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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
z : ℂ
hz : z ∈ Metric.ball c r
r' : NNReal
hr' : ↑r' < r
hr'₀ : 0 < ↑r'
hzr' : z ∈ Metric.ball c ↑r'
⊢ z - c ∈ Metric.ball 0 ↑r'

Tactic:
]

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
z : ℂ
hz : z ∈ Metric.ball c r
r' : NNReal
hr' : ↑r' < r
hr'₀ : 0 < ↑r'
hzr' : z ∈ Metric.ball c ↑r'
⊢ z - c ∈ Metric.ball 0 ↑r'
Module: Mathlib.Analysis.Complex.Polynomial.UnitTrinomial

Goal state before:
pp: case refine_1
p : ℤ[X]
hp : p.IsUnitTrinomial
h : ∀ (z : ℂ), ¬((aeval z) p = 0 ∧ (aeval z) p.mirror = 0)
q : ℤ[X]
hq' : q ∣ p.mirror
hq'' : 0 < (map (algebraMap ℤ ℂ) q).degree
z : ℂ
hz : (aeval z) q = 0
g' : ℤ[X]
hg' : p = q * g'
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.UnitTrinomial

Goal state before:
pp: case refine_2
p : ℤ[X]
hp : p.IsUnitTrinomial
h : ∀ (z : ℂ), ¬((aeval z) p = 0 ∧ (aeval z) p.mirror = 0)
q : ℤ[X]
hq : q ∣ p
hq'' : 0 < (map (algebraMap ℤ ℂ) q).degree
z : ℂ
hz : (aeval z) q = 0
g' : ℤ[X]
hg' : p.mirror = q * g'
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: a b : ℝ
m n : ℕ
hc : Continuous fun u ↦ u ^ m * (1 - u ^ 2) ^ n
x✝ : ℝ
⊢ 1 - sin x✝ ^ 2 = 1 - sin x✝ ^ 2

Tactic:
]

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

Goal state before:
pp: case e'_5
c : ℂ
r : ℝ
z : ℂ
hz : z ∈ Metric.ball c r
f : ℂ → ℂ
hf : DifferentiableOn ℂ f (Metric.ball c r)
n : ℕ
⊢ (↑n !)⁻¹ * ((z - c) ^ n * iteratedDeriv n f c) = (↑n !)⁻¹ * ((z - c) ^ n * iteratedDeriv n f c)

Tactic:
]

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ENNReal
hf : DifferentiableOn ℂ f (Metric.eball c r)
z : ℂ
hz : z ∈ Metric.eball c r
r' : NNReal
hzr' : z ∈ Metric.ball c ↑r'
hr' : ↑r' < r
⊢ HasSum (fun n ↦ (↑n !)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z)

Tactic:
]

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ENNReal
hf : DifferentiableOn ℂ f (Metric.eball c r)
z : ℂ
hz : z ∈ Metric.eball c r
r' : NNReal
hzr' : z ∈ Metric.ball c ↑r'
hr' : ↑r' < r
⊢ HasSum (fun n ↦ (↑n !)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z)
Module: Mathlib.Analysis.Complex.Positivity

Goal state before:
pp: f : ℂ → ℂ
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c
z : ℂ
hz₁ : 0 ≤ z - c
hz₂ : z ∈ Metric.ball c r
H : ∑' (n : ℕ), (↑n.factorial)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z
⊢ 0 ≤ f z

Tactic:
]

Goal state after:
f : ℂ → ℂ
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c
z : ℂ
hz₁ : 0 ≤ z - c
hz₂ : z ∈ Metric.ball c r
H : ∑' (n : ℕ), (↑n.factorial)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = f z
⊢ 0 ≤ f z
Module: Mathlib.Analysis.Complex.Positivity

Goal state before:
pp: f : ℂ → ℂ
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c
z : ℂ
hz₁ : 0 ≤ ↑(z - c).re
hz₂ : z ∈ Metric.ball c r
H : ∑' (n : ℕ), (↑n.factorial)⁻¹ * iteratedDeriv n f c * ↑(z - c).re ^ n = f z
hz' : z - c = ↑(z - c).re
⊢ 0 ≤ f z

Tactic:
]

Goal state after:
f : ℂ → ℂ
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c
z : ℂ
hz₁ : 0 ≤ ↑(z - c).re
hz₂ : z ∈ Metric.ball c r
H : ∑' (n : ℕ), (↑n.factorial)⁻¹ * iteratedDeriv n f c * ↑(z - c).re ^ n = f z
hz' : z - c = ↑(z - c).re
⊢ 0 ≤ f z
Module: Mathlib.Analysis.Complex.Positivity

Goal state before:
pp: f : ℂ → ℂ
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c
z : ℂ
hz₁ : 0 ≤ ↑(z - c).re
hz₂ : z ∈ Metric.ball c r
H : ∑' (n : ℕ), (↑n.factorial)⁻¹ * iteratedDeriv n f c * ↑(z - c).re ^ n = f z
hz' : z - c = ↑(z - c).re
n : ℕ
⊢ 0 ≤ ↑((↑n.factorial)⁻¹ * (iteratedDeriv n f c).re * (z - c).re ^ n)

Tactic:
]

Goal state after:
f : ℂ → ℂ
c : ℂ
r : ℝ
hf : DifferentiableOn ℂ f (Metric.ball c r)
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c
z : ℂ
hz₁ : 0 ≤ ↑(z - c).re
hz₂ : z ∈ Metric.ball c r
H : ∑' (n : ℕ), (↑n.factorial)⁻¹ * iteratedDeriv n f c * ↑(z - c).re ^ n = f z
hz' : z - c = ↑(z - c).re
n : ℕ
⊢ 0 ≤ ↑((↑n.factorial)⁻¹ * (iteratedDeriv n f c).re * (z - c).re ^ n)
Module: Mathlib.Analysis.Complex.Positivity

Goal state before:
pp: f : ℂ → ℂ
hf : Differentiable ℂ f
c : ℂ
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c
z : ℂ
hz : 0 ≤ z - c
⊢ z ∈ Metric.ball c ((z - c).re + 1)

Tactic:
]

Goal state after:
f : ℂ → ℂ
hf : Differentiable ℂ f
c : ℂ
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c
z : ℂ
hz : 0 ≤ z - c
⊢ z ∈ Metric.ball c ((z - c).re + 1)
Module: Mathlib.Analysis.Complex.TaylorSeries

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ENNReal
hf : DifferentiableOn ℂ f (Metric.eball c r)
z : ℂ
hz : z ∈ Metric.eball c r
r' : NNReal
hzr' : z ∈ Metric.ball c ↑r'
hr' : ↑r' < r
⊢ DifferentiableOn ℂ f (Metric.eball c ↑r')

Tactic:
]

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ENNReal
hf : DifferentiableOn ℂ f (Metric.eball c r)
z : ℂ
hz : z ∈ Metric.eball c r
r' : NNReal
hzr' : z ∈ Metric.ball c ↑r'
hr' : ↑r' < r
⊢ DifferentiableOn ℂ f (Metric.eball c ↑r')
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
m n : ℕ
hc : Continuous fun u ↦ u ^ n * (1 - u ^ 2) ^ m
⊢ -∫ (x : ℝ) in b..a, sin x ^ (2 * m + 1) * cos x ^ n = -∫ (x : ℝ) in b..a, sin x ^ (2 * m + 1) * cos x ^ n

Tactic:
]

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

Goal state before:
pp: f : ℂ → ℂ
hf : Differentiable ℂ f
c : ℂ
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c
z : ℂ
hz : 0 ≤ z - c
⊢ ‖↑(z - c).re‖ < (↑(z - c).re).re + 1

Tactic:
]

Goal state after:
f : ℂ → ℂ
hf : Differentiable ℂ f
c : ℂ
h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c
z : ℂ
hz : 0 ≤ z - c
⊢ ‖↑(z - c).re‖ < (↑(z - c).re).re + 1
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
m n : ℕ
hc : Continuous fun u ↦ u ^ n * (1 - u ^ 2) ^ m
x✝ : ℝ
⊢ 1 - cos x✝ ^ 2 = 1 - cos x✝ ^ 2

