mathlib-initiative/mathlib-tactics huggingface dataset:
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
μ : Measure 𝓧
P : Measure Ω
inst✝ : IsFiniteMeasure P
𝓨 : Type u_3
m𝓨 : MeasurableSpace 𝓨
ν : Measure 𝓨
Y : Ω → 𝓨
hX : HasLaw X μ P
hY : HasLaw Y ν P
h : X ⟂ᵢ[P] Y
⊢ map (fun ω ↦ (X ω, Y ω)) P = μ.prod ν

Tactic:
by

Goal state after:
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
μ : Measure 𝓧
P : Measure Ω
inst✝ : IsFiniteMeasure P
𝓨 : Type u_3
m𝓨 : MeasurableSpace 𝓨
ν : Measure 𝓨
Y : Ω → 𝓨
hX : HasLaw X μ P
hY : HasLaw Y ν P
h : HasLaw (fun ω ↦ (X ω, Y ω)) (μ.prod ν) P
⊢ AEMeasurable X P

Tactic:
by

Goal state after:
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
μ : Measure 𝓧
P : Measure Ω
inst✝ : IsFiniteMeasure P
𝓨 : Type u_3
m𝓨 : MeasurableSpace 𝓨
ν : Measure 𝓨
Y : Ω → 𝓨
hX : HasLaw X μ P
hY : HasLaw Y ν P
h : HasLaw (fun ω ↦ (X ω, Y ω)) (μ.prod ν) P
⊢ AEMeasurable Y P

Tactic:
by

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
a b : ℝ
X : Ω → ℝ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hX : AEMeasurable X μ
ha : ∀ᵐ (ω : Ω) ∂μ, a ≤ X ω
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
hX_int₂ : Integrable (fun ω ↦ -X ω ^ 2) μ
hX_int₁ : Integrable (fun ω ↦ (a + b) * X ω) μ
h0 : 0 ≤ -∫ (x : Ω), (X ^ 2) x ∂μ + (a + b) * ∫ (x : Ω), X x ∂μ - a * b
⊢ Var[X; μ] ≤ (a + b) * ∫ (x : Ω), X x ∂μ - a * b - (∫ (x : Ω), X x ∂μ) ^ 2

Tactic:
by

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
a b : ℝ
X : Ω → ℝ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hX : AEMeasurable X μ
ha : ∀ᵐ (ω : Ω) ∂μ, a ≤ X ω
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
hX_int₂ : Integrable (fun ω ↦ -X ω ^ 2) μ
hX_int₁ : Integrable (fun ω ↦ (a + b) * X ω) μ
h0 : 0 ≤ -∫ (x : Ω), (X ^ 2) x ∂μ + (a + b) * ∫ (x : Ω), X x ∂μ - a * b
⊢ (a + b) * ∫ (x : Ω), X x ∂μ - a * b - (∫ (x : Ω), X x ∂μ) ^ 2 = (b - ∫ (x : Ω), X x ∂μ) * (∫ (x : Ω), X x ∂μ - a)

Tactic:
by

Goal state after:
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
μ : Measure 𝓧
P : Measure Ω
inst✝ : IsFiniteMeasure P
𝓨 : Type u_3
m𝓨 : MeasurableSpace 𝓨
ν : Measure 𝓨
Y : Ω → 𝓨
hX : HasLaw X μ P
hY : HasLaw Y ν P
h : HasLaw (fun ω ↦ (X ω, Y ω)) (μ.prod ν) P
⊢ X ⟂ᵢ[P] Y

Tactic:
by

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
⊢ ∀ (i : ι), AEMeasurable (X i) P

Tactic:
by

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
a b : ℝ
X : Ω → ℝ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hX : AEMeasurable X μ
⊢ Var[X; μ] ≤ (b - ∫ (x : Ω), X x ∂μ) * (∫ (x : Ω), X x ∂μ - a)

Tactic:
by

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
a b : ℝ
X : Ω → ℝ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hX : AEMeasurable X μ
⊢ (b - ∫ (x : Ω), X x ∂μ) * (∫ (x : Ω), X x ∂μ - a) = ((b - a) / 2) ^ 2 - (∫ (x : Ω), X x ∂μ - (b + a) / 2) ^ 2

Tactic:
by

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
⊢ map (fun ω i ↦ X i ω) P = Measure.pi μ

Tactic:
by

Goal state after:
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
⊢ ∀ (i : ι), AEMeasurable (X i) P

Tactic:
by

Goal state after:
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'
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
X : Ω → ℝ
Y : Ω' → ℝ
hfμ : MemLp X 2 μ
hgν : MemLp Y 2 ν
⊢ Var[fun p ↦ X p.1 + Y p.2; μ.prod ν] = Var[X; μ] + Var[Y; ν]

Tactic:
by

Goal state after:
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
⊢ iIndepFun X P

Tactic:
by

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

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
inst✝¹ : IsProbabilityMeasure μ
ν : Measure F
inst✝ : IsProbabilityMeasure ν
L : StrongDual ℝ (E × F)
hLμ : MemLp (⇑(L ∘SL ContinuousLinearMap.inl ℝ E F)) 2 μ
hLν : MemLp (⇑(L ∘SL ContinuousLinearMap.inr ℝ E F)) 2 ν
⊢ ⇑L = fun x ↦ (L ∘SL ContinuousLinearMap.inl ℝ E F) x.1 + (L ∘SL ContinuousLinearMap.inr ℝ E F) x.2

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
u : γ → δ
h : IdentDistrib f g μ ν
hu : AEMeasurable u (Measure.map f μ)
⊢ AEMeasurable (u ∘ g) ν

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
u : γ → δ
h : IdentDistrib f g μ ν
hu : AEMeasurable u (Measure.map f μ)
⊢ Measure.map (u ∘ f) μ = Measure.map (u ∘ g) ν

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
h : IdentDistrib f g μ ν
s : Set γ
hs : MeasurableSet s
⊢ μ (f ⁻¹' s) = ν (g ⁻¹' s)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
h : IdentDistrib f g μ ν
p : γ → Prop
pmeas : MeasurableSet {x | p x}
hp : ∀ᵐ (x : α) ∂μ, p (f x)
⊢ ∀ᵐ (x : β) ∂ν, p (g x)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
κ : Measure γ
h₀ : HasLaw f κ μ
h₁ : HasLaw g κ ν
⊢ Measure.map f μ = Measure.map g ν

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
κ : Measure γ
h₀ : IdentDistrib f g μ ν
h₁ : HasLaw f κ μ
⊢ Measure.map g ν = κ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : TopologicalSpace γ
inst✝¹ : PseudoMetrizableSpace γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable g ν

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
𝕜 : Type u_2
inst✝¹⁴ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
𝓧 : Type u_3
𝓨 : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹³ : MeasurableSpace 𝓧
inst✝¹² : MeasurableSpace 𝓨
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : CompleteSpace E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : CompleteSpace F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : CompleteSpace G
X : Ω → 𝓧
Y : Ω → 𝓨
f : 𝓧 → E
g : 𝓨 → F
hXY : X ⟂ᵢ[μ] Y
hX : AEMeasurable X μ
hY : AEMeasurable Y μ
hf : AEStronglyMeasurable f (Measure.map X μ)
hg : AEStronglyMeasurable g (Measure.map Y μ)
B : E →L[𝕜] F →L[𝕜] G
c : ℝ≥0
hc : c ≠ 0
hB : ∀ (x : E) (y : F), ↑c * ‖x‖ * ‖y‖ ≤ ‖(B x) y‖
⊢ ∫ (ω : Ω), (B (f (X ω))) (g (Y ω)) ∂μ = (B (∫ (ω : Ω), f (X ω) ∂μ)) (∫ (ω : Ω), g (Y ω) ∂μ)

Tactic:
by

Goal state after:
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
⊢ essSup f μ = essSup g ν

