mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
F' : ℕ → α → ε
f' : α → ε
bound' : α → ℝ≥0∞
bound_hasFiniteIntegral : HasFiniteIntegral bound' μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F' n a‖ₑ ≤ bound' a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F' n a) atTop (𝓝 (f' a))
⊢ ∫⁻ (a : α), ‖f' a‖ₑ ∂μ < ∞

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
F' : ℕ → α → ε
f' : α → ε
bound' : α → ℝ≥0∞
bound_hasFiniteIntegral : HasFiniteIntegral bound' μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F' n a‖ₑ ≤ bound' a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F' n a) atTop (𝓝 (f' a))
⊢ ∫⁻ (a : α), ‖f' a‖ₑ ∂μ < ∞
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: case mpr
x y : ℝ
h : 0 ≤ y
⊢ |x| ≤ √y → |x| ^ 2 ≤ y

Tactic:
]

Goal state after:
case mpr
x y : ℝ
h : 0 ≤ y
⊢ |x| ≤ √y → |x| ^ 2 ≤ y
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
⊢ ∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ < ∞

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
⊢ ∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ < ∞
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
⊢ HasFiniteIntegral bound μ

pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
⊢ 0 ≤ᵐ[μ] bound

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
⊢ HasFiniteIntegral bound μ

α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
⊢ 0 ≤ᵐ[μ] bound
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
⊢ x ≤ 0 ↔ x ≤ 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
h : 0 ≤ x
⊢ x = 0 ↔ x = 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
⊢ 1 ≤ x ↔ 1 ≤ x

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
⊢ x ≤ 1 ↔ x ≤ 1

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
⊢ 0 ≤ x ∧ 0 ≤ x * (x - 1) ↔ x = 0 ∨ 1 ≤ x

Tactic:
]

Goal state after:
x : ℝ
⊢ 0 ≤ x ∧ 0 ≤ x * (x - 1) ↔ x = 0 ∨ 1 ≤ x
Module: Mathlib.Analysis.Real.Sqrt

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

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: ⊢ √1 < √2

Tactic:
]

Goal state after:
⊢ √1 < √2
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: ⊢ √2 ^ 2 < (3 / 2) ^ 2

Tactic:
]

Goal state after:
⊢ √2 ^ 2 < (3 / 2) ^ 2
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
hg : AEMeasurable g (μ.withDensity f)
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (a : α) in s, (f * g) a ∂μ = ∫⁻ (a : α) in s, (f * g) a ∂μ

pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
hg : AEMeasurable g (μ.withDensity f)
s : Set α
hs : MeasurableSet s
⊢ AEMeasurable g ((μ.restrict s).withDensity f)

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
hg : AEMeasurable g (μ.withDensity f)
s : Set α
hs : MeasurableSet s
⊢ AEMeasurable g ((μ.restrict s).withDensity f)
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
hy : 0 ≤ y
⊢ √x * √y = √x * √y

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
hg : AEMeasurable g (μ.withDensity f)
s : Set α
hs : MeasurableSet s
⊢ AEMeasurable g ((μ.withDensity f).restrict s)

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
hg : AEMeasurable g (μ.withDensity f)
s : Set α
hs : MeasurableSet s
⊢ AEMeasurable g ((μ.withDensity f).restrict s)
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
g : α → ℝ≥0∞
⊢ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ g), ∫⁻ (a : α), g_1 a ∂μ.withDensity f ≤
    ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
g : α → ℝ≥0∞
⊢ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ g), ∫⁻ (a : α), g_1 a ∂μ.withDensity f ≤
    ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ g
⊢ ∫⁻ (a : α), (f * i) a ∂μ ≤ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ g
⊢ ∫⁻ (a : α), (f * i) a ∂μ ≤ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ g
⊢ f * g ≤ f * g

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
⊢ (↑(NNReal.sqrt x.toNNReal))⁻¹ = (↑(NNReal.sqrt x.toNNReal))⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
hx : 0 ≤ x
y : ℝ
⊢ √x * (√y)⁻¹ = √x * (√y)⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
hy : 0 ≤ y
⊢ √(x * y⁻¹) = √x / √y

Tactic:
sqrt_mul' x (inv_nonneg.2 hy),

Goal state after:
x y : ℝ
hy : 0 ≤ y
⊢ √x * √y⁻¹ = √x / √y
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
hy : 0 ≤ y
⊢ √x * (√y)⁻¹ = √x * (√y)⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
⊢ √x / x = √x / x

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
⊢ (√x)⁻¹ = (√x)⁻¹

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
hx : 0 ≤ x
⊢ x < √y ↔ x < √y

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
b : α → ℝ≥0∞ := fun a ↦ 2 * ENNReal.ofReal (bound a)
n : ℕ
a : α
h₁ : ENNReal.ofReal ‖F n a‖ ≤ ENNReal.ofReal (bound a)
h₂ : ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)
⊢ 2 * ENNReal.ofReal (bound a) = b a

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
⊢ |x| ^ 2 < y ↔ |x| ^ 2 < y

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → ℝ≥0∞
⊢ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ ≤
    ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ g), ∫⁻ (a : α), g_1 a ∂μ.withDensity f

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → ℝ≥0∞
⊢ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ ≤
    ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ g), ∫⁻ (a : α), g_1 a ∂μ.withDensity f
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
h : √x < y
hy : 0 < y
⊢ √x < y

Tactic:
]

Goal state after:
x y : ℝ
h : √x < y
hy : 0 < y
⊢ √x < y
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: a : ℕ
⊢ ↑a.sqrt ^ 2 ≤ ↑a

Tactic:
]

Goal state after:
a : ℕ
⊢ ↑a.sqrt ^ 2 ≤ ↑a
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: a : ℕ
⊢ ↑a < (↑a.sqrt + 1) ^ 2

pp: a : ℕ
⊢ 0 ≤ ↑a.sqrt + 1

Tactic:
]

Goal state after:
a : ℕ
⊢ ↑a < (↑a.sqrt + 1) ^ 2

a : ℕ
⊢ 0 ≤ ↑a.sqrt + 1
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
b : α → ℝ≥0∞ := fun a ↦ 2 * ENNReal.ofReal (bound a)
hb : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 (ENNReal.ofReal 0))

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
b : α → ℝ≥0∞ := fun a ↦ 2 * ENNReal.ofReal (bound a)
hb : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 (ENNReal.ofReal 0))
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: a : ℕ
⊢ ↑↑a.sqrt ≤ √↑a ∧ √↑a < ↑↑a.sqrt + 1

Tactic:
]

Goal state after:
a : ℕ
⊢ ↑↑a.sqrt ≤ √↑a ∧ √↑a < ↑↑a.sqrt + 1
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: a : ℕ
⊢ ↑a.sqrt ≤ √↑a ∧ √↑a < ↑a.sqrt + 1

Tactic:
]

Goal state after:
a : ℕ
⊢ ↑a.sqrt ≤ √↑a ∧ √↑a < ↑a.sqrt + 1
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ f * g
x : α
⊢ i x / f x ≤ g x

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ f * g
x : α
⊢ i x / f x ≤ g x
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
h : -1 ≤ x
⊢ 1 + x + (x / 2) ^ 2 = (1 + x / 2) ^ 2

Tactic:
by ring

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
b : α → ℝ≥0∞ := fun a ↦ 2 * ENNReal.ofReal (bound a)
hb : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a
a : α
h : Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
⊢ Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
b : α → ℝ≥0∞ := fun a ↦ 2 * ENNReal.ofReal (bound a)
hb : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a
a : α
h : Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
⊢ Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ f * g
A : (fun x ↦ (f x)⁻¹ * i x) ≤ g
⊢ ∫⁻ (a : α), i a ∂μ ≤ ∫⁻ (a : α), (f * fun i_1 ↦ (f i_1)⁻¹ * i i_1) a ∂μ

Tactic:
]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ f * g
A : (fun x ↦ (f x)⁻¹ * i x) ≤ g
⊢ ∫⁻ (a : α), i a ∂μ ≤ ∫⁻ (a : α), (f * fun i_1 ↦ (f i_1)⁻¹ * i i_1) a ∂μ
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
b : α → ℝ≥0∞ := fun a ↦ 2 * ENNReal.ofReal (bound a)
hb : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a
h : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0)
this : Tendsto (fun n ↦ ∫⁻ (a : α), ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 0)
⊢ Tendsto (fun n ↦ ∫⁻ (a : α), ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 0)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
b : α → ℝ≥0∞ := fun a ↦ 2 * ENNReal.ofReal (bound a)
hb : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a
h : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0)
this : Tendsto (fun n ↦ ∫⁻ (a : α), ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 0)
⊢ Tendsto (fun n ↦ ∫⁻ (a : α), ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 0)
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: z : ℂ
⊢ z.re * z.re ≤ normSq z

Tactic:
]

Goal state after:
z : ℂ
⊢ z.re * z.re ≤ normSq z
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: z w : ℂ
⊢ normSq z + normSq w + 2 * (z * (starRingEnd ℂ) w).re ≤ normSq z + normSq w + 2 * √(normSq z * normSq w)

Tactic:
← normSq_conj w,

Goal state after:
z w : ℂ
⊢ normSq z + normSq ((starRingEnd ℂ) w) + 2 * (z * (starRingEnd ℂ) w).re ≤
    normSq z + normSq ((starRingEnd ℂ) w) + 2 * √(normSq z * normSq ((starRingEnd ℂ) w))
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: z w : ℂ
⊢ normSq z + normSq ((starRingEnd ℂ) w) + 2 * (z * (starRingEnd ℂ) w).re ≤
    normSq z + normSq ((starRingEnd ℂ) w) + 2 * √(normSq (z * (starRingEnd ℂ) w))

Tactic:
]

Goal state after:
z w : ℂ
⊢ normSq z + normSq ((starRingEnd ℂ) w) + 2 * (z * (starRingEnd ℂ) w).re ≤
    normSq z + normSq ((starRingEnd ℂ) w) + 2 * √(normSq (z * (starRingEnd ℂ) w))
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: z : ℂ
⊢ √(normSq z) = √(normSq z)

Tactic:
]

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

Goal state before:
pp: z w : ℂ
⊢ √(normSq z) * √(normSq w) = √(normSq z) * √(normSq w)

Tactic:
]

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

Goal state before:
pp: z w : ℂ
⊢ √(normSq z) / √(normSq w) = √(normSq z) / √(normSq w)

Tactic:
]

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

Goal state before:
pp: n : ℤ
⊢ |↑n| = |↑n|

Tactic:
]

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

Goal state before:
pp: n : ℤ
hn : 0 ≤ n
⊢ ↑n = ↑n

Tactic:
]

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

Goal state before:
pp: x : ℂ
⊢ 1 < ‖x‖ ^ 2 ↔ 1 < ‖x‖ ^ 2

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ f * g
A : (fun x ↦ (f x)⁻¹ * i x) ≤ g
x : α
h'x : f x < ∞
hx : f x ≠ 0
⊢ i x = i x

Tactic:
]

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

Goal state before:
pp: x : ℂ
⊢ 1 ≤ ‖x‖ ^ 2 ↔ 1 ≤ ‖x‖ ^ 2

Tactic:
]

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

Goal state before:
pp: z : ℂ
⊢ z.im ^ 2 = z.im ^ 2

Tactic:
]

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

Goal state before:
pp: z : ℂ
⊢ z.re ^ 2 = z.re ^ 2

Tactic:
]

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

Goal state before:
pp: x y : ℝ
⊢ ‖↑x + ↑y * I‖ = √(normSq (↑x + ↑y * I))

Tactic:
]