Tactic:
]

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

Goal state before:
pp: case succ
f : ℂ → ℂ
c : ℂ
hf : Differentiable ℂ f
z : ℂ
hz : c ≤ z
n : ℕ
h : 0 ≤ iteratedDeriv n (_root_.deriv f) c
⊢ 0 ≤ iteratedDeriv n (_root_.deriv fun x ↦ f x - f c) c

Tactic:
]

Goal state after:
case succ
f : ℂ → ℂ
c : ℂ
hf : Differentiable ℂ f
z : ℂ
hz : c ≤ z
n : ℕ
h : 0 ≤ iteratedDeriv n (_root_.deriv f) c
⊢ 0 ≤ iteratedDeriv n (_root_.deriv fun x ↦ f x - f c) c
Module: Mathlib.Analysis.Complex.TaylorSeries

Goal state before:
pp: case e'_5
c : ℂ
r : ENNReal
z : ℂ
hz : z ∈ Metric.eball c r
f : ℂ → ℂ
hf : DifferentiableOn ℂ f (Metric.eball c r)
n : ℕ
⊢ (↑n !)⁻¹ * ((z - c) ^ n * iteratedDeriv n f c) = (↑n !)⁻¹ * ((z - c) ^ n * iteratedDeriv n f c)

Tactic:
]

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

Goal state before:
pp: case succ
f : ℂ → ℂ
c : ℂ
hf : Differentiable ℂ f
z : ℂ
hz : c ≤ z
n : ℕ
h : 0 ≤ iteratedDeriv n (_root_.deriv f) c
⊢ 0 ≤ iteratedDeriv n (_root_.deriv f) c

Tactic:
]

Goal state after:
case succ
f : ℂ → ℂ
c : ℂ
hf : Differentiable ℂ f
z : ℂ
hz : c ≤ z
n : ℕ
h : 0 ≤ iteratedDeriv n (_root_.deriv f) c
⊢ 0 ≤ iteratedDeriv n (_root_.deriv f) c
Module: Mathlib.Analysis.Complex.TaylorSeries

Goal state before:
pp: case e'_5
c z : ℂ
f : ℂ → ℂ
hf : Differentiable ℂ f
n : ℕ
⊢ (↑n !)⁻¹ * ((z - c) ^ n * iteratedDeriv n f c) = (↑n !)⁻¹ * ((z - c) ^ n * iteratedDeriv n f c)

Tactic:
]

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

Goal state before:
pp: a b c : ℝ
⊢ (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4

Tactic:
by ring

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

Goal state before:
pp: w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : w ∈ ball 0 R
hR : 0 < R
⊢ circleAverage (Complex.re ∘ herglotzRieszKernel 0 w • fun x ↦ log ‖x - ρ‖) 0 R = log ‖w - ρ‖

Tactic:
]

Goal state after:
w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : w ∈ ball 0 R
hR : 0 < R
⊢ circleAverage (Complex.re ∘ herglotzRieszKernel 0 w • fun x ↦ log ‖x - ρ‖) 0 R = log ‖w - ρ‖
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x ↦ cos (2 * x) ^ 2
x✝ : ℝ
⊢ cos x✝ * sin x✝ = 2 * sin x✝ * cos x✝ / 2

Tactic:
]

Goal state after:
a b : ℝ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x ↦ cos (2 * x) ^ 2
x✝ : ℝ
⊢ cos x✝ * sin x✝ = 2 * sin x✝ * cos x✝ / 2
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
hR : 0 < R
⊢ circleAverage (Complex.re ∘ herglotzRieszKernel 0 w • fun x ↦ log ‖x - ρ‖) 0 R = log ‖w - ρ‖

Tactic:
]

Goal state after:
w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
hR : 0 < R
⊢ circleAverage (Complex.re ∘ herglotzRieszKernel 0 w • fun x ↦ log ‖x - ρ‖) 0 R = log ‖w - ρ‖
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: a b : ℝ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x ↦ cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
d : ℝ
⊢ 2 * (2 * d) = 4 * d

Tactic:
by ring

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

Goal state before:
pp: ⊢ ∫ (x : ℝ) in -1..1, √(1 - x ^ 2) = ∫ (x : ℝ) in -1..1, √(1 - x ^ 2)

Tactic:
]

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

Goal state before:
pp: x✝ : ℝ
h : -(π / 2) < x✝ ∧ x✝ ≤ π / 2
⊢ √(1 - sin x✝ ^ 2) * cos x✝ = cos x✝ ^ 2

Tactic:
]

Goal state after:
x✝ : ℝ
h : -(π / 2) < x✝ ∧ x✝ ≤ π / 2
⊢ √(1 - sin x✝ ^ 2) * cos x✝ = cos x✝ ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Integrals.Basic

Goal state before:
pp: x✝ : ℝ
h : -(π / 2) < x✝ ∧ x✝ ≤ π / 2
⊢ cos x✝ * cos x✝ = cos x✝ * cos x✝

Tactic:
]

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

Goal state before:
pp: case inr
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU✝ : U ≠ univ
hU : ∃ a, a ∉ U
this :
  ∀ {U : Set ℂ} (hUo : IsOpen U) (hUc : IsSimplyConnected U) (hU : U ≠ univ),
    0 ∉ U → ∃ f, Injective f ∧ ¬Dense (f '' U) ∧ ∀ z ∈ U, deriv f z ≠ 0
hU₀ : ¬0 ∉ U
⊢ ∃ f, Injective f ∧ ¬Dense (f '' U) ∧ ∀ z ∈ U, deriv f z ≠ 0

Tactic:
]

Goal state after:
case inr
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU✝ : U ≠ univ
hU : ∃ a, a ∉ U
this :
  ∀ {U : Set ℂ} (hUo : IsOpen U) (hUc : IsSimplyConnected U) (hU : U ≠ univ),
    0 ∉ U → ∃ f, Injective f ∧ ¬Dense (f '' U) ∧ ∀ z ∈ U, deriv f z ≠ 0
hU₀ : ¬0 ∉ U
⊢ ∃ f, Injective f ∧ ¬Dense (f '' U) ∧ ∀ z ∈ U, deriv f z ≠ 0
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: case inr
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU✝ : U ≠ univ
hU : ∃ a, a ∉ U
this :
  ∀ {U : Set ℂ} (hUo : IsOpen U) (hUc : IsSimplyConnected U) (hU : U ≠ univ),
    0 ∉ U → ∃ f, Injective f ∧ ¬Dense (f '' U) ∧ ∀ z ∈ U, deriv f z ≠ 0
hU₀ : ¬0 ∉ U
⊢ ∃ f, Injective f ∧ ¬Dense (f '' U) ∧ ∀ z ∈ U, deriv f z ≠ 0

Tactic:
rcases hU with ⟨a, ha⟩

Goal state after:
case inr
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
this :
  ∀ {U : Set ℂ} (hUo : IsOpen U) (hUc : IsSimplyConnected U) (hU : U ≠ univ),
    0 ∉ U → ∃ f, Injective f ∧ ¬Dense (f '' U) ∧ ∀ z ∈ U, deriv f z ≠ 0
hU₀ : ¬0 ∉ U
a : ℂ
ha : a ∉ U
⊢ ∃ f, Injective f ∧ ¬Dense (f '' U) ∧ ∀ z ∈ U, deriv f z ≠ 0
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
⊢ 0 ∉ U

Tactic:
]

Goal state after:
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
⊢ 0 ∉ U
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
hR : 0 < R
r : ℕ → ℝ := fun n ↦ R - (R - ‖w‖) / (↑n + 2)
hr_lt : ∀ (n : ℕ), r n < R
hr_pos : ∀ (n : ℕ), 0 < r n
hr_tendsto : Tendsto r atTop (𝓝 R)
DCT :
  Tendsto (fun n ↦ circleAverage (herglotzLogIntegrand w ρ) 0 (r n)) atTop
    (𝓝 (circleAverage (herglotzLogIntegrand w ρ) 0 R))
n : ℕ
⊢ ‖w‖ * (↑n + 2) + (R - ‖w‖) = ‖w‖ * (↑n + 1) + R

