mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: case e_f
Ω : Type u_1
mΩ : MeasurableSpace Ω
c : ℝ
X : Ω → ℝ
μ : Measure Ω
ω : Ω
⊢ ENNReal.ofReal (c ^ 2) * ‖X ω - ∫ (a : Ω), X a ∂μ‖ₑ ^ 2 = ENNReal.ofReal (c ^ 2) * ‖X ω - ∫ (x : Ω), X x ∂μ‖ₑ ^ 2

Tactic:
]

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
hX : AEMeasurable X μ
⊢ ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) * (X ω - ∫ (x : Ω), X x ∂μ) ∂μ = ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) ^ 2 ∂μ

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
hX : AEMeasurable X μ
⊢ ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) * (X ω - ∫ (x : Ω), X x ∂μ) ∂μ = ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) ^ 2 ∂μ
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
c : ℝ
X : Ω → ℝ
μ : Measure Ω
⊢ c ^ 2 * eVar[X; μ].toReal = c ^ 2 * Var[X; μ]

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
c : ℝ
X : Ω → ℝ
μ : Measure Ω
⊢ c ^ 2 * eVar[X; μ].toReal = c ^ 2 * Var[X; μ]
Module: Mathlib.Probability.Moments.Covariance

Goal state before:
pp: case e_f
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : Integrable X μ
c : ℝ
ω : Ω
⊢ (X ω + c - (∫ (a : Ω), X a ∂μ + ∫ (a : Ω), c ∂μ)) * (Y ω - ∫ (x : Ω), Y x ∂μ) =
    (X ω - ∫ (x : Ω), X x ∂μ) * (Y ω - ∫ (x : Ω), Y x ∂μ)

Tactic:
]

Goal state after:
case e_f
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : Integrable X μ
c : ℝ
ω : Ω
⊢ (X ω + c - (∫ (a : Ω), X a ∂μ + ∫ (a : Ω), c ∂μ)) * (Y ω - ∫ (x : Ω), Y x ∂μ) =
    (X ω - ∫ (x : Ω), X x ∂μ) * (Y ω - ∫ (x : Ω), Y x ∂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

pp: case iSup.hf
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 : ℕ), Measurable fun ω ↦ f ω * f' n ω

pp: case iSup.h_mono
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)
⊢ Monotone fun i ω ↦ f ω * f' i ω

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

case iSup.hf
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 : ℕ), Measurable fun ω ↦ f ω * f' n ω

case iSup.h_mono
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)
⊢ Monotone fun i ω ↦ f ω * f' i ω
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, Y; μ] = cov[X, Y; μ]

Tactic:
]

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

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

Tactic:
]

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

Goal state before:
pp: case pos
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
hX_int : Integrable X μ
⊢ ∫ (ω : Ω), (X ω + c - (∫ (a : Ω), X a ∂μ + μ.real Set.univ • c)) ^ 2 ∂μ = ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) ^ 2 ∂μ

Tactic:
]

Goal state after:
case pos
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
hX_int : Integrable X μ
⊢ ∫ (ω : Ω), (X ω + c - (∫ (a : Ω), X a ∂μ + μ.real Set.univ • c)) ^ 2 ∂μ = ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) ^ 2 ∂μ
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
f g : Ω → ℝ≥0∞
h_meas_f : AEMeasurable f μ
h_meas_g : AEMeasurable g μ
h_indep_fun : f ⟂ᵢ[μ] g
fg_ae : f * g =ᵐ[μ] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g
⊢ ∫⁻ (a : Ω), (AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g) a ∂μ =
    (∫⁻ (a : Ω), AEMeasurable.mk f h_meas_f a ∂μ) * ∫⁻ (a : Ω), AEMeasurable.mk g h_meas_g a ∂μ

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
f g : Ω → ℝ≥0∞
h_meas_f : AEMeasurable f μ
h_meas_g : AEMeasurable g μ
h_indep_fun : f ⟂ᵢ[μ] g
fg_ae : f * g =ᵐ[μ] AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g
⊢ ∫⁻ (a : Ω), (AEMeasurable.mk f h_meas_f * AEMeasurable.mk g h_meas_g) a ∂μ =
    (∫⁻ (a : Ω), AEMeasurable.mk f h_meas_f a ∂μ) * ∫⁻ (a : Ω), AEMeasurable.mk g h_meas_g a ∂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
⊢ 0 = 0

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
⊢ ¬MemLp (fun x ↦ X x + c) 2 μ

Tactic:
]

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
⊢ ¬MemLp (fun x ↦ X x + c) 2 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 (fun ω ↦ X ω + c - c) 2 μ

Tactic:
]

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.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
⊢ ∫ (ω : Ω), (X ω + Y ω - ∫ (x : Ω), X x + Y x ∂μ) * (Z ω - ∫ (x : Ω), Z x ∂μ) ∂μ =
    ∫ (a : Ω),
      (X a - ∫ (x : Ω), X x ∂μ) * (Z a - ∫ (x : Ω), Z x ∂μ) + (Y a - ∫ (x : Ω), Y x ∂μ) * (Z a - ∫ (x : Ω), Z x ∂μ) ∂μ

pp: case hf
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
⊢ Integrable (fun ω ↦ (X ω - ∫ (x : Ω), X x ∂μ) * (Z ω - ∫ (x : Ω), Z x ∂μ)) μ

pp: case hg
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
⊢ Integrable (fun ω ↦ (Y ω - ∫ (x : Ω), Y x ∂μ) * (Z ω - ∫ (x : Ω), Z x ∂μ)) 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
⊢ ∫ (ω : Ω), (X ω + Y ω - ∫ (x : Ω), X x + Y x ∂μ) * (Z ω - ∫ (x : Ω), Z x ∂μ) ∂μ =
    ∫ (a : Ω),
      (X a - ∫ (x : Ω), X x ∂μ) * (Z a - ∫ (x : Ω), Z x ∂μ) + (Y a - ∫ (x : Ω), Y x ∂μ) * (Z a - ∫ (x : Ω), Z x ∂μ) ∂μ

case hf
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
⊢ Integrable (fun ω ↦ (X ω - ∫ (x : Ω), X x ∂μ) * (Z ω - ∫ (x : Ω), Z x ∂μ)) μ

case hg
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
⊢ Integrable (fun 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: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
hX : AEStronglyMeasurable X μ
c : ℝ
⊢ Var[X; μ] = Var[X; μ]

Tactic:
]

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

Goal state before:
pp: case e_f
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
x : Ω
⊢ (X x + Y x - (∫ (a : Ω), X a ∂μ + ∫ (a : Ω), Y a ∂μ)) * (Z x - ∫ (x : Ω), Z x ∂μ) =
    (X x - ∫ (x : Ω), X x ∂μ) * (Z x - ∫ (x : Ω), Z x ∂μ) + (Y x - ∫ (x : Ω), Y x ∂μ) * (Z x - ∫ (x : Ω), Z x ∂μ)

Tactic:
]

Goal state after:
case e_f
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
x : Ω
⊢ (X x + Y x - (∫ (a : Ω), X a ∂μ + ∫ (a : Ω), Y a ∂μ)) * (Z x - ∫ (x : Ω), Z x ∂μ) =
    (X x - ∫ (x : Ω), X x ∂μ) * (Z x - ∫ (x : Ω), Z x ∂μ) + (Y x - ∫ (x : Ω), Y x ∂μ) * (Z x - ∫ (x : Ω), Z 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
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Var[X; μ] + cov[Y, X; μ] + (cov[Y, X; μ] + Var[Y; μ]) = Var[X; μ] + 2 * cov[Y, X; μ] + Var[Y; μ]

pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ AEMeasurable Y μ

pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ AEMeasurable X μ

pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ AEMeasurable (X + Y) μ

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ Var[X; μ] + cov[Y, X; μ] + (cov[Y, X; μ] + Var[Y; μ]) = Var[X; μ] + 2 * cov[Y, X; μ] + Var[Y; μ]

Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ AEMeasurable Y μ

Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ AEMeasurable X μ

Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
⊢ AEMeasurable (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✝⁹ : TopologicalSpace E
inst✝⁸ : ContinuousENorm 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
hX : Integrable X μ
hY : Integrable Y μ
B : E → F → G
cB : Continuous (Function.uncurry B)
C : ℝ≥0
hB : ∀ (x : E) (y : F), ‖B x y‖ₑ ≤ ↑C * ‖x‖ₑ * ‖y‖ₑ
⊢ ∫⁻ (a : Ω), ‖(fun ω ↦ B (X ω) (Y ω)) a‖ₑ ∂μ ≤ ∫⁻ (a : Ω), ↑C * (‖X a‖ₑ * ‖Y a‖ₑ) ∂μ

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁹ : TopologicalSpace E
inst✝⁸ : ContinuousENorm 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
hX : Integrable X μ
hY : Integrable Y μ
B : E → F → G
cB : Continuous (Function.uncurry B)
C : ℝ≥0
hB : ∀ (x : E) (y : F), ‖B x y‖ₑ ≤ ↑C * ‖x‖ₑ * ‖y‖ₑ
⊢ ∫⁻ (a : Ω), ‖(fun ω ↦ B (X ω) (Y ω)) a‖ₑ ∂μ ≤ ∫⁻ (a : Ω), ↑C * (‖X a‖ₑ * ‖Y a‖ₑ) ∂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✝⁹ : TopologicalSpace E
inst✝⁸ : ContinuousENorm 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
hX : Integrable X μ
hY : Integrable Y μ
B : E → F → G
cB : Continuous (Function.uncurry B)
C : ℝ≥0
hB : ∀ (x : E) (y : F), ‖B x y‖ₑ ≤ ↑C * ‖x‖ₑ * ‖y‖ₑ
⊢ ↑C * ((∫⁻ (ω : Ω), ‖X ω‖ₑ ∂μ) * ∫⁻ (ω : Ω), ‖Y ω‖ₑ ∂μ) = ↑C * ((∫⁻ (ω : Ω), ‖X ω‖ₑ ∂μ) * ∫⁻ (ω : Ω), ‖Y ω‖ₑ ∂μ)

Tactic:
]

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

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

Tactic:
]

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

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

pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
s : Finset ι
X : ι → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hX : ∀ i ∈ s, MemLp (X i) 2 μ
⊢ MemLp (∑ i ∈ s, X i) 2 μ

pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
s : Finset ι
X : ι → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hX : ∀ i ∈ s, MemLp (X i) 2 μ
⊢ AEMeasurable (∑ i ∈ s, X i) μ

Tactic:
]

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

Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
s : Finset ι
X : ι → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hX : ∀ i ∈ s, MemLp (X i) 2 μ
⊢ MemLp (∑ i ∈ s, X i) 2 μ

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

Tactic:
]

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
s : Finset ι
X : ι → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hX : ∀ i ∈ s, MemLp (X i) 2 μ
i : ι
hi : i ∈ s
⊢ ∑ i_1 ∈ s, cov[X i, X i_1; μ] = ∑ j ∈ s, cov[X i, X j; μ]

Tactic:
]

Goal state after:
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
𝕜 : Type u_8
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : SeminormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : MeasurableSpace E
inst✝⁶ : OpensMeasurableSpace E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : MeasurableSpace F
inst✝² : OpensMeasurableSpace F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
hX : Integrable X μ
hY : Integrable Y μ
B : E →L[𝕜] F →L[𝕜] G
x : E
y : F
⊢ ‖(B x) y‖ₑ.toReal ≤ (↑‖B‖₊ * ‖x‖ₑ * ‖y‖ₑ).toReal

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
𝕜 : Type u_8
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : SeminormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : MeasurableSpace E
inst✝⁶ : OpensMeasurableSpace E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : MeasurableSpace F
inst✝² : OpensMeasurableSpace F
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
hX : Integrable X μ
hY : Integrable Y μ
B : E →L[𝕜] F →L[𝕜] G
x : E
y : F
⊢ ‖(B x) y‖ₑ.toReal ≤ (↑‖B‖₊ * ‖x‖ₑ * ‖y‖ₑ).toReal
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
⊢ c * cov[X, Y; μ] = c * cov[X, Y; μ]

Tactic:
]

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
⊢ -1 * cov[X, Y; μ] = -cov[X, Y; μ]

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y : Ω → ℝ
μ : Measure Ω
⊢ -1 * cov[X, Y; μ] = -cov[X, Y; μ]
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
inst✝ : MeasurableSingletonClass Ω
x : Ω
⊢ ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂Measure.dirac x) ^ 2 ∂Measure.dirac x = 0

pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
inst✝ : MeasurableSingletonClass Ω
x : Ω
⊢ AEMeasurable X (Measure.dirac x)

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
inst✝ : MeasurableSingletonClass Ω
x : Ω
⊢ ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂Measure.dirac x) ^ 2 ∂Measure.dirac x = 0

Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
inst✝ : MeasurableSingletonClass Ω
x : Ω
⊢ AEMeasurable X (Measure.dirac x)
Module: Mathlib.Probability.Moments.Covariance

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

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X Y Z T : Ω → ℝ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
hX : MemLp X 2 μ
hY : MemLp Y 2 μ
hZ : MemLp Z 2 μ
hT : MemLp T 2 μ
⊢ cov[X, Z; μ] - cov[X, T; μ] - (cov[Y, Z; μ] - cov[Y, T; μ]) =
    cov[X, Z; μ] - cov[X, T; μ] - cov[Y, Z; μ] + cov[Y, T; 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 : TopologicalSpace E
inst✝⁷ : ContinuousENorm E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
inst✝⁴ : NormedAddGroup 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'Y : ¬Y =ᵐ[μ] 0
I : ∫⁻ (ω : Ω), ‖Y ω‖ₑ ∂μ ≠ 0
H : ∫⁻ (a : Ω), ‖X a‖ₑ ∂μ = ∞
J : (fun x ↦ ‖X x‖ₑ) ⟂ᵢ[μ] fun x ↦ ‖Y x‖ₑ
⊢ ∞ = ∞

Tactic:
]

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Y : Ω' → Ω
hX : AEMeasurable X (Measure.map Y μ)
hY : AEMeasurable Y μ
⊢ ∫ (x : Ω'), (X (Y x) - ∫ (x : Ω'), X (Y x) ∂μ) ^ 2 ∂μ = ∫ (ω : Ω'), ((X ∘ Y) ω - ∫ (x : Ω'), (X ∘ Y) x ∂μ) ^ 2 ∂μ

pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Y : Ω' → Ω
hX : AEMeasurable X (Measure.map Y μ)
hY : AEMeasurable Y μ
⊢ AEStronglyMeasurable X (Measure.map Y μ)

pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Y : Ω' → Ω
hX : AEMeasurable X (Measure.map Y μ)
hY : AEMeasurable Y μ
⊢ AEStronglyMeasurable (fun ω ↦ (X ω - ∫ (x : Ω), X x ∂Measure.map Y μ) ^ 2) (Measure.map Y μ)

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Y : Ω' → Ω
hX : AEMeasurable X (Measure.map Y μ)
hY : AEMeasurable Y μ
⊢ ∫ (x : Ω'), (X (Y x) - ∫ (x : Ω'), X (Y x) ∂μ) ^ 2 ∂μ = ∫ (ω : Ω'), ((X ∘ Y) ω - ∫ (x : Ω'), (X ∘ Y) x ∂μ) ^ 2 ∂μ

Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Y : Ω' → Ω
hX : AEMeasurable X (Measure.map Y μ)
hY : AEMeasurable Y μ
⊢ AEStronglyMeasurable X (Measure.map Y μ)

Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Y : Ω' → Ω
hX : AEMeasurable X (Measure.map Y μ)
hY : AEMeasurable Y μ
⊢ AEStronglyMeasurable (fun ω ↦ (X ω - ∫ (x : Ω), X x ∂Measure.map Y μ) ^ 2) (Measure.map 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 : TopologicalSpace E
inst✝⁷ : ContinuousENorm E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
inst✝⁴ : NormedAddGroup 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'Y : ¬Y =ᵐ[μ] 0
I : ∫⁻ (ω : Ω), ‖Y ω‖ₑ ∂μ ≠ 0
H : ∫⁻ (a : Ω), ‖X a‖ₑ ∂μ = ∞
J : (fun x ↦ ‖X x‖ₑ) ⟂ᵢ[μ] fun x ↦ ‖Y x‖ₑ
⊢ ∫⁻ (a : Ω), ↑c * (‖X a‖ₑ * ‖Y a‖ₑ) ∂μ ≤ ∫⁻ (ω : Ω), ‖B (X ω) (Y ω)‖ₑ ∂μ

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁸ : TopologicalSpace E
inst✝⁷ : ContinuousENorm E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
inst✝⁴ : NormedAddGroup 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'Y : ¬Y =ᵐ[μ] 0
I : ∫⁻ (ω : Ω), ‖Y ω‖ₑ ∂μ ≠ 0
H : ∫⁻ (a : Ω), ‖X a‖ₑ ∂μ = ∞
J : (fun x ↦ ‖X x‖ₑ) ⟂ᵢ[μ] fun x ↦ ‖Y x‖ₑ
⊢ ∫⁻ (a : Ω), ↑c * (‖X a‖ₑ * ‖Y a‖ₑ) ∂μ ≤ ∫⁻ (ω : Ω), ‖B (X ω) (Y ω)‖ₑ ∂μ
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
X : Ω → Ω'
hX : MeasurePreserving X μ ν
f : Ω' → ℝ
hf : AEMeasurable f ν
⊢ Var[fun ω ↦ f (X ω); μ] = Var[fun x ↦ f (X x); μ]

Tactic:
]

Goal state after: