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

Goal state before:
pp: case e'_3.e'_5
Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : Measurable X
mY : Measurable Y
⊢ Measure.map X μ = Measure.map X μ

Tactic:
]

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

Goal state before:
pp: case e'_3.e'_6
Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : Measurable X
mY : Measurable Y
⊢ Measure.map Y ν = Measure.map Y ν

Tactic:
]

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

Goal state before:
pp: case refine_1
Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : AEMeasurable X μ
mY : AEMeasurable Y ν
this : (fun ω ↦ AEMeasurable.mk X mX ω.1) ⟂ᵢ[μ.prod ν] fun ω ↦ AEMeasurable.mk Y mY ω.2
⊢ AEMeasurable.mk X mX ∘ Prod.fst =ᵐ[μ.prod ν] X ∘ Prod.fst

Tactic:
]

Goal state after:
case refine_1
Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : AEMeasurable X μ
mY : AEMeasurable Y ν
this : (fun ω ↦ AEMeasurable.mk X mX ω.1) ⟂ᵢ[μ.prod ν] fun ω ↦ AEMeasurable.mk Y mY ω.2
⊢ AEMeasurable.mk X mX ∘ Prod.fst =ᵐ[μ.prod ν] X ∘ Prod.fst
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inr.basic
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m1 m2 m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h1 : m1 ≤ m
h2 : m2 ≤ m
hp1 : IsPiSystem p1
hp2 : IsPiSystem p2
hpm1 : m1 = generateFrom p1
hpm2 : m2 = generateFrom p2
hyp : IndepSets p1 p2 κ μ
h : IsMarkovKernel κ
t1 t2 : Set Ω
ht2 : t2 ∈ {s | MeasurableSet s}
t : Set Ω
ht : t ∈ p1
⊢ MeasurableSet t

Tactic:
]

Goal state after:
case inr.basic
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m1 m2 m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h1 : m1 ≤ m
h2 : m2 ≤ m
hp1 : IsPiSystem p1
hp2 : IsPiSystem p2
hpm1 : m1 = generateFrom p1
hpm2 : m2 = generateFrom p2
hyp : IndepSets p1 p2 κ μ
h : IsMarkovKernel κ
t1 t2 : Set Ω
ht2 : t2 ∈ {s | MeasurableSet s}
t : Set Ω
ht : t ∈ p1
⊢ MeasurableSet t
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: case refine_1.h
Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : AEMeasurable X μ
mY : AEMeasurable Y ν
this : (fun ω ↦ AEMeasurable.mk X mX ω.1) ⟂ᵢ[μ.prod ν] fun ω ↦ AEMeasurable.mk Y mY ω.2
⊢ AEMeasurable.mk X mX =ᵐ[μ] X

Tactic:
]

Goal state after:
case refine_1.h
Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : AEMeasurable X μ
mY : AEMeasurable Y ν
this : (fun ω ↦ AEMeasurable.mk X mX ω.1) ⟂ᵢ[μ.prod ν] fun ω ↦ AEMeasurable.mk Y mY ω.2
⊢ AEMeasurable.mk X mX =ᵐ[μ] X
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case mp
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsZeroOrMarkovKernel κ
hf : Measurable f
hg : Measurable g
s : Set β
t : Set β'
h : ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)

Tactic:
]

Goal state after:
case mp
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsZeroOrMarkovKernel κ
hf : Measurable f
hg : Measurable g
s : Set β
t : Set β'
h : ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: case e'_3
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen[inst✝²] G → μ G ≤ liminf (fun i ↦ (μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i ↦ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
obs : ∀ (i : ℕ), ∫⁻ (x : Ω), ↑‖f x‖₊ ∂μs i ≤ ↑‖f‖₊
x✝ : ℝ≥0∞
hx : ∀ᶠ (n : ℕ) in atTop, x✝ ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs n
i : ℕ
hi : x✝ ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i
x : Ω
aux : ENNReal.ofReal (f x) = ↑(NNReal.mk (f x) ⋯)
⊢ ↑(NNReal.mk (f x) ⋯) = ↑‖f x‖₊

Tactic:
]

Goal state after:
case e'_3
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen[inst✝²] G → μ G ≤ liminf (fun i ↦ (μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i ↦ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
obs : ∀ (i : ℕ), ∫⁻ (x : Ω), ↑‖f x‖₊ ∂μs i ≤ ↑‖f‖₊
x✝ : ℝ≥0∞
hx : ∀ᶠ (n : ℕ) in atTop, x✝ ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs n
i : ℕ
hi : x✝ ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i
x : Ω
aux : ENNReal.ofReal (f x) = ↑(NNReal.mk (f x) ⋯)
⊢ ↑(NNReal.mk (f x) ⋯) = ↑‖f x‖₊
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case mpr
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsZeroOrMarkovKernel κ
hf : Measurable f
hg : Measurable g
s : Set β
t : Set β'
h : IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ
hs : MeasurableSet s
ht : MeasurableSet t
⊢ IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ

Tactic:
]

Goal state after:
case mpr
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
inst✝ : IsZeroOrMarkovKernel κ
hf : Measurable f
hg : Measurable g
s : Set β
t : Set β'
h : IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ
hs : MeasurableSet s
ht : MeasurableSet t
⊢ IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: case refine_2
Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : AEMeasurable X μ
mY : AEMeasurable Y ν
this : (fun ω ↦ AEMeasurable.mk X mX ω.1) ⟂ᵢ[μ.prod ν] fun ω ↦ AEMeasurable.mk Y mY ω.2
⊢ AEMeasurable.mk Y mY ∘ Prod.snd =ᵐ[μ.prod ν] Y ∘ Prod.snd

Tactic:
]

Goal state after:
case refine_2
Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : AEMeasurable X μ
mY : AEMeasurable Y ν
this : (fun ω ↦ AEMeasurable.mk X mX ω.1) ⟂ᵢ[μ.prod ν] fun ω ↦ AEMeasurable.mk Y mY ω.2
⊢ AEMeasurable.mk Y mY ∘ Prod.snd =ᵐ[μ.prod ν] Y ∘ Prod.snd
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m1 m2 m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h1 : m1 ≤ m
h2 : m2 ≤ m
hp1 : IsPiSystem p1
hp2 : IsPiSystem p2
hpm1 : m1 = generateFrom p1
hpm2 : m2 = generateFrom p2
hyp : IndepSets p1 p2 κ μ
h : IsMarkovKernel κ
t1 t2 : Set Ω
ht2 : t2 ∈ {s | MeasurableSet s}
t : Set Ω
ht : MeasurableSet t
iht : ∀ᵐ (a : α) ∂μ, (κ a) (t ∩ t2) = (κ a) t * (κ a) t2
a : α
ha : (κ a) (t ∩ t2) = (κ a) t * (κ a) t2
⊢ tᶜ ∩ t2 = tᶜ ∩ t2

Tactic:
]

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

Goal state before:
pp: case refine_2.h
Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : AEMeasurable X μ
mY : AEMeasurable Y ν
this : (fun ω ↦ AEMeasurable.mk X mX ω.1) ⟂ᵢ[μ.prod ν] fun ω ↦ AEMeasurable.mk Y mY ω.2
⊢ AEMeasurable.mk Y mY =ᵐ[ν] Y

Tactic:
]

Goal state after:
case refine_2.h
Ω : Type u_10
Ω' : Type u_11
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
μ : Measure Ω
ν : Measure Ω'
inst✝³ : IsProbabilityMeasure μ
inst✝² : IsProbabilityMeasure ν
𝓧 : Type u_12
𝓨 : Type u_13
inst✝¹ : MeasurableSpace 𝓧
inst✝ : MeasurableSpace 𝓨
X : Ω → 𝓧
Y : Ω' → 𝓨
mX : AEMeasurable X μ
mY : AEMeasurable Y ν
this : (fun ω ↦ AEMeasurable.mk X mX ω.1) ⟂ᵢ[μ.prod ν] fun ω ↦ AEMeasurable.mk Y mY ω.2
⊢ AEMeasurable.mk Y mY =ᵐ[ν] Y
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
β : ι → Type u_10
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
h : ∀ (s : Finset ι), iIndepFun (s.restrict f) μ
⊢ ∀ (s : Finset ι) {f' : ι → Set Ω}, (∀ i ∈ s, MeasurableSet (f' i)) → μ (⋂ i ∈ s, f' i) = ∏ i ∈ s, μ (f' i)

Tactic:
]

Goal state after:
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
β : ι → Type u_10
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
h : ∀ (s : Finset ι), iIndepFun (s.restrict f) μ
⊢ ∀ (s : Finset ι) {f' : ι → Set Ω}, (∀ i ∈ s, MeasurableSet (f' i)) → μ (⋂ i ∈ s, f' i) = ∏ i ∈ s, μ (f' i)
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
f' : Ω → β
g' : Ω → β'
hfg : IndepFun f g κ μ
hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[κ a] f'
hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[κ a] g'
A : Set β
hA : MeasurableSet A
B : Set β'
hB : MeasurableSet B
a : α
hf' : f =ᵐ[κ a] f'
hg' : g =ᵐ[κ a] g'
hfg' : (κ a) (f ⁻¹' A ∩ g ⁻¹' B) = (κ a) (f ⁻¹' A) * (κ a) (g ⁻¹' B)
h1 : f ⁻¹' A =ᵐ[κ a] f' ⁻¹' A
h2 : g ⁻¹' B =ᵐ[κ a] g' ⁻¹' B
⊢ (κ a) (f ⁻¹' A ∩ g ⁻¹' B) = (κ a) (f ⁻¹' A) * (κ a) (g ⁻¹' B)

Tactic:
]

Goal state after:
α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
f' : Ω → β
g' : Ω → β'
hfg : IndepFun f g κ μ
hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[κ a] f'
hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[κ a] g'
A : Set β
hA : MeasurableSet A
B : Set β'
hB : MeasurableSet B
a : α
hf' : f =ᵐ[κ a] f'
hg' : g =ᵐ[κ a] g'
hfg' : (κ a) (f ⁻¹' A ∩ g ⁻¹' B) = (κ a) (f ⁻¹' A) * (κ a) (g ⁻¹' B)
h1 : f ⁻¹' A =ᵐ[κ a] f' ⁻¹' A
h2 : g ⁻¹' B =ᵐ[κ a] g' ⁻¹' B
⊢ (κ a) (f ⁻¹' A ∩ g ⁻¹' B) = (κ a) (f ⁻¹' A) * (κ a) (g ⁻¹' B)
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
β : ι → Type u_10
m : (i : ι) → MeasurableSpace (β i)
f✝ : (i : ι) → Ω → β i
h : ∀ (s : Finset ι), iIndepFun (s.restrict f✝) μ
s : Finset ι
f : ι → Set Ω
hs : ∀ i ∈ s, MeasurableSet (f i)
this : ⋂ i ∈ s, f i = ⋂ i, f ↑i
⊢ μ (⋂ i, f ↑i) = ∏ i, μ (f ↑i)

Tactic:
]

Goal state after:
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
β : ι → Type u_10
m : (i : ι) → MeasurableSpace (β i)
f✝ : (i : ι) → Ω → β i
h : ∀ (s : Finset ι), iIndepFun (s.restrict f✝) μ
s : Finset ι
f : ι → Set Ω
hs : ∀ i ∈ s, MeasurableSet (f i)
this : ⋂ i ∈ s, f i = ⋂ i, f ↑i
⊢ μ (⋂ i, f ↑i) = ∏ i, μ (f ↑i)
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : Fintype ι
β : ι → Type u_11
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf : ∀ (i : ι), AEMeasurable (f i) μ
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → μ (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, μ (f i ⁻¹' sets i)
this : IsProbabilityMeasure μ
⊢ Measure.map (fun ω i ↦ f i ω) μ = Measure.pi fun i ↦ Measure.map (f i) μ

Tactic:
]

Goal state after:
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : Fintype ι
β : ι → Type u_11
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf : ∀ (i : ι), AEMeasurable (f i) μ
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → μ (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, μ (f i ⁻¹' sets i)
this : IsProbabilityMeasure μ
⊢ Measure.map (fun ω i ↦ f i ω) μ = Measure.pi fun i ↦ Measure.map (f i) μ
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m1 m2 m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h1 : m1 ≤ m
h2 : m2 ≤ m
hp1 : IsPiSystem p1
hp2 : IsPiSystem p2
hpm1 : m1 = generateFrom p1
hpm2 : m2 = generateFrom p2
hyp : IndepSets p1 p2 κ μ
h : IsMarkovKernel κ
t1 t2 : Set Ω
ht2 : t2 ∈ {s | MeasurableSet s}
t : Set Ω
ht : MeasurableSet t
iht : ∀ᵐ (a : α) ∂μ, (κ a) (t ∩ t2) = (κ a) t * (κ a) t2
a : α
ha : (κ a) (t ∩ t2) = (κ a) t * (κ a) t2
this : tᶜ ∩ t2 = t2 \ (t ∩ t2)
⊢ (κ a) t2 - (κ a) t2 * (κ a) t = (κ a) t2 - (κ a) t2 * (κ a) t

Tactic:
]

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

Goal state before:
pp: case left.e_f
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : Fintype ι
β : ι → Type u_11
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf : ∀ (i : ι), AEMeasurable (f i) μ
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → μ (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, μ (f i ⁻¹' sets i)
this : IsProbabilityMeasure μ
s : (i : ι) → Set (β i)
hm : ∀ (i : ι), MeasurableSet (s i)
x : ι
⊢ μ (f x ⁻¹' s x) = μ (f x ⁻¹' s x)

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: case inr.iUnion
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m1 m2 m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h1 : m1 ≤ m
h2 : m2 ≤ m
hp1 : IsPiSystem p1
hp2 : IsPiSystem p2
hpm1 : m1 = generateFrom p1
hpm2 : m2 = generateFrom p2
hyp : IndepSets p1 p2 κ μ
h✝ : IsMarkovKernel κ
t1 t2 : Set Ω
ht2 : t2 ∈ {s | MeasurableSet s}
f : ℕ → Set Ω
hf_disj : Pairwise (Function.onFun Disjoint f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), (κ a) (f i ∩ t2) = (κ a) (f i) * (κ a) t2
⊢ ∀ᵐ (a : α) ∂μ, (κ a) ((⋃ i, f i) ∩ t2) = (κ a) (⋃ i, f i) * (κ a) t2

Tactic:
]

Goal state after:
case inr.iUnion
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m1 m2 m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h1 : m1 ≤ m
h2 : m2 ≤ m
hp1 : IsPiSystem p1
hp2 : IsPiSystem p2
hpm1 : m1 = generateFrom p1
hpm2 : m2 = generateFrom p2
hyp : IndepSets p1 p2 κ μ
h✝ : IsMarkovKernel κ
t1 t2 : Set Ω
ht2 : t2 ∈ {s | MeasurableSet s}
f : ℕ → Set Ω
hf_disj : Pairwise (Function.onFun Disjoint f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), (κ a) (f i ∩ t2) = (κ a) (f i) * (κ a) t2
⊢ ∀ᵐ (a : α) ∂μ, (κ a) ((⋃ i, f i) ∩ t2) = (κ a) (⋃ i, f i) * (κ a) t2
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: case right
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : Fintype ι
β : ι → Type u_11
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf : ∀ (i : ι), AEMeasurable (f i) μ
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → μ (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, μ (f i ⁻¹' sets i)
this : IsProbabilityMeasure μ
s : (i : ι) → Set (β i)
hm : ∀ (i : ι), MeasurableSet (s i)
⊢ μ (⋂ i, f i ⁻¹' s i) = μ ((fun c x ↦ f x c) ⁻¹' univ.pi s)

Tactic:
]

Goal state after:
case right
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : Fintype ι
β : ι → Type u_11
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf : ∀ (i : ι), AEMeasurable (f i) μ
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → μ (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, μ (f i ⁻¹' sets i)
this : IsProbabilityMeasure μ
s : (i : ι) → Set (β i)
hm : ∀ (i : ι), MeasurableSet (s i)
⊢ μ (⋂ i, f i ⁻¹' s i) = μ ((fun c x ↦ f x c) ⁻¹' univ.pi s)
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
ι : Type u_2
μ : ProbabilityMeasure Ω
μs : ι → ProbabilityMeasure Ω
L : Filter ι
inst✝ : L.IsCountablyGenerated
h_opens : ∀ (G : Set Ω), IsOpen[inst✝²] G → μ G ≤ liminf (fun i ↦ (μs i) G) L
u : ℕ → ι
hu : Tendsto u atTop L
G : Set Ω
hG : IsOpen[inst✝²] G
⊢ liminf (fun i ↦ (μs i) G) L ≤ liminf (fun i ↦ (μs i) G) (map u atTop)

Tactic:
]

Goal state after:
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
ι : Type u_2
μ : ProbabilityMeasure Ω
μs : ι → ProbabilityMeasure Ω
L : Filter ι
inst✝ : L.IsCountablyGenerated
h_opens : ∀ (G : Set Ω), IsOpen[inst✝²] G → μ G ≤ liminf (fun i ↦ (μs i) G) L
u : ℕ → ι
hu : Tendsto u atTop L
G : Set Ω
hG : IsOpen[inst✝²] G
⊢ liminf (fun i ↦ (μs i) G) L ≤ liminf (fun i ↦ (μs i) G) (map u atTop)
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : Fintype ι
β : ι → Type u_11
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf : ∀ (i : ι), AEMeasurable (f i) μ
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → μ (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, μ (f i ⁻¹' sets i)
this : IsProbabilityMeasure μ
h₀ :
  ∀ {s : (i : ι) → Set (β i)},
    (∀ (i : ι), MeasurableSet (s i)) →
      ∏ i, μ (f i ⁻¹' s i) = ∏ i, (Measure.map (f i) μ) (s i) ∧
        μ (⋂ i, f i ⁻¹' s i) = (Measure.map (fun ω i ↦ f i ω) μ) (univ.pi s)
h' : (i : ι) → Set (β i)
hm : ∀ (i : ι), MeasurableSet (h' i)
⊢ μ (⋂ i, f i ⁻¹' h' i) = ∏ i, μ (f i ⁻¹' h' i)

Tactic:
]

Goal state after:
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : Fintype ι
β : ι → Type u_11
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
hf : ∀ (i : ι), AEMeasurable (f i) μ
h :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → μ (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, μ (f i ⁻¹' sets i)
this : IsProbabilityMeasure μ
h₀ :
  ∀ {s : (i : ι) → Set (β i)},
    (∀ (i : ι), MeasurableSet (s i)) →
      ∏ i, μ (f i ⁻¹' s i) = ∏ i, (Measure.map (f i) μ) (s i) ∧
        μ (⋂ i, f i ⁻¹' s i) = (Measure.map (fun ω i ↦ f i ω) μ) (univ.pi s)
h' : (i : ι) → Set (β i)
hm : ∀ (i : ι), MeasurableSet (h' i)
⊢ μ (⋂ i, f i ⁻¹' h' i) = ∏ i, μ (f i ⁻¹' h' i)
Module: Mathlib.MeasureTheory.Measure.Portmanteau

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ι → ProbabilityMeasure Ω
L : Filter ι
inst✝ : L.IsCountablyGenerated
h : ∀ (F : Set Ω), IsClosed[inst✝²] F → limsup (fun i ↦ ↑(μs i) F) L ≤ ↑μ F
⊢ ∀ (F : Set Ω), IsClosed[inst✝²] F → limsup (fun i ↦ ↑(μs i) F) L ≤ ↑μ F

Tactic:
]

Goal state after:
Ω : Type u_1
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ι → ProbabilityMeasure Ω
L : Filter ι
inst✝ : L.IsCountablyGenerated
h : ∀ (F : Set Ω), IsClosed[inst✝²] F → limsup (fun i ↦ ↑(μs i) F) L ≤ ↑μ F
⊢ ∀ (F : Set Ω), IsClosed[inst✝²] F → limsup (fun i ↦ ↑(μs i) F) L ≤ ↑μ F
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : Fintype ι
β : ι → Type u_11
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
inst✝ : IsProbabilityMeasure μ
hf : ∀ (i : ι), AEMeasurable (f i) μ
⊢ (Measure.map (fun ω i ↦ f i ω) μ = Measure.pi fun i ↦ Measure.map (f i) μ) →
    ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
      (∀ i ∈ S, MeasurableSet (sets i)) → μ (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, μ (f i ⁻¹' sets i)

Tactic:
]

Goal state after:
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : Fintype ι
β : ι → Type u_11
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
inst✝ : IsProbabilityMeasure μ
hf : ∀ (i : ι), AEMeasurable (f i) μ
⊢ (Measure.map (fun ω i ↦ f i ω) μ = Measure.pi fun i ↦ Measure.map (f i) μ) →
    ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
      (∀ i ∈ S, MeasurableSet (sets i)) → μ (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, μ (f i ⁻¹' sets i)
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: case left.e_f
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : Fintype ι
β : ι → Type u_11
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
inst✝ : IsProbabilityMeasure μ
hf : ∀ (i : ι), AEMeasurable (f i) μ
s : (i : ι) → Set (β i)
hm : ∀ (i : ι), MeasurableSet (s i)
x : ι
⊢ μ (f x ⁻¹' s x) = μ (f x ⁻¹' s x)

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m1 m2 m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h1 : m1 ≤ m
h2 : m2 ≤ m
hp1 : IsPiSystem p1
hp2 : IsPiSystem p2
hpm1 : m1 = generateFrom p1
hpm2 : m2 = generateFrom p2
hyp : IndepSets p1 p2 κ μ
h✝ : IsMarkovKernel κ
t1 t2 : Set Ω
ht2 : t2 ∈ {s | MeasurableSet s}
f : ℕ → Set Ω
hf_disj : Pairwise (Function.onFun Disjoint f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), (κ a) (f i ∩ t2) = (κ a) (f i) * (κ a) t2
a : α
ha : ∀ (i : ℕ), (κ a) (f i ∩ t2) = (κ a) (f i) * (κ a) t2
⊢ ∑' (i : ℕ), (κ a) (t2 ∩ f i) = (∑' (i : ℕ), (κ a) (f i)) * (κ a) t2

Tactic:
]

Goal state after:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m1 m2 m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h1 : m1 ≤ m
h2 : m2 ≤ m
hp1 : IsPiSystem p1
hp2 : IsPiSystem p2
hpm1 : m1 = generateFrom p1
hpm2 : m2 = generateFrom p2
hyp : IndepSets p1 p2 κ μ
h✝ : IsMarkovKernel κ
t1 t2 : Set Ω
ht2 : t2 ∈ {s | MeasurableSet s}
f : ℕ → Set Ω
hf_disj : Pairwise (Function.onFun Disjoint f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
h : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), (κ a) (f i ∩ t2) = (κ a) (f i) * (κ a) t2
a : α
ha : ∀ (i : ℕ), (κ a) (f i ∩ t2) = (κ a) (f i) * (κ a) t2
⊢ ∑' (i : ℕ), (κ a) (t2 ∩ f i) = (∑' (i : ℕ), (κ a) (f i)) * (κ a) t2