Tactic:
by ring

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

Goal state before:
pp: w ρ : ℂ
R : ℝ
hρ : ‖ρ‖ = R
hw : ‖w‖ < R
hR : 0 < R
r : ℕ → ℝ := fun n ↦ R - (R - ‖w‖) / (↑n + 2)
hr_lt : ∀ (n : ℕ), r n < R
hr_pos : ∀ (n : ℕ), 0 < r n
hr_tendsto : Tendsto r atTop (𝓝 R)
DCT :
  Tendsto (fun n ↦ circleAverage (herglotzLogIntegrand w ρ) 0 (r n)) atTop
    (𝓝 (circleAverage (herglotzLogIntegrand w ρ) 0 R))
n : ℕ
⊢ R * (↑n + 1) + R = R * (↑n + 2)

Tactic:
by ring

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

Goal state before:
pp: w ρ c : ℂ
R : ℝ
hρ : ρ ∈ sphere c R
hw : w ∈ ball c R
z : ℂ
⊢ z + 0 + c - ρ = z - (ρ - c)

Tactic:
by ring

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

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z
x : ℂ
hx : x ∈ U
⊢ f '' U ∈ map f (𝓝 x)

Tactic:
]

Goal state after:
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z
x : ℂ
hx : x ∈ U
⊢ f '' U ∈ map f (𝓝 x)
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: case h
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z
x : ℂ
hx : x ∈ U
this : f '' U ∈ 𝓝 (f x)
⊢ ∀ᶠ (x : ℂ) in 𝓝 (f x), -x ∉ f '' U

Tactic:
]

Goal state after:
case h
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z
x : ℂ
hx : x ∈ U
this : f '' U ∈ 𝓝 (f x)
⊢ ∀ᶠ (x : ℂ) in 𝓝 (f x), -x ∉ f '' U
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: w ρ c : ℂ
R : ℝ
hρ : ρ ∈ sphere c R
hw : w ∈ ball c R
this :
  (fun z ↦ (herglotzRieszKernel 0 (w - c) z).re * log ‖z - (ρ - c)‖) =
    Complex.re ∘ herglotzRieszKernel 0 (w - c) • fun x ↦ log ‖x - (ρ - c)‖
⊢ log ‖w - c - (ρ - c)‖ = log ‖w - ρ‖

Tactic:
]

Goal state after:
w ρ c : ℂ
R : ℝ
hρ : ρ ∈ sphere c R
hw : w ∈ ball c R
this :
  (fun z ↦ (herglotzRieszKernel 0 (w - c) z).re * log ‖z - (ρ - c)‖) =
    Complex.re ∘ herglotzRieszKernel 0 (w - c) • fun x ↦ log ‖x - (ρ - c)‖
⊢ log ‖w - c - (ρ - c)‖ = log ‖w - ρ‖
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z
x : ℂ
hx : x ∈ U
this : f '' U ∈ 𝓝 (f x)
a : ℂ
ha : a ∈ U
b : ℂ
hb : b ∈ U
hab : f b = -f a
⊢ a = (fun x ↦ x ^ 2) (-f a)

Tactic:
]

Goal state after:
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
hU₀ : 0 ∉ U
f : ℂ → ℂ
hfc : ContinuousOn f U
hf_inv : LeftInverse (fun x ↦ x ^ 2) f
hf₀ : ∀ z ∈ U, f z ≠ 0
hdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z
x : ℂ
hx : x ∈ U
this : f '' U ∈ 𝓝 (f x)
a : ℂ
ha : a ∈ U
b : ℂ
hb : b ∈ U
hab : f b = -f a
⊢ a = (fun x ↦ x ^ 2) (-f a)
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
⊢ circleAverage (∑ i ∈ ?s, fun x ↦ ↑(D i) * log ‖x - i‖) c R =
    circleAverage (∑ u ∈ h.toFinset, fun x ↦ ↑(D u) * log ‖x - u‖) c R

pp: case s
R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
⊢ Finset ℂ

pp: case h
R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
⊢ (Function.support fun u x ↦ ↑(D u) * log ‖x - u‖) ⊆ ↑?s