Tactic:
by

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

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
inst✝¹ : IsProbabilityMeasure μ
ν : Measure F
inst✝ : IsProbabilityMeasure ν
L : StrongDual ℝ (E × F)
hLμ : MemLp (⇑(L ∘SL ContinuousLinearMap.inl ℝ E F)) 2 μ
hLν : MemLp (⇑(L ∘SL ContinuousLinearMap.inr ℝ E F)) 2 ν
⊢ Var[⇑L; μ.prod ν] = Var[⇑(L ∘SL ContinuousLinearMap.inl ℝ E F); μ] + Var[⇑(L ∘SL ContinuousLinearMap.inr ℝ E F); ν]

Tactic:
by

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
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : ContinuousENorm E
inst✝⁸ : MeasurableSpace E
inst✝⁷ : OpensMeasurableSpace E
inst✝⁶ : TopologicalSpace F
inst✝⁵ : ContinuousENorm F
inst✝⁴ : MeasurableSpace F
inst✝³ : OpensMeasurableSpace F
inst✝² : SMul E F
inst✝¹ : ContinuousSMul E F
inst✝ : ENormSMulClass E F
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
hX : Integrable X μ
hY : Integrable Y μ
⊢ Continuous (Function.uncurry fun x1 x2 ↦ x1 • x2)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
f : α → ℝ≥0∞
g : β → ℝ≥0∞
h : IdentDistrib f g μ ν
⊢ ∫⁻ (x : α), f x ∂μ = ∫⁻ (x : β), g x ∂ν

Tactic:
by

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
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : ContinuousENorm E
inst✝⁸ : MeasurableSpace E
inst✝⁷ : OpensMeasurableSpace E
inst✝⁶ : TopologicalSpace F
inst✝⁵ : ContinuousENorm F
inst✝⁴ : MeasurableSpace F
inst✝³ : OpensMeasurableSpace F
inst✝² : SMul E F
inst✝¹ : ContinuousSMul E F
inst✝ : ENormSMulClass E F
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
hX : Integrable X μ
hY : Integrable Y μ
⊢ ∀ (x : E) (y : F), ‖x • y‖ₑ ≤ ↑1 * ‖x‖ₑ * ‖y‖ₑ

Tactic:
by

Goal state after:
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✝² : IsProbabilityMeasure μ'
mE : MeasurableSpace E
Z : Ω' → E
l : Filter ι
inst✝¹ : TopologicalSpace E
inst✝ : OpensMeasurableSpace E
hZ : AEMeasurable Z μ'
x✝ : ι
⊢ AEMeasurable Z μ'

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω : ι → Type u_5
m : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝³ : ∀ (i : ι), IsProbabilityMeasure (μ i)
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝² : IsProbabilityMeasure μ'
mE : MeasurableSpace E
X : (i : ι) → Ω i → E
Z : Ω' → E
l : Filter ι
inst✝¹ : TopologicalSpace E
inst✝ : OpensMeasurableSpace E
i : ι
hX : ∀ (j : ι), IdentDistrib (X i) (X j) (μ i) (μ j)
hZ : IdentDistrib (X i) Z (μ i) μ'
⊢ Tendsto (fun n ↦ ⟨Measure.map (X n) (μ n), ⋯⟩) l (𝓝 ⟨Measure.map Z μ', ⋯⟩)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω : ι → Type u_5
m : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝³ : ∀ (i : ι), IsProbabilityMeasure (μ i)
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝² : IsProbabilityMeasure μ'
mE : MeasurableSpace E
X Y : (i : ι) → Ω i → E
Z : Ω' → E
l : Filter ι
inst✝¹ : TopologicalSpace E
inst✝ : OpensMeasurableSpace E
T : Ω' → E
hXY : ∀ (i : ι), X i =ᵐ[μ i] Y i
hZT : Z =ᵐ[μ'] T
h : TendstoInDistribution X l Z μ μ'
⊢ Tendsto (fun n ↦ ⟨Measure.map (Y n) (μ n), ⋯⟩) l (𝓝 ⟨Measure.map T μ', ⋯⟩)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω : ι → Type u_5
m : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝³ : ∀ (i : ι), IsProbabilityMeasure (μ i)
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝² : IsProbabilityMeasure μ'
mE : MeasurableSpace E
X : (i : ι) → Ω i → E
Z : Ω' → E
l : Filter ι
inst✝¹ : TopologicalSpace E
inst✝ : IsEmpty E
⊢ Tendsto (fun n ↦ ⟨Measure.map (X n) (μ n), ⋯⟩) l (𝓝 ⟨Measure.map Z μ', ⋯⟩)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
𝓧 : Type u_3
𝓨 : Type u_4
E : Type u_5
inst✝⁴ : MeasurableSpace 𝓧
inst✝³ : MeasurableSpace 𝓨
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
X : Ω → 𝓧
Y : Ω → 𝓨
f : 𝓧 → 𝕜
g : 𝓨 → E
hXY : X ⟂ᵢ[μ] Y
hX : AEMeasurable X μ
hY : AEMeasurable Y μ
hf : AEStronglyMeasurable f (Measure.map X μ)
hg : AEStronglyMeasurable g (Measure.map Y μ)
hE : CompleteSpace E
⊢ 1 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
𝓧 : Type u_3
𝓨 : Type u_4
E : Type u_5
inst✝⁴ : MeasurableSpace 𝓧
inst✝³ : MeasurableSpace 𝓨
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
X : Ω → 𝓧
Y : Ω → 𝓨
f : 𝓧 → 𝕜
g : 𝓨 → E
hXY : X ⟂ᵢ[μ] Y
hX : AEMeasurable X μ
hY : AEMeasurable Y μ
hf : AEStronglyMeasurable f (Measure.map X μ)
hg : AEStronglyMeasurable g (Measure.map Y μ)
hE : CompleteSpace E
⊢ ∀ (x : 𝕜) (y : E), ↑1 * ‖x‖ * ‖y‖ ≤ ‖((ContinuousLinearMap.lsmul ℝ 𝕜) x) y‖

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω'' : Type u_4
Ω : ι → Type u_5
m : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
m'' : MeasurableSpace Ω''
μ'' : Measure Ω''
inst✝⁴ : IsProbabilityMeasure μ''
mE : MeasurableSpace E
l : Filter ι
inst✝³ : TopologicalSpace E
inst✝² : HasOuterApproxClosed E
inst✝¹ : BorelSpace E
X : (i : ι) → Ω i → E
Z : Ω' → E
W : Ω'' → E
inst✝ : l.NeBot
h1 : TendstoInDistribution X l Z μ μ'
h2 : TendstoInDistribution X l W μ μ''
⊢ Measure.map Z μ' = Measure.map W μ''

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: case neg
Ω : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
𝓧 : Type u_3
𝓨 : Type u_4
E : Type u_5
inst✝⁴ : MeasurableSpace 𝓧
inst✝³ : MeasurableSpace 𝓨
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
X : Ω → 𝓧
Y : Ω → 𝓨
f : 𝓧 → 𝕜
g : 𝓨 → E
hXY : X ⟂ᵢ[μ] Y
hX : AEMeasurable X μ
hY : AEMeasurable Y μ
hf : AEStronglyMeasurable f (Measure.map X μ)
hg : AEStronglyMeasurable g (Measure.map Y μ)
hE : ¬CompleteSpace E
⊢ ∫ (ω : Ω), f (X ω) • g (Y ω) ∂μ = (∫ (ω : Ω), f (X ω) ∂μ) • ∫ (ω : Ω), g (Y ω) ∂μ

Tactic:
simp [integral, hE]

Goal state after:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
𝓧 : Type u_3
𝓨 : Type u_4
E : Type u_5
inst✝⁴ : MeasurableSpace 𝓧
inst✝³ : MeasurableSpace 𝓨
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
X : Ω → 𝓧
Y : Ω → 𝓨
f : 𝓧 → 𝕜
g : 𝓨 → E
hXY : X ⟂ᵢ[μ] Y
hX : AEMeasurable X μ
hY : AEMeasurable Y μ
hf : AEStronglyMeasurable f (Measure.map X μ)
hg : AEStronglyMeasurable g (Measure.map Y μ)
⊢ ∫ (ω : Ω), f (X ω) • g (Y ω) ∂μ = (∫ (ω : Ω), f (X ω) ∂μ) • ∫ (ω : Ω), g (Y ω) ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : Ω → 𝕜
Y : Ω → E
hXY : X ⟂ᵢ[μ] Y
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
hE : CompleteSpace E
⊢ 1 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : NormedAddCommGroup γ
inst✝¹ : NormedSpace ℝ γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : AEStronglyMeasurable f μ
⊢ AEStronglyMeasurable id (Measure.map f μ)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : Ω → 𝕜
Y : Ω → E
hXY : X ⟂ᵢ[μ] Y
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
hE : CompleteSpace E
⊢ ∀ (x : 𝕜) (y : E), ↑1 * ‖x‖ * ‖y‖ ≤ ‖((ContinuousLinearMap.lsmul ℝ 𝕜) x) y‖

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: case neg
Ω : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : Ω → 𝕜
Y : Ω → E
hXY : X ⟂ᵢ[μ] Y
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
hE : ¬CompleteSpace E
⊢ ∫ (x : Ω), (X • Y) x ∂μ = (∫ (x : Ω), X x ∂μ) • ∫ (x : Ω), Y x ∂μ

Tactic:
simp [integral, hE]

Goal state after:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : Ω → 𝕜
Y : Ω → E
hXY : X ⟂ᵢ[μ] Y
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
⊢ ∫ (x : Ω), (X • Y) x ∂μ = (∫ (x : Ω), X x ∂μ) • ∫ (x : Ω), Y x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝² : NormedAddCommGroup γ
inst✝¹ : NormedSpace ℝ γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
⊢ ∫ (x : α), f x ∂μ = ∫ (x : β), g x ∂ν

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: ι : Type u_1
E : Type u_2
Ω' : Type u_3
Ω : ι → Type u_5
m : (i : ι) → MeasurableSpace (Ω i)
μ : (i : ι) → Measure (Ω i)
inst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)
m' : MeasurableSpace Ω'
μ' : Measure Ω'
inst✝⁵ : IsProbabilityMeasure μ'
mE : MeasurableSpace E
X : (i : ι) → Ω i → E
Z : Ω' → E
l : Filter ι
inst✝⁴ : TopologicalSpace E
F : Type u_6
inst✝³ : OpensMeasurableSpace E
inst✝² : TopologicalSpace F
inst✝¹ : MeasurableSpace F
inst✝ : BorelSpace F
g : E → F
hg : Continuous[inst✝⁴, inst✝²] g
h : TendstoInDistribution X l Z μ μ'
⊢ Tendsto (fun n ↦ ⟨Measure.map (g ∘ X n) (μ n), ⋯⟩) l (𝓝 ⟨Measure.map (g ∘ Z) μ', ⋯⟩)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝¹ : NormedAddCommGroup γ
inst✝ : OpensMeasurableSpace γ
h : IdentDistrib f g μ ν
p : ℝ≥0∞
⊢ eLpNorm f p μ = eLpNorm g p ν

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝¹ : NormedAddCommGroup γ
inst✝ : BorelSpace γ
p : ℝ≥0∞
h : IdentDistrib f g μ ν
hf : MemLp f p μ
⊢ MemLp g p ν

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ : Measure α
ν : Measure β
f : α → γ
g : β → γ
inst✝¹ : NormedAddCommGroup γ
inst✝ : BorelSpace γ
h : IdentDistrib f g μ ν
hf : Integrable f μ
⊢ Integrable g ν

Tactic:
by

Goal state after:
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))
f : BoundedContinuousFunction E NNReal
⊢ AEMeasurable (fun ω ↦ ↑(f ω)) (Measure.map Z μ')

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
f : α → ℝ
g : β → ℝ
h : IdentDistrib f g μ ν
⊢ eVar[f; μ] = eVar[g; ν]

Tactic:
by

Goal state after:
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
f : α → ℝ
g : β → ℝ
h : IdentDistrib f g μ ν
⊢ Var[f; μ] = Var[g; ν]

Tactic:
by

Goal state after:
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))
f : BoundedContinuousFunction E NNReal
i : ι
⊢ AEMeasurable (fun ω ↦ ↑(f ω)) (Measure.map (X i) μ')

Tactic:
by

Goal state after: