mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: case iSup
Ω : Type u_1
Mf mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
c : ℝ≥0∞
T : Set Ω
h_meas_T : MeasurableSet T
h_ind : IndepSets {s | MeasurableSet s} {T} μ
h_mul_indicator : ∀ (g : Ω → ℝ≥0∞), Measurable g → Measurable fun a ↦ g a * T.indicator (fun x ↦ c) a
f : ℕ → Ω → ℝ≥0∞
h_meas_f : ∀ (n : ℕ), Measurable (f n)
h_mono_f : Monotone f
h_ind_f :
  ∀ (n : ℕ),
    ∫⁻ (ω : Ω), f n ω * T.indicator (fun x ↦ c) ω ∂μ = (∫⁻ (ω : Ω), f n ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ
h_measM_f : ∀ (n : ℕ), Measurable (f n)
⊢ ∫⁻ (ω : Ω), (⨆ n, f n ω) * T.indicator (fun x ↦ c) ω ∂μ =
    (∫⁻ (ω : Ω), ⨆ n, f n ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ

Tactic:
[

Goal state after:
case iSup
Ω : Type u_1
Mf mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
c : ℝ≥0∞
T : Set Ω
h_meas_T : MeasurableSet T
h_ind : IndepSets {s | MeasurableSet s} {T} μ
h_mul_indicator : ∀ (g : Ω → ℝ≥0∞), Measurable g → Measurable fun a ↦ g a * T.indicator (fun x ↦ c) a
f : ℕ → Ω → ℝ≥0∞
h_meas_f : ∀ (n : ℕ), Measurable (f n)
h_mono_f : Monotone f
h_ind_f :
  ∀ (n : ℕ),
    ∫⁻ (ω : Ω), f n ω * T.indicator (fun x ↦ c) ω ∂μ = (∫⁻ (ω : Ω), f n ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ
h_measM_f : ∀ (n : ℕ), Measurable (f n)
⊢ ∫⁻ (ω : Ω), (⨆ n, f n ω) * T.indicator (fun x ↦ c) ω ∂μ =
    (∫⁻ (ω : Ω), ⨆ n, f n ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: case iSup
Ω : Type u_1
Mf mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
c : ℝ≥0∞
T : Set Ω
h_meas_T : MeasurableSet T
h_ind : IndepSets {s | MeasurableSet s} {T} μ
h_mul_indicator : ∀ (g : Ω → ℝ≥0∞), Measurable g → Measurable fun a ↦ g a * T.indicator (fun x ↦ c) a
f : ℕ → Ω → ℝ≥0∞
h_meas_f : ∀ (n : ℕ), Measurable (f n)
h_mono_f : Monotone f
h_ind_f :
  ∀ (n : ℕ),
    ∫⁻ (ω : Ω), f n ω * T.indicator (fun x ↦ c) ω ∂μ = (∫⁻ (ω : Ω), f n ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ
h_measM_f : ∀ (n : ℕ), Measurable (f n)
⊢ ⨆ n, ∫⁻ (a : Ω), f n a * T.indicator (fun x ↦ c) a ∂μ =
    ⨆ i, (∫⁻ (a : Ω), f i a ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ

Tactic:
[

Goal state after:
case iSup
Ω : Type u_1
Mf mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
c : ℝ≥0∞
T : Set Ω
h_meas_T : MeasurableSet T
h_ind : IndepSets {s | MeasurableSet s} {T} μ
h_mul_indicator : ∀ (g : Ω → ℝ≥0∞), Measurable g → Measurable fun a ↦ g a * T.indicator (fun x ↦ c) a
f : ℕ → Ω → ℝ≥0∞
h_meas_f : ∀ (n : ℕ), Measurable (f n)
h_mono_f : Monotone f
h_ind_f :
  ∀ (n : ℕ),
    ∫⁻ (ω : Ω), f n ω * T.indicator (fun x ↦ c) ω ∂μ = (∫⁻ (ω : Ω), f n ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ
h_measM_f : ∀ (n : ℕ), Measurable (f n)
⊢ ⨆ n, ∫⁻ (a : Ω), f n a * T.indicator (fun x ↦ c) a ∂μ =
    ⨆ i, (∫⁻ (a : Ω), f i a ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x ↦ c) ω ∂μ
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ ∫ (a : Ω), X a * Y a ∂μ - ∫ (a : Ω), X a * ∫ (x : Ω), Y x ∂μ ∂μ -
      (∫ (a : Ω), (∫ (x : Ω), X x ∂μ) * Y a ∂μ - ∫ (a : Ω), (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ ∂μ) =
    ∫ (x : Ω), (X * Y) x ∂μ - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ ∫ (a : Ω), X a * Y a ∂μ - ∫ (a : Ω), X a * ∫ (x : Ω), Y x ∂μ ∂μ -
      (∫ (a : Ω), (∫ (x : Ω), X x ∂μ) * Y a ∂μ - ∫ (a : Ω), (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ ∂μ) =
    ∫ (x : Ω), (X * Y) x ∂μ - (∫ (x : Ω), X x ∂μ) * ∫ (x : Ω), Y x ∂μ
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
h : X =ᵐ[μ] Y
⊢ eVar[X; μ] = eVar[Y; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
h : X =ᵐ[μ] Y
⊢ eVar[X; μ] = eVar[Y; μ]
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
h : X =ᵐ[μ] Y
⊢ Var[X; μ] = Var[Y; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
h : X =ᵐ[μ] Y
⊢ Var[X; μ] = Var[Y; μ]
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
this : ∫⁻ (x : Ω), ‖X x - ∫ (x : Ω), X x ∂μ‖ₑ ^ 2 ∂μ < ∞
⊢ eVar[X; μ] < ∞

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
this : ∫⁻ (x : Ω), ‖X x - ∫ (x : Ω), X x ∂μ‖ₑ ^ 2 ∂μ < ∞
⊢ eVar[X; μ] < ∞
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: case hfi
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
h : ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) ^ 2 ∂μ = 0
⊢ Integrable (fun ω ↦ (X ω - ∫ (x : Ω), X x ∂μ) ^ 2) μ

Tactic:
[

Goal state after:
case hfi
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
h : ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) ^ 2 ∂μ = 0
⊢ Integrable (fun ω ↦ (X ω - ∫ (x : Ω), X x ∂μ) ^ 2) μ
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: case add
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
f' g : Ω → ℝ≥0∞
a✝¹ : Disjoint (Function.support f') (Function.support g)
h_measMg_f' : Measurable f'
a✝ : Measurable g
h_ind_f' : ∫⁻ (ω : Ω), f ω * f' ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' ω ∂μ
h_ind_g' : ∫⁻ (ω : Ω), f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ
h_measM_f' : Measurable f'
⊢ ∫⁻ (ω : Ω), f ω * (f' + g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), (f' + g) ω ∂μ

Tactic:
[

Goal state after:
case add
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
f' g : Ω → ℝ≥0∞
a✝¹ : Disjoint (Function.support f') (Function.support g)
h_measMg_f' : Measurable f'
a✝ : Measurable g
h_ind_f' : ∫⁻ (ω : Ω), f ω * f' ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' ω ∂μ
h_ind_g' : ∫⁻ (ω : Ω), f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ
h_measM_f' : Measurable f'
⊢ ∫⁻ (ω : Ω), f ω * (f' + g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), (f' + g) ω ∂μ
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
⊢ cov[X, Y; μ] = cov[Y, X; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
⊢ cov[X, Y; μ] = cov[Y, X; μ]
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : Integrable X μ
c : ℝ
⊢ cov[fun ω ↦ X ω + c, Y; μ] = cov[X, Y; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : Integrable X μ
c : ℝ
⊢ cov[fun ω ↦ X ω + c, Y; μ] = cov[X, Y; μ]
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: case iSup
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
f' : ℕ → Ω → ℝ≥0∞
h_meas_f' : ∀ (n : ℕ), Measurable (f' n)
h_mono_f' : Monotone f'
h_ind_f' : ∀ (n : ℕ), ∫⁻ (ω : Ω), f ω * f' n ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' n ω ∂μ
h_measM_f' : ∀ (n : ℕ), Measurable (f' n)
⊢ ∫⁻ (ω : Ω), f ω * ⨆ n, f' n ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), ⨆ n, f' n ω ∂μ

Tactic:
[

Goal state after:
case iSup
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
f' : ℕ → Ω → ℝ≥0∞
h_meas_f' : ∀ (n : ℕ), Measurable (f' n)
h_mono_f' : Monotone f'
h_ind_f' : ∀ (n : ℕ), ∫⁻ (ω : Ω), f ω * f' n ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' n ω ∂μ
h_measM_f' : ∀ (n : ℕ), Measurable (f' n)
⊢ ∫⁻ (ω : Ω), f ω * ⨆ n, f' n ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), ⨆ n, f' n ω ∂μ
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : Integrable X μ
c : ℝ
⊢ cov[fun ω ↦ c + X ω, Y; μ] = cov[X, Y; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : Integrable X μ
c : ℝ
⊢ cov[fun ω ↦ c + X ω, Y; μ] = cov[X, Y; μ]
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: case iSup
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
f' : ℕ → Ω → ℝ≥0∞
h_meas_f' : ∀ (n : ℕ), Measurable (f' n)
h_mono_f' : Monotone f'
h_ind_f' : ∀ (n : ℕ), ∫⁻ (ω : Ω), f ω * f' n ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' n ω ∂μ
h_measM_f' : ∀ (n : ℕ), Measurable (f' n)
⊢ ⨆ n, ∫⁻ (a : Ω), f a * f' n a ∂μ = ⨆ i, (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (a : Ω), f' i a ∂μ

Tactic:
[

Goal state after:
case iSup
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
f' : ℕ → Ω → ℝ≥0∞
h_meas_f' : ∀ (n : ℕ), Measurable (f' n)
h_mono_f' : Monotone f'
h_ind_f' : ∀ (n : ℕ), ∫⁻ (ω : Ω), f ω * f' n ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' n ω ∂μ
h_measM_f' : ∀ (n : ℕ), Measurable (f' n)
⊢ ⨆ n, ∫⁻ (a : Ω), f a * f' n a ∂μ = ⨆ i, (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (a : Ω), f' i a ∂μ
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hY : Integrable Y μ
c : ℝ
⊢ cov[X, fun ω ↦ c + Y ω; μ] = cov[X, Y; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hY : Integrable Y μ
c : ℝ
⊢ cov[X, fun ω ↦ c + Y ω; μ] = cov[X, Y; μ]
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y Z : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
hZ : MemLp Z 2 μ
⊢ cov[X + Y, Z; μ] = cov[X, Z; μ] + cov[Y, Z; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y Z : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
hZ : MemLp Z 2 μ
⊢ cov[X + Y, Z; μ] = cov[X, Z; μ] + cov[Y, Z; μ]
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y Z : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
hZ : MemLp Z 2 μ
⊢ cov[X + Y, Z; μ] = cov[X, Z; μ] + cov[Y, Z; μ]

Tactic:
simp_rw [covariance, Pi.add_apply]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y Z : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
hZ : MemLp Z 2 μ
⊢ ∫ (ω : Ω), (X ω + Y ω - ∫ (x : Ω), X x + Y x ∂μ) * (Z ω - ∫ (x : Ω), Z x ∂μ) ∂μ =
    ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) * (Z ω - ∫ (x : Ω), Z x ∂μ) ∂μ +
      ∫ (ω : Ω), (Y ω - ∫ (x : Ω), Y x ∂μ) * (Z ω - ∫ (x : Ω), Z x ∂μ) ∂μ
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : AEStronglyMeasurable X μ
c : ℝ
hX_Lp : ¬MemLp X 2 μ
h_memLp : MemLp (fun x ↦ X x + c) 2 μ
this : X = fun ω ↦ X ω + c - c
⊢ MemLp X 2 μ

Tactic:
this

Goal state after:
case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : AEStronglyMeasurable X μ
c : ℝ
hX_Lp : ¬MemLp X 2 μ
h_memLp : MemLp (fun x ↦ X x + c) 2 μ
this : X = fun ω ↦ X ω + c - c
⊢ MemLp (fun ω ↦ X ω + c - c) 2 μ
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : AEStronglyMeasurable X μ
c : ℝ
⊢ Var[fun ω ↦ c + X ω; μ] = Var[X; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : AEStronglyMeasurable X μ
c : ℝ
⊢ Var[fun ω ↦ c + X ω; μ] = Var[X; μ]
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : AEStronglyMeasurable X μ
c : ℝ
⊢ Var[fun ω ↦ X ω - c; μ] = Var[X; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : AEStronglyMeasurable X μ
c : ℝ
⊢ Var[fun ω ↦ X ω - c; μ] = Var[X; μ]
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : AEStronglyMeasurable X μ
c : ℝ
⊢ Var[fun ω ↦ c - X ω; μ] = Var[X; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : AEStronglyMeasurable X μ
c : ℝ
⊢ Var[fun ω ↦ c - X ω; μ] = Var[X; μ]
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
c : ℝ
⊢ cov[c • X, Y; μ] = c * cov[X, Y; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
c : ℝ
⊢ cov[c • X, Y; μ] = c * cov[X, Y; μ]
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
c : ℝ
⊢ cov[fun ω ↦ X ω / c, Y; μ] = cov[X, Y; μ] / c

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
c : ℝ
⊢ cov[fun ω ↦ X ω / c, Y; μ] = cov[X, Y; μ] / c
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
c : ℝ
⊢ cov[X, fun ω ↦ Y ω / c; μ] = cov[X, Y; μ] / c

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
c : ℝ
⊢ cov[X, fun ω ↦ Y ω / c; μ] = cov[X, Y; μ] / c
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y Z : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
hZ : MemLp Z 2 μ
⊢ cov[X - Y, Z; μ] = cov[X, Z; μ] - cov[Y, Z; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y Z : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
hZ : MemLp Z 2 μ
⊢ cov[X - Y, Z; μ] = cov[X, Z; μ] - cov[Y, Z; μ]
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y Z : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
hZ : MemLp Z 2 μ
⊢ cov[X, Y - Z; μ] = cov[X, Y; μ] - cov[X, Z; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y Z : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
hZ : MemLp Z 2 μ
⊢ cov[X, Y - Z; μ] = cov[X, Y; μ] - cov[X, Z; μ]
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
X : Ω → ℝ
Y : Ω' ≃ᵐ Ω
⊢ Var[X; Measure.map (⇑Y) μ] = Var[X ∘ ⇑Y; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
X : Ω → ℝ
Y : Ω' ≃ᵐ Ω
⊢ Var[X; Measure.map (⇑Y) μ] = Var[X ∘ ⇑Y; μ]
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁸ : NormedAddGroup E
inst✝⁷ : MeasurableSpace E
inst✝⁶ : OpensMeasurableSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : ContinuousENorm F
inst✝³ : MeasurableSpace F
inst✝² : OpensMeasurableSpace F
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousENorm G
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
B : E → F → G
c : ℝ≥0
hc : c ≠ 0
hB : ∀ (x : E) (y : F), ↑c * ‖x‖ₑ * ‖y‖ₑ ≤ ‖B x y‖ₑ
h'XY : Integrable (fun ω ↦ B (X ω) (Y ω)) μ
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
h'X : ¬X =ᵐ[μ] 0
⊢ Integrable Y μ

Tactic:
refine hXY.symm.integrable_left_of_integrable_op (Function.swap B) c hc (fun y x ↦ ?_)
    h'XY hY hX h'X
  grw [mul_right_comm, hB]

Goal state after:
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
Y : Ω → ℝ
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
s : Finset ι
inst✝ : IsFiniteMeasure μ
hX : ∀ i ∈ s, MemLp (X i) 2 μ
hY : MemLp Y 2 μ
⊢ ∑ i ∈ s, cov[X i, Y; μ] = ∑ i ∈ s, cov[Y, X i; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
Y : Ω → ℝ
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
s : Finset ι
inst✝ : IsFiniteMeasure μ
hX : ∀ i ∈ s, MemLp (X i) 2 μ
hY : MemLp Y 2 μ
⊢ ∑ i ∈ s, cov[X i, Y; μ] = ∑ i ∈ s, cov[Y, X i; μ]
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: case pos.e_a
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
hX : AEStronglyMeasurable X μ
hℒ : MemLp X 2 μ
⊢ ENNReal.ofReal (∫ (x : Ω), (X ^ 2) x ∂μ) = ∫⁻ (ω : Ω), ‖X ω‖ₑ ^ 2 ∂μ

Tactic:
[

Goal state after:
case pos.e_a
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → ℝ
hX : AEStronglyMeasurable X μ
hℒ : MemLp X 2 μ
⊢ ENNReal.ofReal (∫ (x : Ω), (X ^ 2) x ∂μ) = ∫⁻ (ω : Ω), ‖X ω‖ₑ ^ 2 ∂μ
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
Ω' : Type u_2
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
X Y : Ω → ℝ
Z : Ω' ≃ᵐ Ω
⊢ cov[X, Y; Measure.map (⇑Z) μ] = cov[X ∘ ⇑Z, Y ∘ ⇑Z; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
Ω' : Type u_2
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
X Y : Ω → ℝ
Z : Ω' ≃ᵐ Ω
⊢ cov[X, Y; Measure.map (⇑Z) μ] = cov[X ∘ ⇑Z, Y ∘ ⇑Z; μ]
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
Ω' : Type u_2
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Z : Ω' → Ω
hX : AEStronglyMeasurable X (Measure.map Z μ)
hY : AEStronglyMeasurable Y (Measure.map Z μ)
hZ : AEMeasurable Z μ
⊢ cov[X, Y; Measure.map Z μ] = cov[X ∘ Z, Y ∘ Z; μ]

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
Ω' : Type u_2
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Z : Ω' → Ω
hX : AEStronglyMeasurable X (Measure.map Z μ)
hY : AEStronglyMeasurable Y (Measure.map Z μ)
hZ : AEMeasurable Z μ
⊢ cov[X, Y; Measure.map Z μ] = cov[X ∘ Z, Y ∘ Z; μ]
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: case e'_4.e_a
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
hX : AEStronglyMeasurable X μ
c : ℝ≥0
hc : c ≠ 0
A : ↑c ≠ 0
B : AEStronglyMeasurable (fun x ↦ ∫ (x : Ω), X x ∂μ) μ
⊢ eVar[X; μ] = ((∫⁻ (x : Ω), ‖X x - ∫ (x : Ω), X x ∂μ‖ₑ ^ 2 ∂μ) ^ 2⁻¹) ^ 2

Tactic:
[

Goal state after:
case e'_4.e_a
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
hX : AEStronglyMeasurable X μ
c : ℝ≥0
hc : c ≠ 0
A : ↑c ≠ 0
B : AEStronglyMeasurable (fun x ↦ ∫ (x : Ω), X x ∂μ) μ
⊢ eVar[X; μ] = ((∫⁻ (x : Ω), ‖X x - ∫ (x : Ω), X x ∂μ‖ₑ ^ 2 ∂μ) ^ 2⁻¹) ^ 2
Module: Mathlib.Probability.HasLaw

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
P : Measure Ω
c : ℝ≥0∞
inst✝ : MeasurableSingletonClass 𝓧
x : 𝓧
hX : HasLaw X (c • dirac x) P
⊢ ∀ᵐ (y : 𝓧) ∂c • dirac x, y = x

Tactic:
apply Measure.ae_smul_measure (by simp)

Goal state after:
Module: Mathlib.Probability.HasLaw

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_3
inst✝ : Fintype ι
𝓧 : ι → Type u_4
m𝓧 : (i : ι) → MeasurableSpace (𝓧 i)
μ : (i : ι) → Measure (𝓧 i)
X : (i : ι) → Ω → 𝓧 i
hX : ∀ (i : ι), HasLaw (X i) (μ i) P
h : iIndepFun X P
⊢ (Measure.pi fun i ↦ map (X i) P) = Measure.pi μ

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_3
inst✝ : Fintype ι
𝓧 : ι → Type u_4
m𝓧 : (i : ι) → MeasurableSpace (𝓧 i)
μ : (i : ι) → Measure (𝓧 i)
X : (i : ι) → Ω → 𝓧 i
hX : ∀ (i : ι), HasLaw (X i) (μ i) P
h : iIndepFun X P
⊢ (Measure.pi fun i ↦ map (X i) P) = Measure.pi μ
Module: Mathlib.Probability.HasLaw

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝¹ : IsProbabilityMeasure P
ι : Type u_3
inst✝ : Fintype ι
𝓧 : ι → Type u_4
m𝓧 : (i : ι) → MeasurableSpace (𝓧 i)
μ : (i : ι) → Measure (𝓧 i)
X : (i : ι) → Ω → 𝓧 i
hX : ∀ (i : ι), HasLaw (X i) (μ i) P
h : HasLaw (fun ω i ↦ X i ω) (Measure.pi μ) P
⊢ Measure.pi μ = Measure.pi fun i ↦ map (X i) P

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝¹ : IsProbabilityMeasure P
ι : Type u_3
inst✝ : Fintype ι
𝓧 : ι → Type u_4
m𝓧 : (i : ι) → MeasurableSpace (𝓧 i)
μ : (i : ι) → Measure (𝓧 i)
X : (i : ι) → Ω → 𝓧 i
hX : ∀ (i : ι), HasLaw (X i) (μ i) P
h : HasLaw (fun ω i ↦ X i ω) (Measure.pi μ) P
⊢ Measure.pi μ = Measure.pi fun i ↦ map (X i) P
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝³ : ConditionallyCompleteLinearOrder γ
inst✝² : TopologicalSpace γ
inst✝¹ : OpensMeasurableSpace γ
inst✝ : OrderClosedTopology γ
h : IdentDistrib f g μ ν
I : ∀ (a : γ), μ {x | a < f x} = ν {x | a < g x}
⊢ essSup f μ = essSup g ν

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝³ : ConditionallyCompleteLinearOrder γ
inst✝² : TopologicalSpace γ
inst✝¹ : OpensMeasurableSpace γ
inst✝ : OrderClosedTopology γ
h : IdentDistrib f g μ ν
I : ∀ (a : γ), μ {x | a < f x} = ν {x | a < g x}
⊢ essSup f μ = essSup g ν
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝³ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝² : TopologicalSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : OpensMeasurableSpace E
X : ι → Ω' → E
hX₁ : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
hX₂ : ∀ᵐ (ω : Ω') ∂μ', Tendsto (fun i ↦ X i ω) l (𝓝 (Z ω))
⊢ Tendsto (fun n ↦ ⟨Measure.map (X n) μ', ⋯⟩) l (𝓝 ⟨Measure.map Z μ', ⋯⟩)

Tactic:
[

Goal state after:
ι : Type u_1
E : Type u_2
Ω' : Type u_3
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝³ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝² : TopologicalSpace E
inst✝¹ : l.IsCountablyGenerated
inst✝ : OpensMeasurableSpace E
X : ι → Ω' → E
hX₁ : ∀ (i : ι), AEMeasurable (X i) μ'
hZ : AEMeasurable Z μ'
hX₂ : ∀ᵐ (ω : Ω') ∂μ', Tendsto (fun i ↦ X i ω) l (𝓝 (Z ω))
⊢ Tendsto (fun n ↦ ⟨Measure.map (X n) μ', ⋯⟩) l (𝓝 ⟨Measure.map Z μ', ⋯⟩)
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ
ε : ℝ
hε : 0 < ε
hι : Nonempty ι
C : ℝ
hC₁ : 0 < C
hC₂ : eLpNorm ({x | C ≤ ↑‖f j x‖₊}.indicator (f j)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
⊢ x ∈ {x | ⟨C, ⋯⟩ ≤ ‖f i x‖₊} ↔ x ∈ {x | C ≤ ‖f i x‖}

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i : ι), IdentDistrib (f i) (f j) μ μ
ε : ℝ
hε : 0 < ε
hι : Nonempty ι
C : ℝ
hC₁ : 0 < C
hC₂ : eLpNorm ({x | C ≤ ↑‖f j x‖₊}.indicator (f j)) p μ ≤ ENNReal.ofReal ε
i : ι
x : α
⊢ x ∈ {x | ⟨C, ⋯⟩ ≤ ‖f i x‖₊} ↔ x ∈ {x | C ≤ ‖f i x‖}