Tactic:
]

Goal state after:
case h
R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
⊢ (Function.support fun u x ↦ ↑(D u) * log ‖x - u‖) ⊆ ↑h.toFinset
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
⊢ ∑ i ∈ h.toFinset, circleAverage (fun x ↦ ↑(D i) * log ‖x - i‖) c R =
    ∑ i ∈ h.toFinset, circleAverage (fun x ↦ ↑(D i) * log ‖x - i‖) c R

pp: R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
⊢ ∀ i ∈ h.toFinset, CircleIntegrable (fun x ↦ ↑(D i) * log ‖x - i‖) c R

Tactic:
]

Goal state after:
R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
⊢ ∀ i ∈ h.toFinset, CircleIntegrable (fun x ↦ ↑(D i) * log ‖x - i‖) c R
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
⊢ ε < ‖f z - x‖

pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
⊢ 0 < ‖f z - x‖

Tactic:
]

Goal state after:
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
⊢ ε < ‖f z - x‖

U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
⊢ 0 < ‖f z - x‖
Module: Mathlib.Analysis.Complex.SqrtDeriv

Goal state before:
pp: z : ℂ
hz : z ∈ slitPlane
⊢ z ^ (-1 / 2) / 2 = z ^ (-1 / 2) / 2

Tactic:
]

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

Goal state before:
pp: case e_a
R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
u : ℂ
hu : u ∈ h.toFinset
⊢ log R = log R

pp: case e_a
R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
u : ℂ
hu : u ∈ h.toFinset
⊢ u ∈ closedBall c |R|

Tactic:
]

Goal state after:
case e_a
R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
u : ℂ
hu : u ∈ h.toFinset
⊢ u ∈ closedBall c |R|
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
⊢ ∑ u ∈ h.toFinset, ↑(D u) * log R = ∑ i ∈ ?s, ↑(D i) * log R

pp: case s
R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
⊢ Finset ℂ

pp: case h
R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
⊢ (Function.support fun u ↦ ↑(D u) * log R) ⊆ ↑?s

Tactic:
]

Goal state after:
case h
R : ℝ
c : ℂ
D : Function.locallyFinsuppWithin (closedBall c |R|) ℤ
h : D.support.Finite
⊢ (Function.support fun u ↦ ↑(D u) * log R) ⊆ ↑h.toFinset
Module: Mathlib.Analysis.Complex.RiemannMapping

Goal state before:
pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
hdz : DifferentiableAt ℂ f z
⊢ (0 * (f z - x) - ↑ε * deriv f z) / (f z - x) ^ 2 ≠ 0

pp: U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
hdz : DifferentiableAt ℂ f z
⊢ f z - x ≠ 0

Tactic:
]

Goal state after:
U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
hdz : DifferentiableAt ℂ f z
⊢ (0 * (f z - x) - ↑ε * deriv f z) / (f z - x) ^ 2 ≠ 0

U : Set ℂ
hUo : IsOpen U
hUc : IsSimplyConnected U
hU : U ≠ univ
f : ℂ → ℂ
hf_inj : Injective f
hfd : ¬Dense (f '' U)
hdf : ∀ z ∈ U, deriv f z ≠ 0
x : ℂ
ε : ℝ
hε₀ : 0 < ε
hε : ∀ a ∈ U, ε < dist (f a) x
hfx : ∀ z ∈ U, f z ≠ x
z : ℂ
hz : z ∈ U
hdz : DifferentiableAt ℂ f z
⊢ f z - x ≠ 0
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: case pos
c : ℂ
R : ℝ
D : ℂ → ℤ
hR : R ≠ 0
hD : Function.HasFiniteSupport D
h : c ∈ Function.support D
this : ∀ {g : ℂ → ℝ}, (Function.support fun u ↦ ↑(D u) * g u) ⊆ ↑(Finite.toFinset hD)
x : ℂ
hx : x ∈ Finite.toFinset hD ∧ x ≠ c
⊢ ↑(D x) * (log R - log ‖c - x‖) = ↑(D x) * (log R + -log ‖c - x‖)

Tactic:
]

Goal state after:
case pos
c : ℂ
R : ℝ
D : ℂ → ℤ
hR : R ≠ 0
hD : Function.HasFiniteSupport D
h : c ∈ Function.support D
this : ∀ {g : ℂ → ℝ}, (Function.support fun u ↦ ↑(D u) * g u) ⊆ ↑(Finite.toFinset hD)
x : ℂ
hx : x ∈ Finite.toFinset hD ∧ x ≠ c
⊢ ↑(D x) * (log R - log ‖c - x‖) = ↑(D x) * (log R + -log ‖c - x‖)
Module: Mathlib.Order.Monotone.Odd

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝⁵ : AddCommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedAddMonoid G
inst✝² : AddCommGroup H
inst✝¹ : PartialOrder H
inst✝ : IsOrderedAddMonoid H
f : G → H
h₁ : ∀ (x : G), f (-x) = -f x
h₂ : StrictMonoOn f (Ici 0)
x : G
hx : x ∈ Iic 0
y : G
hy : y ∈ Iic 0
hxy : x < y
⊢ f (-y) < f (-x)

Tactic:
]

Goal state after:
G : Type u_1
H : Type u_2
inst✝⁵ : AddCommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedAddMonoid G
inst✝² : AddCommGroup H
inst✝¹ : PartialOrder H
inst✝ : IsOrderedAddMonoid H
f : G → H
h₁ : ∀ (x : G), f (-x) = -f x
h₂ : StrictMonoOn f (Ici 0)
x : G
hx : x ∈ Iic 0
y : G
hy : y ∈ Iic 0
hxy : x < y
⊢ f (-y) < f (-x)
Module: Mathlib.Order.Monotone.Odd

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝⁵ : AddCommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedAddMonoid G
inst✝² : AddCommGroup H
inst✝¹ : PartialOrder H
inst✝ : IsOrderedAddMonoid H
f : G → H
h₁ : ∀ (x : G), f (-x) = -f x
h₂ : MonotoneOn f (Ici 0)
x : G
hx : x ∈ Iic 0
y : G
hy : y ∈ Iic 0
hxy : x ≤ y
⊢ f (-y) ≤ f (-x)

Tactic:
]

Goal state after:
G : Type u_1
H : Type u_2
inst✝⁵ : AddCommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedAddMonoid G
inst✝² : AddCommGroup H
inst✝¹ : PartialOrder H
inst✝ : IsOrderedAddMonoid H
f : G → H
h₁ : ∀ (x : G), f (-x) = -f x
h₂ : MonotoneOn f (Ici 0)
x : G
hx : x ∈ Iic 0
y : G
hy : y ∈ Iic 0
hxy : x ≤ y
⊢ f (-y) ≤ f (-x)
Module: Mathlib.Analysis.Complex.JensenFormula

Goal state before:
pp: case neg
c : ℂ
R : ℝ
D : ℂ → ℤ
hR : R ≠ 0
hD : Function.HasFiniteSupport D
h : D c = 0
x : ℂ
h₁ : ¬c = x
⊢ ↑(D x) * (log R - log ‖c - x‖) = ↑(D x) * (log R + -log ‖c - x‖)

Tactic:
]

Goal state after:
case neg
c : ℂ
R : ℝ
D : ℂ → ℤ
hR : R ≠ 0
hD : Function.HasFiniteSupport D
h : D c = 0
x : ℂ
h₁ : ¬c = x
⊢ ↑(D x) * (log R - log ‖c - x‖) = ↑(D x) * (log R + -log ‖c - x‖)
Module: Mathlib.Order.Monotone.Union

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
a : α
f : α → β
h₁ : StrictMonoOn f (Iic a)
h₂ : StrictMonoOn f (Ici a)
⊢ StrictMonoOn f (Iic a ∪ Ici a)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
a : α
f : α → β
h₁ : StrictMonoOn f (Iic a)
h₂ : StrictMonoOn f (Ici a)
⊢ StrictMonoOn f (Iic a ∪ Ici a)