioc (-Real.pi) Real.pi

pp: z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
hx : x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0})
⊢ (starRingEnd ℂ) (g x) ≠ 0

Tactic:
]

Goal state after:
case e_6
z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
hx : x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0})
⊢ (if (g x).arg = Real.pi then Real.pi else -(g x).arg) + (g x).arg ∈ Set.Ioc (-Real.pi) Real.pi

z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
hx : x ∈ g ⁻¹' (slitPlane ∩ {y | y ≠ 0})
⊢ (starRingEnd ℂ) (g x) ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: x : ℝ
hx : x ≠ 0
⊢ DifferentiableAt ℝ (fun x ↦ x * log x) x

Tactic:
]

Goal state after:
x : ℝ
hx : x ≠ 0
⊢ DifferentiableAt ℝ (fun x ↦ x * log x) x
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: x : ℝ
hx : 0 < x
⊢ deriv (fun x ↦ x * log x) x = (fun x ↦ log x + 1) x

Tactic:
]

Goal state after:
x : ℝ
hx : 0 < x
⊢ deriv (fun x ↦ x * log x) x = (fun x ↦ log x + 1) x
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: ⊢ ∃ a, ∀ (b : ℝ), a ≤ b → deriv (fun x ↦ x * log x) b = log b + 1

Tactic:
]

Goal state after:
⊢ ∃ a, ∀ (b : ℝ), a ≤ b → deriv (fun x ↦ x * log x) b = log b + 1
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: ⊢ ∃ a, ∀ (b : ℝ), a ≤ b → derivWithin (fun x ↦ x * log x) (Set.Ioi b) b = log b + 1

Tactic:
]

Goal state after:
⊢ ∃ a, ∀ (b : ℝ), a ≤ b → derivWithin (fun x ↦ x * log x) (Set.Ioi b) b = log b + 1
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions

Goal state before:
pp: case right.e_6
z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
hx : x ∈ g ⁻¹' (slitPlane ∩ {0}ᶜ)
⊢ log ((starRingEnd ℂ) (g x)) + log (g x) = log ((starRingEnd ℂ) (g x)) + log (g x)

pp: case right.e_6.h
z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
hx : x ∈ g ⁻¹' (slitPlane ∩ {0}ᶜ)
⊢ (g x).arg ≠ Real.pi

Tactic:
]

Goal state after:
case right.e_6.h
z : ℂ
g : ℂ → ℂ
h₁g : AnalyticAt ℂ g z
h₂g : g z ≠ 0
h₃g : g z ∈ slitPlane
t₀ : g ⁻¹' (slitPlane ∩ {y | y ≠ 0}) ∈ 𝓝 z
x : ℂ
hx : x ∈ g ⁻¹' (slitPlane ∩ {0}ᶜ)
⊢ (g x).arg ≠ Real.pi
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: case pos
x : ℝ
hx : x = 0
⊢ deriv (deriv fun x ↦ x * log x) 0 = 0

Tactic:
]

Goal state after:
case pos
x : ℝ
hx : x = 0
⊢ deriv (deriv fun x ↦ x * log x) 0 = 0
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: x : ℝ
hx : ¬x = 0
this : ∀ᶠ (y : ℝ) in 𝓝 x, deriv (fun x ↦ x * log x) y = log y + 1
⊢ x⁻¹ = x⁻¹

Tactic:
]

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

Goal state before:
pp: x : ℝ
hx : 0 < x
⊢ 0 < x⁻¹

Tactic:
]

Goal state after:
x : ℝ
hx : 0 < x
⊢ 0 < x⁻¹
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions

Goal state before:
pp: f : ℂ → ℂ
z✝ : ℂ
h₁f : AnalyticAt ℂ f z✝
h₂f : f z✝ ≠ 0
z : ℂ
⊢ Real.log (normSq (f z)) / 2 = 2⁻¹ * Real.log (normSq (f z))

pp: f : ℂ → ℂ
z✝ : ℂ
h₁f : AnalyticAt ℂ f z✝
h₂f : f z✝ ≠ 0
z : ℂ
⊢ 0 ≤ normSq (f z)

Tactic:
]

Goal state after:
f : ℂ → ℂ
z✝ : ℂ
h₁f : AnalyticAt ℂ f z✝
h₂f : f z✝ ≠ 0
z : ℂ
⊢ Real.log (normSq (f z)) / 2 = 2⁻¹ * Real.log (normSq (f z))

f : ℂ → ℂ
z✝ : ℂ
h₁f : AnalyticAt ℂ f z✝
h₂f : f z✝ ≠ 0
z : ℂ
⊢ 0 ≤ normSq (f z)
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: case neg
x y : ℝ
hx : ¬x = 0
hy : ¬y = 0
⊢ -(x * y * (log x + log y)) = y * -(x * log x) + x * -(y * log y)

Tactic:
]

Goal state after:
case neg
x y : ℝ
hx : ¬x = 0
hy : ¬y = 0
⊢ -(x * y * (log x + log y)) = y * -(x * log x) + x * -(y * log y)
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions

Goal state before:
pp: f : ℂ → ℂ
z : ℂ
h₁f : AnalyticAt ℂ f z
h₂f : f z ≠ 0
this : (fun x ↦ Real.log ‖f x‖) = 2⁻¹ • Real.log ∘ ⇑normSq ∘ f
⊢ HarmonicAt (2⁻¹ • Real.log ∘ ⇑normSq ∘ f) z

Tactic:
]

Goal state after:
f : ℂ → ℂ
z : ℂ
h₁f : AnalyticAt ℂ f z
h₂f : f z ≠ 0
this : (fun x ↦ Real.log ‖f x‖) = 2⁻¹ • Real.log ∘ ⇑normSq ∘ f
⊢ HarmonicAt (2⁻¹ • Real.log ∘ ⇑normSq ∘ f) z
Module: Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions

Goal state before:
pp: case neg
f : ℂ → ℂ
z : ℂ
h₁f : AnalyticAt ℂ f z
h₂f : f z ≠ 0
this : (fun x ↦ Real.log ‖f x‖) = 2⁻¹ • Real.log ∘ ⇑normSq ∘ f
h₃f : f z ∉ slitPlane
⊢ HarmonicAt (Real.log ∘ ⇑normSq ∘ (-f)) z

Tactic:
]

Goal state after:
case neg
f : ℂ → ℂ
z : ℂ
h₁f : AnalyticAt ℂ f z
h₂f : f z ≠ 0
this : (fun x ↦ Real.log ‖f x‖) = 2⁻¹ • Real.log ∘ ⇑normSq ∘ f
h₃f : f z ∉ slitPlane
⊢ HarmonicAt (Real.log ∘ ⇑normSq ∘ (-f)) z
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: x : ℝ
hx : x ≠ 0
⊢ -(log x + 1) = -log x - 1

Tactic:
]

Goal state after:
x : ℝ
hx : x ≠ 0
⊢ -(log x + 1) = -log x - 1
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: x : ℝ
hx : x ≠ 0
⊢ DifferentiableAt ℝ negMulLog x

Tactic:
]

Goal state after:
x : ℝ
hx : x ≠ 0
⊢ DifferentiableAt ℝ negMulLog x
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: a b r : ℝ
h : -1 < r
c : ℝ
hc : 0 ≤ c
⊢ IntegrableOn (fun x ↦ x ^ r) (Ioc 0 c) volume

Tactic:
]

Goal state after:
a b r : ℝ
h : -1 < r
c : ℝ
hc : 0 ≤ c
⊢ IntegrableOn (fun x ↦ x ^ r) (Ioc 0 c) volume
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: x : ℝ
⊢ deriv^[2] (fun x ↦ -(x * log x)) x = -x⁻¹

Tactic:
]

Goal state after:
x : ℝ
⊢ deriv^[2] (fun x ↦ -(x * log x)) x = -x⁻¹
Module: Mathlib.Analysis.SpecialFunctions.Log.NegMulLog

Goal state before:
pp: x : ℝ
h : ∀ (x : ℝ), deriv (deriv fun x ↦ x * log x) x = x⁻¹
⊢ -x⁻¹ = -x⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc

Goal state before:
pp: ⊢ sinc 0 = 1

Tactic:
by simp [sinc]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc

Goal state before:
pp: case neg
x : ℝ
hx : ¬x = 0
⊢ |sin x| / |x| ≤ 1

Tactic:
]

Goal state after:
case neg
x : ℝ
hx : ¬x = 0
⊢ |sin x| / |x| ≤ 1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc

Goal state before:
pp: case neg
x : ℝ
hx : ¬x = 0
⊢ |sin x| ≤ |x|

Tactic:
]

Goal state after:
case neg
x : ℝ
hx : ¬x = 0
⊢ |sin x| ≤ |x|
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc

Goal state before:
pp: case inl
x : ℝ
hx : x < 0
⊢ sin x * x⁻¹ * -x ≤ 1

pp: case inl
x : ℝ
hx : x < 0
⊢ 0 < -x

Tactic:
]

Goal state after:
case inl
x : ℝ
hx : x < 0
⊢ sin x * x⁻¹ * -x ≤ 1

case inl
x : ℝ
hx : x < 0
⊢ 0 < -x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc

Goal state before:
pp: case inl
x : ℝ
hx : x < 0
⊢ -1 ≤ sin x

Tactic:
]

Goal state after:
case inl
x : ℝ
hx : x < 0
⊢ -1 ≤ sin 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'
x : ℝ
hx : x ∈ Ici 0
⊢ ↑x ∈ {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'
x : ℝ
hx : x ∈ Ici 0
⊢ ↑x ∈ {z | 0 ≤ z.re}
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc

Goal state before:
pp: case inr.inr
x : ℝ
hx : 0 < x
⊢ sin x ≤ 1

Tactic:
]

Goal state after:
case inr.inr
x : ℝ
hx : 0 < x
⊢ sin x ≤ 1
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: x : ℝ
hx : x ≠ 0
⊢ sin x / x ≤ |x|⁻¹

Tactic:
]

Goal state after:
x : ℝ
hx : x ≠ 0
⊢ sin x / x ≤ |x|⁻¹
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
z : ℂ
hz : z ∈ sphere 0 R
⊢ ((z + 0) / (z - 0)).re • f z = f z

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
z : ℂ
hz : z ∈ sphere 0 R
⊢ ((z + 0) / (z - 0)).re • f z = f z
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc

Goal state before:
pp: x : ℝ
⊢ ContinuousAt (dslope sin 0) x

Tactic:
]

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

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

Tactic:
]

Goal state after:
case neg
x : ℝ
hx : ¬x = 0
⊢ ContinuousAt sin x
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
m : IntegrableOn (cos (r * π) • fun x ↦ x ^ r) (Ι 0 (-c)) volume
⊢ IntegrableOn (fun x ↦ (-x) ^ r) (Ι 0 (-c)) volume

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
m : IntegrableOn (cos (r * π) • fun x ↦ x ^ r) (Ι 0 (-c)) volume
⊢ IntegrableOn (fun x ↦ (-x) ^ r) (Ι 0 (-c)) volume
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
m : IntegrableOn (cos (r * π) • fun x ↦ x ^ r) (Ι 0 (-c)) volume
x : ℝ
hx : x ∈ Ioc 0 (-c)
⊢ (cos (r * π) • fun x ↦ x ^ r) x = (fun x ↦ (-x) ^ r) x

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
m : IntegrableOn (cos (r * π) • fun x ↦ x ^ r) (Ι 0 (-c)) volume
x : ℝ
hx : x ∈ Ioc 0 (-c)
⊢ (cos (r * π) • fun x ↦ x ^ r) x = (fun x ↦ (-x) ^ r) x
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

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

Tactic:
]

Goal state after:
case refine_2
s t : ℝ
ht : 0 < t
h : -1 < s
⊢ IntervalIntegrable (fun x ↦ x ^ s) volume 0 t
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
y : ℝ
hy : 0 ≤ y
⊢ ‖0‖ ≤ ‖0‖

Tactic:
]

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

Goal state before:
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
⊢ 0 = -(-1)⁻¹ / log (-1) ^ 2

Tactic:
]

Goal state after:
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
⊢ 0 = -(-1)⁻¹ / log (-1) ^ 2
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
⊢ DifferentiableOn ℂ (fun z ↦ ↑q / (↑q * z - W)) (closedBall 0 R)

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
⊢ DifferentiableOn ℂ (fun z ↦ ↑q / (↑q * z - W)) (closedBall 0 R)
Module: Mathlib.Analysis.SpecialFunctions.Log.InvLog

Goal state before:
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
⊢ 0 = -0⁻¹ / log 0 ^ 2

Tactic:
]

Goal state after:
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
⊢ 0 = -0⁻¹ / log 0 ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Log.InvLog

Goal state before:
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
⊢ 0 = -1⁻¹ / log 1 ^ 2

Tactic:
]

Goal state after:
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
⊢ 0 = -1⁻¹ / log 1 ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: s t : ℝ
ht : 0 < t
h : s ≤ -1
H : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume
I : 0 < min 1 t
H' : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 (min 1 t)) volume
x : ℝ
hx : x ∈ Ioo 0 (min 1 t)
⊢ s ≤ -1

pp: s t : ℝ
ht : 0 < t
h : s ≤ -1
H : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume
I : 0 < min 1 t
H' : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 (min 1 t)) volume
x : ℝ
hx : x ∈ Ioo 0 (min 1 t)
⊢ x < 1

Tactic:
]

Goal state after:
s t : ℝ
ht : 0 < t
h : s ≤ -1
H : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume
I : 0 < min 1 t
H' : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 (min 1 t)) volume
x : ℝ
hx : x ∈ Ioo 0 (min 1 t)
⊢ s ≤ -1

s t : ℝ
ht : 0 < t
h : s ≤ -1
H : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume
I : 0 < min 1 t
H' : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 (min 1 t)) volume
x : ℝ
hx : x ∈ Ioo 0 (min 1 t)
⊢ x < 1
Module: Mathlib.Analysis.SpecialFunctions.Log.InvLog

Goal state before:
pp: ⊢ Tendsto (fun x ↦ (log x)⁻¹) atTop (nhds 0)

Tactic:
]

Goal state after:
⊢ Tendsto (fun x ↦ (log x)⁻¹) atTop (nhds 0)
Module: Mathlib.Analysis.SpecialFunctions.Integrability.Basic

Goal state before:
pp: s t : ℝ
ht : 0 < t
h : s ≤ -1
H : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume
I : 0 < min 1 t
H' : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 (min 1 t)) volume
this : IntegrableOn (fun x ↦ x⁻¹) (Ioo 0 (min 1 t)) volume
⊢ IntegrableOn (fun x ↦ x⁻¹) (Ioo 0 (min 1 t)) volume

Tactic:
]

Goal state after:
s t : ℝ
ht : 0 < t
h : s ≤ -1
H : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 t) volume
I : 0 < min 1 t
H' : IntegrableOn (fun x ↦ x ^ s) (Ioo 0 (min 1 t)) volume
this : IntegrableOn (fun x ↦ x⁻¹) (Ioo 0 (min 1 t)) volume
⊢ IntegrableOn (fun x ↦ x⁻¹) (Ioo 0 (min 1 t)) volume