mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: case hbc
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ ∫ (x : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), ‖1 - charFun (μ n) (x • z)‖ ∂volume ≤ ∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 2

Tactic:
intervalIntegral.integral_of_le hr'

Goal state after:
case hbc
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ ∫ (x : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), ‖1 - charFun (μ n) (x • z)‖ ∂volume ≤
    ∫ (x : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), 2 ∂volume
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: case hbc
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ ∫ (x : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), ‖1 - charFun (μ n) (x • z)‖ ∂volume ≤
    ∫ (x : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), 2 ∂volume

Tactic:
]

Goal state after:
case hbc
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
n : ℕ
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ ∫ (x : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), ‖1 - charFun (μ n) (x • z)‖ ∂volume ≤
    ∫ (x : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), 2 ∂volume
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
y : E
this : ProperSpace E
hy : ¬y = 0
h' : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r * ‖y‖⁻¹ < ‖x‖}) atTop (𝓝 0)
r : ℝ
μ : Measure E
x : E
hx : r < ‖⟪y, x⟫_𝕜‖
⊢ r < ‖x‖ * ‖y‖

pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
y : E
this : ProperSpace E
hy : ¬y = 0
h' : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r * ‖y‖⁻¹ < ‖x‖}) atTop (𝓝 0)
r : ℝ
μ : Measure E
x : E
hx : r < ‖⟪y, x⟫_𝕜‖
⊢ 0 < ‖y‖

Tactic:
]

Goal state after:
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
y : E
this : ProperSpace E
hy : ¬y = 0
h' : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r * ‖y‖⁻¹ < ‖x‖}) atTop (𝓝 0)
r : ℝ
μ : Measure E
x : E
hx : r < ‖⟪y, x⟫_𝕜‖
⊢ r < ‖x‖ * ‖y‖

E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
y : E
this : ProperSpace E
hy : ¬y = 0
h' : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r * ‖y‖⁻¹ < ‖x‖}) atTop (𝓝 0)
r : ℝ
μ : Measure E
x : E
hx : r < ‖⟪y, x⟫_𝕜‖
⊢ 0 < ‖y‖
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
y : E
this : ProperSpace E
hy : ¬y = 0
h' : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r * ‖y‖⁻¹ < ‖x‖}) atTop (𝓝 0)
r : ℝ
μ : Measure E
x : E
hx : r < ‖⟪y, x⟫_𝕜‖
⊢ r < ‖y‖ * ‖x‖

Tactic:
]

Goal state after:
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
y : E
this : ProperSpace E
hy : ¬y = 0
h' : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r * ‖y‖⁻¹ < ‖x‖}) atTop (𝓝 0)
r : ℝ
μ : Measure E
x : E
hx : r < ‖⟪y, x⟫_𝕜‖
⊢ r < ‖y‖ * ‖x‖
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
z : E
n : ℕ
h_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {Measure.map (fun x ↦ ⟪z, x⟫_𝕜) (μ n)}, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
⊢ Tendsto (fun r ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop (𝓝 0)

Tactic:
]

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
z : E
n : ℕ
h_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {Measure.map (fun x ↦ ⟪z, x⟫_𝕜) (μ n)}, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
⊢ Tendsto (fun r ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
z : E
n : ℕ
h_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {Measure.map (fun x ↦ ⟪z, x⟫_𝕜) (μ n)}, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
r : ℝ
⊢ (μ n) ((fun x ↦ ⟪z, x⟫_𝕜) ⁻¹' {x | r < ‖x‖}) = (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}

pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
z : E
n : ℕ
h_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {Measure.map (fun x ↦ ⟪z, x⟫_𝕜) (μ n)}, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
r : ℝ
⊢ MeasurableSet {x | r < ‖x‖}

Tactic:
]

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
z : E
n : ℕ
h_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {Measure.map (fun x ↦ ⟪z, x⟫_𝕜) (μ n)}, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
r : ℝ
⊢ (μ n) ((fun x ↦ ⟪z, x⟫_𝕜) ⁻¹' {x | r < ‖x‖}) = (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}

E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
z : E
n : ℕ
h_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {Measure.map (fun x ↦ ⟪z, x⟫_𝕜) (μ n)}, μ_1 {x | r < ‖x‖}) atTop (𝓝 0)
r : ℝ
⊢ MeasurableSet {x | r < ‖x‖}
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖⟪y, x⟫_𝕜‖}) atTop (𝓝 0)
z : E
⊢ Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop) atTop (𝓝 0)

Tactic:
]

