mathlib-initiative/mathlib-tactics huggingface dataset:
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 ↑(Real.log (normSq (g x))) = reCLM (log ((starRingEnd ℂ) (g x) * g x))

Tactic:
rw [ofReal_log, normSq_eq_conj_mul_self]

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.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
⊢ ⇑reCLM ∘ ⇑ofRealCLM ∘ Real.log ∘ ⇑normSq ∘ g =ᶠ[𝓝 z] ⇑reCLM ∘ log ∘ (⇑conjCLE ∘ g * g)

Tactic:
by

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

Goal state before:
pp: ⊢ Set.univ = Set.Iio 0 ∪ Set.Ioi 0 ∪ {0}

Tactic:
by

Goal state after:
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) + ↑r * cexp (↑φ * I)) / (↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I))).re

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 (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
hz_re : 0 ≤ z.re
hz_im : z.im ≤ 0
⊢ ‖f z‖ ≤ C

Tactic:
by

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

Goal state before:
pp: ⊢ Continuous fun x ↦ x * log x

Tactic:
by

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

Goal state before:
pp: R : ℝ
w c z : ℂ
hz : z ∈ sphere c R
hw : w ∈ ball c R
⊢ (R - ‖w - c‖) / (R + ‖w - c‖) ≤ ((z - c + (w - c)) / (z - c - (w - c))).re

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: R : ℝ
w c : ℂ
hw : w ∈ ball c R
⊢ ContinuousOn (fun z ↦ z - c + (w - c)) (sphere c |R|)

Tactic:
by

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

Goal state before:
pp: R : ℝ
w c : ℂ
hw : w ∈ ball c R
⊢ ContinuousOn (fun z ↦ z - c - (w - c)) (sphere c |R|)

Tactic:
by

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

Goal state before:
pp: R : ℝ
w c : ℂ
hw : w ∈ ball c R
⊢ ContinuousOn (herglotzRieszKernel c w) (sphere c |R|)

Tactic:
by

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

Goal state before:
pp: x : ℝ
hx : x ≠ 0
⊢ HasDerivAt (fun x ↦ x * log x) (log x + 1) x

Tactic:
by

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

Goal state before:
pp: R : ℝ
w : ℂ
g : ℂ → ℝ
hg : CircleIntegrable g 0 R
hw : w ∈ ball 0 R
⊢ CircleIntegrable (fun ζ ↦ ↑(g ζ)) 0 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
C : ℝ
f : ℂ → E
hd : DiffContOnCl ℂ f {z | 0 < z.re}
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
C' : ℝ
hC' : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C'
z : ℂ
hz : 0 ≤ z.re
c : ℝ
hc : c < 2
B : ℝ
hO : f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
⊢ ‖f z‖ ≤ max C C'

Tactic:
rcases le_total z.im 0 with h | h

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}
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
C' : ℝ
hC' : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C'
z : ℂ
hz : 0 ≤ z.re
c : ℝ
hc : c < 2
B : ℝ
hO : f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
h : z.im ≤ 0
⊢ ‖f z‖ ≤ max C C'

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}
hre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)
him : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
C' : ℝ
hC' : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C'
z : ℂ
hz : 0 ≤ z.re
c : ℝ
hc : c < 2
B : ℝ
hO : f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)
h : 0 ≤ z.im
⊢ ‖f z‖ ≤ max C C'
Module: Mathlib.Analysis.Complex.Poisson

Goal state before:
pp: R : ℝ
w : ℂ
g : ℂ → ℝ
hg : CircleIntegrable g 0 R
hw : w ∈ ball 0 R
h₁ : CircleIntegrable (fun ζ ↦ ↑(g ζ)) 0 R
h₂ : CircleIntegrable (fun ζ ↦ herglotzRieszKernel 0 w ζ • ↑(g ζ)) 0 R
⊢ circleAverage (⇑reCLM ∘ fun ζ ↦ herglotzRieszKernel 0 w ζ • ↑(g ζ)) 0 R =
    circleAverage (re ∘ herglotzRieszKernel 0 w • g) 0 R