Goal state after:
x y : ℝ
⊢ ‖↑x + ↑y * I‖ = √(normSq (↑x + ↑y * I))
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : ∫⁻ (a : α), ENNReal.ofReal (bound a) ∂μ < ∞
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
b : α → ℝ≥0∞ := fun a ↦ 2 * ENNReal.ofReal (bound a)
hb : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a
h : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0)
⊢ ∫⁻ (a : α), b a ∂μ ≠ ∞

pp: case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
b : α → ℝ≥0∞ := fun a ↦ 2 * ENNReal.ofReal (bound a)
hb : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a
h : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0)
⊢ 0 ≤ᵐ[μ] bound

Tactic:
]

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : ∫⁻ (a : α), ENNReal.ofReal (bound a) ∂μ < ∞
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
b : α → ℝ≥0∞ := fun a ↦ 2 * ENNReal.ofReal (bound a)
hb : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a
h : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0)
⊢ ∫⁻ (a : α), b a ∂μ ≠ ∞

case refine_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
F : ℕ → α → β
f : α → β
bound : α → ℝ
F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ
bound_hasFiniteIntegral : HasFiniteIntegral bound μ
h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) atTop (𝓝 (f a))
f_measurable : AEStronglyMeasurable f μ
b : α → ℝ≥0∞ := fun a ↦ 2 * ENNReal.ofReal (bound a)
hb : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a
h : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0)
⊢ 0 ≤ᵐ[μ] bound
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : ∀ᵐ (x : α) ∂μ.restrict s, f x < ∞
⊢ ∫⁻ (a : α) in s, (f * g) a ∂μ = ∫⁻ (a : α) in s, (f * g) a ∂μ

Tactic:
]

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

Goal state before:
pp: z : ℂ
⊢ √(z.re * z.re + z.im * z.im) = √(z.re * z.re + z.im * z.im)

Tactic:
]

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

Goal state before:
pp: x : ℝ
hx : x ∈ Ici 0
⊢ x = x

Tactic:
]

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

Goal state before:
pp: z : ℂ
⊢ z.im ^ 2 ≤ z.re ^ 2 + z.im ^ 2

Tactic:
]

Goal state after:
z : ℂ
⊢ z.im ^ 2 ≤ z.re ^ 2 + z.im ^ 2
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: z : ℂ
⊢ z.im ≠ 0 ↔ z.im ≠ 0

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h'f : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
⊢ ∫⁻ (a : α), g a ∂μ.withDensity (AEMeasurable.mk f hf) = ∫⁻ (a : α), g a ∂μ.withDensity f'

Tactic:
]

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

Goal state before:
pp: case h₁
x y : ℝ
m : ℝ := max |x| |y|
hm₀ : 0 ≤ m
⊢ |x| ≤ m

Tactic:
]

Goal state after:
case h₁
x y : ℝ
m : ℝ := max |x| |y|
hm₀ : 0 ≤ m
⊢ |x| ≤ m
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: case h₂
x y : ℝ
m : ℝ := max |x| |y|
hm₀ : 0 ≤ m
⊢ |y| ≤ m

Tactic:
]

Goal state after:
case h₂
x y : ℝ
m : ℝ := max |x| |y|
hm₀ : 0 ≤ m
⊢ |y| ≤ m
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: x y : ℝ
m : ℝ := max |x| |y|
hm₀ : 0 ≤ m
⊢ √2 * m = √2 * m

pp: x y : ℝ
m : ℝ := max |x| |y|
hm₀ : 0 ≤ m
⊢ 0 ≤ m

pp: case hx
x y : ℝ
m : ℝ := max |x| |y|
hm₀ : 0 ≤ m
⊢ 0 ≤ 2

Tactic:
]

Goal state after:
x y : ℝ
m : ℝ := max |x| |y|
hm₀ : 0 ≤ m
⊢ 0 ≤ m

case hx
x y : ℝ
m : ℝ := max |x| |y|
hm₀ : 0 ≤ m
⊢ 0 ≤ 2