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), Tendsto (fun r ↦ ⨆ μ_1 ∈ Set.range μ, μ_1 {x | r < ‖⟪y, x⟫_𝕜‖}) atTop (𝓝 0)
z : E
⊢ Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : ¬y = 0
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : ¬y = 0
h' : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | ‖y‖⁻¹ * r < ‖⟪(↑‖y‖)⁻¹ • y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
r : ℝ
n : ℕ
x : E
⊢ r < ‖⟪y, x⟫_𝕜‖ ↔ ‖y‖⁻¹ * r < ‖(starRingEnd 𝕜) (↑‖y‖)⁻¹ * ⟪y, x⟫_𝕜‖

Tactic:
]

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : ¬y = 0
h' : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | ‖y‖⁻¹ * r < ‖⟪(↑‖y‖)⁻¹ • y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
r : ℝ
n : ℕ
x : E
⊢ r < ‖⟪y, x⟫_𝕜‖ ↔ ‖y‖⁻¹ * r < ‖(starRingEnd 𝕜) (↑‖y‖)⁻¹ * ⟪y, x⟫_𝕜‖
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : ¬y = 0
h' : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | ‖y‖⁻¹ * r < ‖⟪(↑‖y‖)⁻¹ • y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
r : ℝ
n : ℕ
x : E
⊢ r < ‖⟪y, x⟫_𝕜‖ ↔ ‖y‖⁻¹ * r < ‖⟪(↑‖y‖)⁻¹ • y, x⟫_𝕜‖

Tactic:
rw [inner_smul_left]

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : ¬y = 0
h' : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | ‖y‖⁻¹ * r < ‖⟪(↑‖y‖)⁻¹ • y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
r : ℝ
n : ℕ
x : E
⊢ r < ‖⟪y, x⟫_𝕜‖ ↔ ‖y‖⁻¹ * r < ‖(starRingEnd 𝕜) (↑‖y‖)⁻¹ * ⟪y, x⟫_𝕜‖
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
r : ℝ
hr : 0 < r
⊢ 0 ≤ 2 * r⁻¹

Tactic:
]

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
r : ℝ
hr : 0 < r
⊢ 0 ≤ 2 * r⁻¹
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : ¬y = 0
h' : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | ‖y‖⁻¹ * r < ‖⟪(↑‖y‖)⁻¹ • y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
r : ℝ
n : ℕ
x : E
⊢ r < ‖⟪y, x⟫_𝕜‖ ↔ r < ‖⟪y, x⟫_𝕜‖

pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : ¬y = 0
h' : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | ‖y‖⁻¹ * r < ‖⟪(↑‖y‖)⁻¹ • y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
r : ℝ
n : ℕ
x : E
⊢ 0 < ‖y‖⁻¹

Tactic:
]

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : ¬y = 0
h' : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | ‖y‖⁻¹ * r < ‖⟪(↑‖y‖)⁻¹ • y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
r : ℝ
n : ℕ
x : E
⊢ 0 < ‖y‖⁻¹
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ Tendsto (fun n ↦ ∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - charFun (μ n) (t • z)) atTop
    (𝓝 (∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - f (t • z)))

Tactic:
intervalIntegral.integral_of_le hr'

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ Tendsto (fun n ↦ ∫ (t : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), 1 - charFun (μ n) (t • z) ∂volume) atTop
    (𝓝 (∫ (t : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), 1 - f (t • z) ∂volume))
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n).real {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
C : ℝ≥0
hμ : ∀ᶠ (n : ℕ) in atTop, (μ n) Set.univ ≤ ↑C
z : E
hz : ‖z‖ = 1
r : ℝ
⊢ limsup (fun n ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop = limsup (fun n ↦ (μ n) {x | r < ‖⟪z, x⟫_𝕜‖}) atTop

pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n).real {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
C : ℝ≥0
hμ : ∀ᶠ (n : ℕ) in atTop, (μ n) Set.univ ≤ ↑C
z : E
hz : ‖z‖ = 1
r : ℝ
⊢ ∀ᶠ (a : ℕ) in atTop, (μ a) {x | r < ‖⟪z, x⟫_𝕜‖} ≤ ↑C

Tactic:
]

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n).real {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
C : ℝ≥0
hμ : ∀ᶠ (n : ℕ) in atTop, (μ n) Set.univ ≤ ↑C
z : E
hz : ‖z‖ = 1
r : ℝ
⊢ ∀ᶠ (a : ℕ) in atTop, (μ a) {x | r < ‖⟪z, x⟫_𝕜‖} ≤ ↑C
Module: Mathlib.MeasureTheory.Measure.MeasuredSets

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s t : MeasuredSets μ
⊢ μ ↑s - μ ↑t ≤ edist s t

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s t : MeasuredSets μ
⊢ μ ↑s - μ ↑t ≤ edist s t
Module: Mathlib.MeasureTheory.Measure.MeasuredSets

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s t : MeasuredSets μ
⊢ μ.real ↑s - μ.real ↑t ≤ (μ ↑s - μ ↑t).toReal

pp: case hb
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s t : MeasuredSets μ
⊢ edist s t ≠ ∞

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s t : MeasuredSets μ
⊢ μ.real ↑s - μ.real ↑t ≤ (μ ↑s - μ ↑t).toReal

case hb
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s t : MeasuredSets μ
⊢ edist s t ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
g : E →ᵇ ℝ
ε : ℝ
εpos : ε > 0
I1 : ∀ᶠ (n : ℕ) in atTop, dist (∫ (x : E), g x ∂∑ i ∈ Finset.range (n + 1), ↑(ν i)) (∫ (x : E), g x ∂↑μ) < ε / 3
I2 : ∀ᶠ (n : ℕ) in atTop, ‖g‖ * ↑(u n) < ε / 3
n : ℕ
hn : dist (∫ (x : E), g x ∂∑ i ∈ Finset.range (n + 1), ↑(ν i)) (∫ (x : E), g x ∂↑μ) < ε / 3
h'n : ‖g‖ * ↑(u n) < ε / 3
this :
  Tendsto (fun ρ ↦ ∫ (x : E), g x ∂↑(∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i))) (↑f)
    (𝓝 (∫ (x : E), g x ∂↑(∑ i ∈ Finset.range (n + 1), ν i)))
ρ : FiniteMeasure E
hρ :
  dist (∫ (x : E), g x ∂↑(∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i)))
      (∫ (x : E), g x ∂↑(∑ i ∈ Finset.range (n + 1), ν i)) <
    ε / 3
h'ρ : ρ.mass ≤ C ∧ ∀ (n : ℕ), ρ (K n)ᶜ ≤ u n
⊢ ε / 3 + ε / 3 + ε / 3 = ε

Tactic:
by ring

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

Goal state before:
pp: case refine_1
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
⊢ ((Measure.sum fun n ↦ ↑(ν n)) univ).toNNReal ≤ (↑C).toNNReal

Tactic:
]

Goal state after:
case refine_1
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
⊢ ((Measure.sum fun n ↦ ↑(ν n)) univ).toNNReal ≤ (↑C).toNNReal
Module: Mathlib.MeasureTheory.Measure.MeasuredSets

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s✝ : Set α
hs✝ : MeasurableSet s✝
ε✝ : ℝ≥0∞
hε : 0 < ε✝
s : Set α
hs : MeasurableSet s
h's : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ t ∈ C, μ (t ∆ s) < ε
ε : ℝ≥0∞
εpos : 0 < ε
t : Set α
tC : t ∈ C
ht : μ (t ∆ s) < ε / 2
D : Set (Set α)
D_count : D.Countable
DC : D ⊆ C
Dne : D.Nonempty
f : ℕ → Set α
hD : μ (⋂ n, (accumulate f n)ᶜ) = 0
hf : D = range f
fC : ∀ (n : ℕ), accumulate f n ∈ C
this : ⋃₀ D = ⋃ n, accumulate f n
⊢ Tendsto (fun n ↦ μ (accumulate f n)ᶜ) atTop (𝓝 0)

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetRing C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s✝ : Set α
hs✝ : MeasurableSet s✝
ε✝ : ℝ≥0∞
hε : 0 < ε✝
s : Set α
hs : MeasurableSet s
h's : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ t ∈ C, μ (t ∆ s) < ε
ε : ℝ≥0∞
εpos : 0 < ε
t : Set α
tC : t ∈ C
ht : μ (t ∆ s) < ε / 2
D : Set (Set α)
D_count : D.Countable
DC : D ⊆ C
Dne : D.Nonempty
f : ℕ → Set α
hD : μ (⋂ n, (accumulate f n)ᶜ) = 0
hf : D = range f
fC : ∀ (n : ℕ), accumulate f n ∈ C
this : ⋃₀ D = ⋃ n, accumulate f n
⊢ Tendsto (fun n ↦ μ (accumulate f n)ᶜ) atTop (𝓝 0)