Tactic:
by

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

Goal state before:
pp: ⊢ (deriv fun x ↦ x * log x) =ᶠ[𝓝[>] 0] fun x ↦ log x + 1

Tactic:
by

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

Goal state before:
pp: R : ℝ
w : ℂ
g : ℂ → ℝ
hg : CircleIntegrable g 0 R
hw : w ∈ ball 0 R
⊢ (circleAverage (fun ζ ↦ herglotzRieszKernel 0 w ζ • ↑(g ζ)) 0 R).re =
    circleAverage (re ∘ herglotzRieszKernel 0 w • g) 0 R

Tactic:
by

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
⊢ ⇑reCLM ∘ log ∘ (⇑conjCLE ∘ g * g) =ᶠ[𝓝 z] ⇑reCLM ∘ (log ∘ ⇑conjCLE ∘ g + log ∘ g)

Tactic:
by

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

Goal state before:
pp: ⊢ Tendsto (deriv fun x ↦ x * log x) (𝓝[>] 0) atBot

Tactic:
by

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

Goal state before:
pp: ⊢ Tendsto (fun x ↦ deriv (fun x ↦ x * log x) x) atTop atTop

Tactic:
by

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

Goal state before:
pp: ⊢ Tendsto (fun x ↦ derivWithin (fun x ↦ x * log x) (Set.Ioi x) x) atTop atTop

Tactic:
by

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

Goal state before:
pp: ⊢ Disjoint (𝓝 (deriv (fun x ↦ x * log x) 0)) Filter.atBot

Tactic:
by

Goal state after:
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
⊢ deriv (deriv fun x ↦ x * log x) x = x⁻¹

Tactic:
by

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
⊢ ⇑reCLM ∘ (log ∘ ⇑conjCLE ∘ g + log ∘ g) =ᶠ[𝓝 z] ⇑reCLM ∘ (⇑conjCLE ∘ log ∘ g + log ∘ g)

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: ⊢ StrictConvexOn ℝ (Set.Ici 0) fun x ↦ x * log x

Tactic:
by

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

Goal state before:
pp: ⊢ negMulLog = fun x ↦ -(x * log x)

Tactic:
by

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

Goal state before:
pp: ⊢ negMulLog 0 = 0

Tactic:
by

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

Goal state before:
pp: ⊢ negMulLog 1 = 0

Tactic:
by

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

Goal state before:
pp: x : ℝ
h1 : 0 ≤ x
h2 : x ≤ 1
⊢ 0 ≤ x.negMulLog

Tactic:
by

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
⊢ HarmonicAt (Real.log ∘ ⇑normSq ∘ g) z

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: x y : ℝ
⊢ (x * y).negMulLog = y * x.negMulLog + x * y.negMulLog

Tactic:
by

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

Goal state before:
pp: ⊢ Continuous negMulLog

Tactic:
by

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

Goal state before:
pp: ⊢ DifferentiableOn ℝ negMulLog {0}ᶜ

Tactic:
by

Goal state after:
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
h₃f : f z ∉ slitPlane
⊢ ⇑normSq ∘ f = ⇑normSq ∘ (-f)

Tactic:
by

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

Goal state before:
pp: x : ℝ
hx : x ≠ 0
⊢ x ∈ {0}ᶜ

Tactic:
by

Goal state after:
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
h₃f : f z ∉ slitPlane
⊢ (-f) z ≠ 0

Tactic:
by

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

Goal state before:
pp: f : ℂ → ℂ
z : ℂ
h₁f : AnalyticAt ℂ f z
h₂f : f z ≠ 0
⊢ HarmonicAt (fun x ↦ Real.log ‖f x‖) z

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: x : ℝ
hx : x ≠ 0
⊢ deriv negMulLog x = -log x - 1

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
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
C' : ℝ
hC' : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C'
z : ℂ
hz : 0 ≤ z.re
⊢ ‖f z‖ ≤ max C C'

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: a b r : ℝ
h : -1 < r
c : ℝ
hc : 0 ≤ c
x : ℝ
hx : x ∈ Ioo 0 c
⊢ r + 1 ≠ 0

Tactic:
by

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

Goal state before:
pp: a b r : ℝ
h : -1 < r
c : ℝ
hc : 0 ≤ c
⊢ ∀ x ∈ Ioo 0 c, HasDerivAt (fun x ↦ x ^ (r + 1) / (r + 1)) (x ^ r) x

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: x : ℝ
⊢ deriv^[2] negMulLog x = -x⁻¹

Tactic:
by

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

Goal state before:
pp: ⊢ StrictConcaveOn ℝ (Set.Ici 0) negMulLog

Tactic:
by

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

Goal state before:
pp: x : ℝ
h0 : 0 ≤ x
h1 : x ≠ 1
⊢ x.negMulLog < 1 - x

Tactic:
by

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

Goal state before:
pp: x : ℝ
h0 : 0 ≤ x
⊢ x.negMulLog ≤ 1 - x

Tactic:
by

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

Goal state before:
pp: a b r : ℝ
h : -1 < r
c : ℝ
hc : 0 ≤ c
hderiv : ∀ x ∈ Ioo 0 c, HasDerivAt (fun x ↦ x ^ (r + 1) / (r + 1)) (x ^ r) x
⊢ ContinuousOn (fun x ↦ x ^ (r + 1) / (r + 1)) (Icc 0 c)

Tactic:
refine (continuousOn_id.rpow_const ?_).div_const _

Goal state after:
a b r : ℝ
h : -1 < r
c : ℝ
hc : 0 ≤ c
hderiv : ∀ x ∈ Ioo 0 c, HasDerivAt (fun x ↦ x ^ (r + 1) / (r + 1)) (x ^ r) x
⊢ ∀ x ∈ Icc 0 c, id x ≠ 0 ∨ 0 ≤ r + 1
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc

Goal state before:
pp: ⊢ sinc 0 = 1

Tactic:
by

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

Goal state before:
pp: x : ℝ
hx : x ≠ 0
⊢ sinc x = sin x / x

Tactic:
simp [sinc, hx]

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

Goal state before:
pp: x : ℝ
hx : x ≠ 0
⊢ sinc x = sin x / x

Tactic:
by

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

Goal state before:
pp: ⊢ sinc = dslope sin 0

Tactic:
by

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

Goal state before:
pp: x : ℝ
⊢ sinc (-x) = sinc x

Tactic:
by

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

Goal state before:
pp: x : ℝ
⊢ |sinc x| ≤ 1

Tactic:
by

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

Goal state before:
pp: R φ θ r : ℝ
⊢ ↑R * cexp (↑θ * I) / (↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I)) -
      ↑r * cexp (↑θ * I) / (↑r * cexp (↑θ * I) - ↑R * cexp (↑φ * I)) =
    ↑((↑R * cexp (↑θ * I) + ↑r * cexp (↑φ * I)) / (↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I))).re

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: x : ℝ
⊢ sin x / x ≤ |x|⁻¹

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
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'
⊢ ContinuousOn (fun x ↦ f ↑x) (Ici 0)

Tactic:
by

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

Goal state before:
pp: 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
⊢ 0 ≤ -c

Tactic:
by

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

Goal state before:
pp: x : ℝ
hx : x ≠ 0
⊢ sinc x ≤ |x|⁻¹

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
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

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
⊢ 0 < q

Tactic:
by

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

Goal state before:
pp: ⊢ Continuous sinc

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
⊢ q < 1

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
x : ℂ
h : ‖x‖ ≤ R
this : ‖↑q * x‖ < ‖W‖
⊢ ↑q * x - W ≠ 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
x : ℂ
h : ‖x‖ ≤ R
⊢ ‖↑q * x‖ = q * ‖x‖

Tactic:
by

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

Goal state before:
pp: H : Tendsto (fun t ↦ (log t * t)⁻¹) (nhdsWithin 0 {0}ᶜ) (nhds (deriv (fun x ↦ (log x)⁻¹) 0))
⊢ Set.Iio 0 ⊆ {0}ᶜ

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
x : ℂ
h : ‖x‖ ≤ R
⊢ q * ‖x‖ ≤ q * R

Tactic:
by

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

Goal state before:
pp: H : Tendsto (fun t ↦ (log t * t)⁻¹) (nhdsWithin 0 {0}ᶜ) (nhds (deriv (fun x ↦ (log x)⁻¹) 0))
⊢ Disjoint (nhds (deriv (fun x ↦ (log x)⁻¹) 0)) atTop

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
x : ℂ
h : ‖x‖ ≤ R
⊢ q * R < 1 * R

Tactic:
by

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

Goal state before:
pp: 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 ∈ Ι 0 (-c)
⊢ 0 ≤ -c

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
x : ℂ
h : ‖x‖ ≤ R
⊢ 1 * R = ‖W‖

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
f : ℂ → E
R : ℝ
w : ℂ
inst✝ : CompleteSpace E
hf : DiffContOnCl ℂ f (ball 0 R)
hw : w ∈ ball 0 R
hR : 0 < R
h₁w : w ≠ 0
W : ℂ := ↑R * cexp (↑w.arg * I)
q : ℝ := ‖w‖ / R
h₁q : 0 < q
h₂q : q < 1
x : ℂ
h : ‖x‖ ≤ R
⊢ ↑q * x - W ≠ 0

Tactic:
by

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

Goal state before:
pp: ⊢ ¬DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) 0

Tactic:
by

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

Goal state before:
pp: ⊢ HasDerivAt log 1 1

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: ⊢ ¬ContinuousAt (fun x ↦ (log x)⁻¹) 1

Tactic:
by

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

Goal state before:
pp: 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)
⊢ -x < 0

Tactic:
by

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

Goal state before:
pp: H : ContinuousAt (fun x ↦ (log x)⁻¹) (-1)
⊢ ContinuousAt (fun x ↦ (log x)⁻¹) 1

Tactic:
by

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

Goal state before:
pp: a b r : ℝ
h : -1 < r
⊢ IntervalIntegrable (fun x ↦ x ^ r) volume a b

Tactic:
by

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

Goal state before:
pp: 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
⊢ x ≠ -1 ∧ x ≠ 0 ∧ x ≠ 1 ∨ x = -1 ∨ x = 0 ∨ x = 1

Tactic:
by

Goal state after:
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‖ ≤ ‖f ↑0‖

Tactic:
h₀ 0 le_rfl

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
y : ℝ
hy : 0 ≤ y
⊢ ‖0‖ ≤ ‖0‖
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
⊢ ∮ (z : ℂ) in C(0, R), z⁻¹ • (↑q * z / (↑q * z - W)) • f z = ∮ (z : ℂ) in C(0, R), (↑q / (↑q * z - W)) • f z

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: x : ℝ
hx₀ : x ≠ 0
hx₁ : x ≠ 1
hx₂ : x ≠ -1
⊢ DifferentiableAt ℝ (fun x ↦ (log x)⁻¹) x

Tactic:
by

Goal state after:
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
⊢ IntegrableOn (fun x ↦ x⁻¹) (Ioo 0 (min 1 t)) volume

Tactic:
by

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

Goal state before:
pp: x : ℝ
hx₀ : x ≠ 0
hx₁ : x ≠ 1
hx₂ : x ≠ -1
⊢ HasDerivAt (fun x ↦ (log x)⁻¹) (-x⁻¹ / log x ^ 2) x

Tactic:
by

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

Goal state before:
pp: ⊢ {-1, 0, 1}ᶜ ⊆ {0}ᶜ

Tactic:
by

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

Goal state before:
pp: ⊢ ∀ x ∈ {-1, 0, 1}ᶜ, log x ≠ 0

Tactic:
by

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

Goal state before:
pp: ⊢ Set.Ioi 1 ⊆ {-1, 0, 1}ᶜ

Tactic:
by

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

Goal state before:
pp: ⊢ (fun x ↦ (log x)⁻¹) =o[atTop] fun x ↦ 1

Tactic:
by

Goal state after: