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

Goal state before:
pp: Ω : Type u_1
S : Type u_2
T : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
F : T → Type u_5
inst✝¹⁰ : (s : S) → TopologicalSpace (E s)
inst✝⁹ : (s : S) → MeasurableSpace (E s)
inst✝⁸ : ∀ (s : S), BorelSpace (E s)
inst✝⁷ : ∀ (s : S), HasOuterApproxClosed (E s)
inst✝⁶ : (t : T) → TopologicalSpace (F t)
inst✝⁵ : (t : T) → MeasurableSpace (F t)
inst✝⁴ : ∀ (t : T), BorelSpace (F t)
inst✝³ : ∀ (t : T), HasOuterApproxClosed (F t)
X : (s : S) → Ω → E s
Y : (t : T) → Ω → F t
inst✝² : IsFiniteMeasure P
inst✝¹ : Finite S
inst✝ : Finite T
mX : ∀ (s : S), AEMeasurable (X s) P
mY : ∀ (t : T), AEMeasurable (Y t) P
h :
  ∀ (f : ((s : S) → E s) →ᵇ ℝ) (g : ((t : T) → F t) →ᵇ ℝ),
    ∫ (x : Ω), (fun ω ↦ (f fun x ↦ X x ω) * g fun x ↦ Y x ω) x ∂P =
      (∫ (x : Ω), (fun ω ↦ f fun x ↦ X x ω) x ∂P) * ∫ (x : Ω), (fun ω ↦ g fun x ↦ Y x ω) x ∂P
this✝ : Fintype S
this : Fintype T
f : (s : S) → E s →ᵇ ℝ
g : (t : T) → F t →ᵇ ℝ
t : T
⊢ Continuous (Function.eval t)

Tactic:
by fun_prop

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

Goal state before:
pp: Ω : Type u_2
ι : Type u_3
s : ι → MeasurableSpace Ω
m m0 : MeasurableSpace Ω
μ : Measure Ω
β : Type u_4
p : Set ι → Prop
f : Filter ι
ns : β → Set ι
inst✝¹ : StandardBorelSpace Ω
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iCondIndep m hm s μ
hf : ∀ (t : Set ι), p t → tᶜ ∈ f
hns : Directed (fun x1 x2 ↦ x1 ⊆ x2) ns
hnsp : ∀ (a : β), p (ns a)
hns_univ : ∀ (n : ι), ∃ a, n ∈ ns a
t : Set Ω
ht_tail : MeasurableSet t
⊢ ∀ᵐ (ω : Ω) ∂μ, μ[t.indicator fun ω ↦ 1 | m] ω = 0 ∨ μ[t.indicator fun ω ↦ 1 | m] ω = 1

Tactic:
have h := ae_of_ae_trim hm
    (Kernel.measure_zero_or_one_of_measurableSet_limsup h_le h_indep hf hns hnsp hns_univ ht_tail)

Goal state after:
Ω : Type u_2
ι : Type u_3
s : ι → MeasurableSpace Ω
m m0 : MeasurableSpace Ω
μ : Measure Ω
β : Type u_4
p : Set ι → Prop
f : Filter ι
ns : β → Set ι
inst✝¹ : StandardBorelSpace Ω
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iCondIndep m hm s μ
hf : ∀ (t : Set ι), p t → tᶜ ∈ f
hns : Directed (fun x1 x2 ↦ x1 ⊆ x2) ns
hnsp : ∀ (a : β), p (ns a)
hns_univ : ∀ (n : ι), ∃ a, n ∈ ns a
t : Set Ω
ht_tail : MeasurableSet t
h : ∀ᵐ (x : Ω) ∂μ, ((condExpKernel μ m) x) t = 0 ∨ ((condExpKernel μ m) x) t = 1
⊢ ∀ᵐ (ω : Ω) ∂μ, μ[t.indicator fun ω ↦ 1 | m] ω = 0 ∨ μ[t.indicator fun ω ↦ 1 | m] ω = 1
Module: Mathlib.Probability.Independence.ZeroOne

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
inst✝² : SemilatticeSup ι
inst✝¹ : NoMaxOrder ι
inst✝ : Nonempty ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
⊢ Indep (limsup s atTop) (limsup s atTop) κ μα

Tactic:
let ns : ι → Set ι := Set.Iic

Goal state after:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
inst✝² : SemilatticeSup ι
inst✝¹ : NoMaxOrder ι
inst✝ : Nonempty ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
ns : ι → Set ι := Set.Iic
⊢ Indep (limsup s atTop) (limsup s atTop) κ μα
Module: Mathlib.Probability.Kernel.Category.Stoch

Goal state before:
pp: X : SFinKer
⊢ Measurable Prod.snd

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.Stoch

Goal state before:
pp: X : SFinKer
⊢ Measurable fun x ↦ (PUnit.unit, x)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.Stoch

Goal state before:
pp: X : SFinKer
⊢ Measurable Prod.fst

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.Stoch

Goal state before:
pp: X : SFinKer
⊢ Measurable fun x ↦ (x, PUnit.unit)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X Y Z : SFinKer
⊢ Measurable ⇑prodAssoc

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X Y Z : SFinKer
⊢ Measurable ⇑prodAssoc.symm

Tactic:
by fun_prop

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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
π : ι → Set (Set Ω)
hπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
h_eq' :
  ∀ (s : Finset ι) (f : ι → Set Ω),
    (∀ i ∈ s, f i ∈ π i) →
      ∀ i ∈ s, (fun ω ↦ (((condExpKernel μ m') ω) (f i)).toReal) =ᵐ[μ] μ[(f i).indicator fun ω ↦ 1 | m']
h_eq✝ :
  ∀ (s : Finset ι) (f : ι → Set Ω),
    (∀ i ∈ s, f i ∈ π i) →
      ∀ᵐ (ω : Ω) ∂μ, ∀ i ∈ s, (((condExpKernel μ m') ω) (f i)).toReal = μ[(f i).indicator fun ω ↦ 1 | m'] ω
h_inter_eq✝ :
  ∀ (s : Finset ι) (f : ι → Set Ω),
    (∀ i ∈ s, f i ∈ π i) →
      (fun ω ↦ (((condExpKernel μ m') ω) (⋂ i ∈ s, f i)).toReal) =ᵐ[μ] μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m']
s : Finset ι
f : ι → Set Ω
h✝ : μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ i ∈ s, μ[(f i).indicator fun ω ↦ 1 | m']
hf : ∀ i ∈ s, f i ∈ π i
ω : Ω
h_eq : ∀ i ∈ s, (((condExpKernel μ m') ω) (f i)).toReal = μ[(f i).indicator fun ω ↦ 1 | m'] ω
h_inter_eq : (((condExpKernel μ m') ω) (⋂ i ∈ s, f i)).toReal = μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m'] ω
h : μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m'] ω = (∏ i ∈ s, μ[(f i).indicator fun ω ↦ 1 | m']) ω
h_ne_top : ((condExpKernel μ m') ω) (⋂ i ∈ s, f i) ≠ ∞
this : ∏ i ∈ s, ((condExpKernel μ m') ω) (f i) ≠ ∞
⊢ ENNReal.ofReal (∏ c ∈ s, μ[(f c).indicator fun ω ↦ 1 | m'] ω) =
    ENNReal.ofReal (∏ i ∈ s, (((condExpKernel μ m') ω) (f i)).toReal)

Tactic:
congr 1

Goal state after:
Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
π : ι → Set (Set Ω)
hπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
h_eq' :
  ∀ (s : Finset ι) (f : ι → Set Ω),
    (∀ i ∈ s, f i ∈ π i) →
      ∀ i ∈ s, (fun ω ↦ (((condExpKernel μ m') ω) (f i)).toReal) =ᵐ[μ] μ[(f i).indicator fun ω ↦ 1 | m']
h_eq✝ :
  ∀ (s : Finset ι) (f : ι → Set Ω),
    (∀ i ∈ s, f i ∈ π i) →
      ∀ᵐ (ω : Ω) ∂μ, ∀ i ∈ s, (((condExpKernel μ m') ω) (f i)).toReal = μ[(f i).indicator fun ω ↦ 1 | m'] ω
h_inter_eq✝ :
  ∀ (s : Finset ι) (f : ι → Set Ω),
    (∀ i ∈ s, f i ∈ π i) →
      (fun ω ↦ (((condExpKernel μ m') ω) (⋂ i ∈ s, f i)).toReal) =ᵐ[μ] μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m']
s : Finset ι
f : ι → Set Ω
h✝ : μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ i ∈ s, μ[(f i).indicator fun ω ↦ 1 | m']
hf : ∀ i ∈ s, f i ∈ π i
ω : Ω
h_eq : ∀ i ∈ s, (((condExpKernel μ m') ω) (f i)).toReal = μ[(f i).indicator fun ω ↦ 1 | m'] ω
h_inter_eq : (((condExpKernel μ m') ω) (⋂ i ∈ s, f i)).toReal = μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m'] ω
h : μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m'] ω = (∏ i ∈ s, μ[(f i).indicator fun ω ↦ 1 | m']) ω
h_ne_top : ((condExpKernel μ m') ω) (⋂ i ∈ s, f i) ≠ ∞
this : ∏ i ∈ s, ((condExpKernel μ m') ω) (f i) ≠ ∞
⊢ ∏ c ∈ s, μ[(f c).indicator fun ω ↦ 1 | m'] ω = ∏ i ∈ s, (((condExpKernel μ m') ω) (f i)).toReal
Module: Mathlib.Probability.Independence.BoundedContinuousFunction

Goal state before:
pp: Ω : Type u_1
T : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
F : T → Type u_5
G : Type u_6
inst✝⁹ : (t : T) → TopologicalSpace (F t)
inst✝⁸ : (t : T) → MeasurableSpace (F t)
inst✝⁷ : ∀ (t : T), BorelSpace (F t)
inst✝⁶ : ∀ (t : T), HasOuterApproxClosed (F t)
inst✝⁵ : TopologicalSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : HasOuterApproxClosed G
Y : (t : T) → Ω → F t
Z : Ω → G
inst✝¹ : Fintype T
inst✝ : IsFiniteMeasure P
mZ : AEMeasurable Z P
mY : ∀ (t : T), AEMeasurable (Y t) P
h :
  ∀ (f : G →ᵇ ℝ) (g : (t : T) → F t →ᵇ ℝ),
    ∫ (x : Ω), (⇑f ∘ Z * ∏ t, ⇑(g t) ∘ Y t) x ∂P = (∫ (x : Ω), (⇑f ∘ Z) x ∂P) * ∫ (x : Ω), (∏ t, ⇑(g t) ∘ Y t) x ∂P
f : G →ᵇ ℝ
g : (j : T) → F j →ᵇ ℝ
⊢ Measurable fun y ↦ ∏ j, (g j) (y j)

Tactic:
by fun_prop

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

Goal state before:
pp: Ω : Type u_1
T : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
F : T → Type u_5
G : Type u_6
inst✝⁹ : (t : T) → TopologicalSpace (F t)
inst✝⁸ : (t : T) → MeasurableSpace (F t)
inst✝⁷ : ∀ (t : T), BorelSpace (F t)
inst✝⁶ : ∀ (t : T), HasOuterApproxClosed (F t)
inst✝⁵ : TopologicalSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : HasOuterApproxClosed G
Y : (t : T) → Ω → F t
Z : Ω → G
inst✝¹ : Fintype T
inst✝ : IsFiniteMeasure P
mZ : AEMeasurable Z P
mY : ∀ (t : T), AEMeasurable (Y t) P
h :
  ∀ (f : G →ᵇ ℝ) (g : (t : T) → F t →ᵇ ℝ),
    ∫ (x : Ω), (⇑f ∘ Z * ∏ t, ⇑(g t) ∘ Y t) x ∂P = (∫ (x : Ω), (⇑f ∘ Z) x ∂P) * ∫ (x : Ω), (∏ t, ⇑(g t) ∘ Y t) x ∂P
f : G →ᵇ ℝ
g : (j : T) → F j →ᵇ ℝ
⊢ Measurable fun p ↦ f p.1 * ∏ j, (g j) (p.2 j)

Tactic:
by fun_prop

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

Goal state before:
pp: Ω : Type u_1
T : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
F : T → Type u_5
G : Type u_6
inst✝⁹ : (t : T) → TopologicalSpace (F t)
inst✝⁸ : (t : T) → MeasurableSpace (F t)
inst✝⁷ : ∀ (t : T), BorelSpace (F t)
inst✝⁶ : ∀ (t : T), HasOuterApproxClosed (F t)
inst✝⁵ : TopologicalSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : HasOuterApproxClosed G
Y : (t : T) → Ω → F t
Z : Ω → G
inst✝¹ : IsFiniteMeasure P
inst✝ : Finite T
mZ : AEMeasurable Z P
mY : ∀ (t : T), AEMeasurable (Y t) P
h :
  ∀ (f : G →ᵇ ℝ) (g : ((t : T) → F t) →ᵇ ℝ),
    ∫ (x : Ω), (fun ω ↦ f (Z ω) * g fun x ↦ Y x ω) x ∂P =
      (∫ (x : Ω), (⇑f ∘ Z) x ∂P) * ∫ (x : Ω), (fun ω ↦ g fun x ↦ Y x ω) x ∂P
this : Fintype T
f : G →ᵇ ℝ
g : (t : T) → F t →ᵇ ℝ
t : T
⊢ Continuous (Function.eval t)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X₁✝ X₂✝ X₃✝ Y₁✝ Y₂✝ Y₃✝ : SFinKer
κ₁ : X₁✝ ⟶ Y₁✝
κ₂ : X₂✝ ⟶ Y₂✝
η : X₃✝ ⟶ Y₃✝
a✝ : (X₁✝.carrier × X₂✝.carrier) × X₃✝.carrier
s✝ : Set (Y₁✝.carrier × Y₂✝.carrier × Y₃✝.carrier)
hs : MeasurableSet s✝
⊢ Measurable ⇑prodAssoc

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Proper

Goal state before:
pp: case refine_2
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f₁ f₂ : X → ℝ≥0∞
a✝¹ : Measurable f₁
a✝ : Measurable f₂
hf₁ : ∫⁻ (x : X), B.indicator 1 x * f₁ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₁ x ∂π x₀
hf₂ : ∫⁻ (x : X), B.indicator 1 x * f₂ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₂ x ∂π x₀
⊢ ∫⁻ (x : X), B.indicator 1 x * f₁ x + B.indicator 1 x * f₂ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₁ x + f₂ x ∂π x₀

Tactic:
lintegral_add_right,

Goal state after:
case refine_2
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f₁ f₂ : X → ℝ≥0∞
a✝¹ : Measurable f₁
a✝ : Measurable f₂
hf₁ : ∫⁻ (x : X), B.indicator 1 x * f₁ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₁ x ∂π x₀
hf₂ : ∫⁻ (x : X), B.indicator 1 x * f₂ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₂ x ∂π x₀
⊢ ∫⁻ (a : X), B.indicator 1 a * f₁ a ∂π x₀ + ∫⁻ (a : X), B.indicator 1 a * f₂ a ∂π x₀ =
    B.indicator 1 x₀ * ∫⁻ (x : X), f₁ x + f₂ x ∂π x₀

case refine_2.hg
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f₁ f₂ : X → ℝ≥0∞
a✝¹ : Measurable f₁
a✝ : Measurable f₂
hf₁ : ∫⁻ (x : X), B.indicator 1 x * f₁ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₁ x ∂π x₀
hf₂ : ∫⁻ (x : X), B.indicator 1 x * f₂ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₂ x ∂π x₀
⊢ Measurable fun x ↦ B.indicator 1 x * f₂ x
Module: Mathlib.Probability.Kernel.Proper

Goal state before:
pp: case refine_2
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f₁ f₂ : X → ℝ≥0∞
a✝¹ : Measurable f₁
a✝ : Measurable f₂
hf₁ : ∫⁻ (x : X), B.indicator 1 x * f₁ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₁ x ∂π x₀
hf₂ : ∫⁻ (x : X), B.indicator 1 x * f₂ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₂ x ∂π x₀
⊢ ∫⁻ (a : X), B.indicator 1 a * f₁ a ∂π x₀ + ∫⁻ (a : X), B.indicator 1 a * f₂ a ∂π x₀ =
    B.indicator 1 x₀ * ∫⁻ (x : X), f₁ x + f₂ x ∂π x₀

pp: case refine_2.hg
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f₁ f₂ : X → ℝ≥0∞
a✝¹ : Measurable f₁
a✝ : Measurable f₂
hf₁ : ∫⁻ (x : X), B.indicator 1 x * f₁ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₁ x ∂π x₀
hf₂ : ∫⁻ (x : X), B.indicator 1 x * f₂ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₂ x ∂π x₀
⊢ Measurable fun x ↦ B.indicator 1 x * f₂ x

Tactic:
lintegral_add_right,

Goal state after:
case refine_2
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f₁ f₂ : X → ℝ≥0∞
a✝¹ : Measurable f₁
a✝ : Measurable f₂
hf₁ : ∫⁻ (x : X), B.indicator 1 x * f₁ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₁ x ∂π x₀
hf₂ : ∫⁻ (x : X), B.indicator 1 x * f₂ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₂ x ∂π x₀
⊢ ∫⁻ (a : X), B.indicator 1 a * f₁ a ∂π x₀ + ∫⁻ (a : X), B.indicator 1 a * f₂ a ∂π x₀ =
    B.indicator 1 x₀ * (∫⁻ (a : X), f₁ a ∂π x₀ + ∫⁻ (a : X), f₂ a ∂π x₀)

case refine_2.hg
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f₁ f₂ : X → ℝ≥0∞
a✝¹ : Measurable f₁
a✝ : Measurable f₂
hf₁ : ∫⁻ (x : X), B.indicator 1 x * f₁ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₁ x ∂π x₀
hf₂ : ∫⁻ (x : X), B.indicator 1 x * f₂ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₂ x ∂π x₀
⊢ Measurable f₂

case refine_2.hg
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f₁ f₂ : X → ℝ≥0∞
a✝¹ : Measurable f₁
a✝ : Measurable f₂
hf₁ : ∫⁻ (x : X), B.indicator 1 x * f₁ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₁ x ∂π x₀
hf₂ : ∫⁻ (x : X), B.indicator 1 x * f₂ x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f₂ x ∂π x₀
⊢ Measurable fun x ↦ B.indicator 1 x * f₂ x
Module: Mathlib.Probability.Independence.BoundedContinuousFunction

Goal state before:
pp: Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
H : Type u_7
inst✝⁹ : (s : S) → TopologicalSpace (E s)
inst✝⁸ : (s : S) → MeasurableSpace (E s)
inst✝⁷ : ∀ (s : S), BorelSpace (E s)
inst✝⁶ : ∀ (s : S), HasOuterApproxClosed (E s)
inst✝⁵ : TopologicalSpace H
inst✝⁴ : MeasurableSpace H
inst✝³ : BorelSpace H
inst✝² : HasOuterApproxClosed H
X : (s : S) → Ω → E s
U : Ω → H
inst✝¹ : Fintype S
inst✝ : IsFiniteMeasure P
mX : ∀ (s : S), AEMeasurable (X s) P
mU : AEMeasurable U P
h :
  ∀ (f : (s : S) → E s →ᵇ ℝ) (g : H →ᵇ ℝ),
    ∫ (x : Ω), ((∏ s, ⇑(f s) ∘ X s) * ⇑g ∘ U) x ∂P = (∫ (x : Ω), (∏ s, ⇑(f s) ∘ X s) x ∂P) * ∫ (x : Ω), (⇑g ∘ U) x ∂P
f : (i : S) → E i →ᵇ ℝ
g : H →ᵇ ℝ
⊢ Measurable fun x ↦ ∏ i, (f i) (x i)

Tactic:
by fun_prop

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

Goal state before:
pp: Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
H : Type u_7
inst✝⁹ : (s : S) → TopologicalSpace (E s)
inst✝⁸ : (s : S) → MeasurableSpace (E s)
inst✝⁷ : ∀ (s : S), BorelSpace (E s)
inst✝⁶ : ∀ (s : S), HasOuterApproxClosed (E s)
inst✝⁵ : TopologicalSpace H
inst✝⁴ : MeasurableSpace H
inst✝³ : BorelSpace H
inst✝² : HasOuterApproxClosed H
X : (s : S) → Ω → E s
U : Ω → H
inst✝¹ : Fintype S
inst✝ : IsFiniteMeasure P
mX : ∀ (s : S), AEMeasurable (X s) P
mU : AEMeasurable U P
h :
  ∀ (f : (s : S) → E s →ᵇ ℝ) (g : H →ᵇ ℝ),
    ∫ (x : Ω), ((∏ s, ⇑(f s) ∘ X s) * ⇑g ∘ U) x ∂P = (∫ (x : Ω), (∏ s, ⇑(f s) ∘ X s) x ∂P) * ∫ (x : Ω), (⇑g ∘ U) x ∂P
f : (i : S) → E i →ᵇ ℝ
g : H →ᵇ ℝ
⊢ Measurable fun p ↦ (∏ i, (f i) (p.1 i)) * g p.2

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X✝ Y✝ : SFinKer
κ : X✝ ⟶ Y✝
⊢ Measurable Prod.snd

Tactic:
by fun_prop

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

Goal state before:
pp: Ω : Type u_1
S : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
E : S → Type u_4
H : Type u_7
inst✝⁹ : (s : S) → TopologicalSpace (E s)
inst✝⁸ : (s : S) → MeasurableSpace (E s)
inst✝⁷ : ∀ (s : S), BorelSpace (E s)
inst✝⁶ : ∀ (s : S), HasOuterApproxClosed (E s)
inst✝⁵ : TopologicalSpace H
inst✝⁴ : MeasurableSpace H
inst✝³ : BorelSpace H
inst✝² : HasOuterApproxClosed H
X : (s : S) → Ω → E s
U : Ω → H
inst✝¹ : IsFiniteMeasure P
inst✝ : Finite S
mX : ∀ (s : S), AEMeasurable (X s) P
mU : AEMeasurable U P
h :
  ∀ (f : ((s : S) → E s) →ᵇ ℝ) (g : H →ᵇ ℝ),
    ∫ (x : Ω), (fun ω ↦ (f fun x ↦ X x ω) * g (U ω)) x ∂P =
      (∫ (x : Ω), (fun ω ↦ f fun x ↦ X x ω) x ∂P) * ∫ (x : Ω), (⇑g ∘ U) x ∂P
this : Fintype S
f : (s : S) → E s →ᵇ ℝ
g : H →ᵇ ℝ
s : S
⊢ Continuous (Function.eval s)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X✝ Y✝ : SFinKer
κ : X✝ ⟶ Y✝
a✝ : PUnit.{u + 1} × X✝.carrier
s✝ : Set Y✝.carrier
hs : MeasurableSet s✝
this : IsSFiniteKernel κ.hom
⊢ Measurable Prod.snd

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Proper

Goal state before:
pp: case refine_3
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f' : ℕ → X → ℝ≥0∞
hf'_meas : ∀ (n : ℕ), Measurable (f' n)
hf'_mono : Monotone f'
hf' : ∀ (n : ℕ), ∫⁻ (x : X), B.indicator 1 x * f' n x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f' n x ∂π x₀
⊢ ∫⁻ (x : X), ⨆ i, B.indicator 1 x * f' i x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), ⨆ n, f' n x ∂π x₀

Tactic:
rw [lintegral_iSup (by measurability), lintegral_iSup hf'_meas hf'_mono, ENNReal.mul_iSup]

Goal state after:
case refine_3
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f' : ℕ → X → ℝ≥0∞
hf'_meas : ∀ (n : ℕ), Measurable (f' n)
hf'_mono : Monotone f'
hf' : ∀ (n : ℕ), ∫⁻ (x : X), B.indicator 1 x * f' n x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f' n x ∂π x₀
⊢ ⨆ n, ∫⁻ (a : X), B.indicator 1 a * f' n a ∂π x₀ = ⨆ i, B.indicator 1 x₀ * ∫⁻ (a : X), f' i a ∂π x₀

case refine_3
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
B : Set X
f : X → ℝ≥0∞
x₀ : X
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hB : MeasurableSet B
f' : ℕ → X → ℝ≥0∞
hf'_meas : ∀ (n : ℕ), Measurable (f' n)
hf'_mono : Monotone f'
hf' : ∀ (n : ℕ), ∫⁻ (x : X), B.indicator 1 x * f' n x ∂π x₀ = B.indicator 1 x₀ * ∫⁻ (x : X), f' n x ∂π x₀
⊢ Monotone fun i x ↦ B.indicator 1 x * f' i x
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X✝ Y✝ : SFinKer
κ : X✝ ⟶ Y✝
⊢ Measurable Prod.fst

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X✝ Y✝ : SFinKer
κ : X✝ ⟶ Y✝
a✝ : X✝.carrier × PUnit.{u + 1}
s✝ : Set Y✝.carrier
hs : MeasurableSet s✝
this : IsSFiniteKernel κ.hom
⊢ Measurable Prod.fst

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: W X Y Z : SFinKer
⊢ Measurable id

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: W X Y Z : SFinKer
⊢ Measurable ⇑prodAssoc

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X Y : SFinKer
⊢ Measurable id

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X Y : SFinKer
⊢ Measurable Prod.snd

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X Y : SFinKer
⊢ Measurable Prod.fst

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X Y : SFinKer
⊢ Measurable (Prod.snd ∘ id)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X Y : SFinKer
⊢ Measurable (Prod.fst ∘ id)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Proper

Goal state before:
pp: case refine_2
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
f g : X → ℝ≥0∞
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hg : Measurable g
x₀ : X
g₁ g₂ : X → ℝ≥0∞
a✝ : Measurable g₁
hg₂_meas : Measurable g₂
hg₁ : ∫⁻ (x : X), g₁ x * f x ∂π x₀ = g₁ x₀ * ∫⁻ (x : X), f x ∂π x₀
hg₂ : ∫⁻ (x : X), g₂ x * f x ∂π x₀ = g₂ x₀ * ∫⁻ (x : X), f x ∂π x₀
⊢ ∫⁻ (x : X), g₁ x * f x + g₂ x * f x ∂π x₀ = g₁ x₀ * ∫⁻ (x : X), f x ∂π x₀ + g₂ x₀ * ∫⁻ (x : X), f x ∂π x₀

Tactic:
lintegral_add_right,

Goal state after:
case refine_2
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
f g : X → ℝ≥0∞
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hg : Measurable g
x₀ : X
g₁ g₂ : X → ℝ≥0∞
a✝ : Measurable g₁
hg₂_meas : Measurable g₂
hg₁ : ∫⁻ (x : X), g₁ x * f x ∂π x₀ = g₁ x₀ * ∫⁻ (x : X), f x ∂π x₀
hg₂ : ∫⁻ (x : X), g₂ x * f x ∂π x₀ = g₂ x₀ * ∫⁻ (x : X), f x ∂π x₀
⊢ ∫⁻ (a : X), g₁ a * f a ∂π x₀ + ∫⁻ (a : X), g₂ a * f a ∂π x₀ =
    g₁ x₀ * ∫⁻ (x : X), f x ∂π x₀ + g₂ x₀ * ∫⁻ (x : X), f x ∂π x₀

case refine_2.hg
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
f g : X → ℝ≥0∞
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hg : Measurable g
x₀ : X
g₁ g₂ : X → ℝ≥0∞
a✝ : Measurable g₁
hg₂_meas : Measurable g₂
hg₁ : ∫⁻ (x : X), g₁ x * f x ∂π x₀ = g₁ x₀ * ∫⁻ (x : X), f x ∂π x₀
hg₂ : ∫⁻ (x : X), g₂ x * f x ∂π x₀ = g₂ x₀ * ∫⁻ (x : X), f x ∂π x₀
⊢ Measurable fun x ↦ g₂ x * f x
Module: Mathlib.Probability.Kernel.Proper

Goal state before:
pp: case refine_2.hg
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
f g : X → ℝ≥0∞
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hg : Measurable g
x₀ : X
g₁ g₂ : X → ℝ≥0∞
a✝ : Measurable g₁
hg₂_meas : Measurable g₂
hg₁ : ∫⁻ (x : X), g₁ x * f x ∂π x₀ = g₁ x₀ * ∫⁻ (x : X), f x ∂π x₀
hg₂ : ∫⁻ (x : X), g₂ x * f x ∂π x₀ = g₂ x₀ * ∫⁻ (x : X), f x ∂π x₀
⊢ Measurable fun x ↦ g₂ x * f x

Tactic:
exact (hg₂_meas.mono h𝓑𝓧 le_rfl).mul hf

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X : SFinKer
⊢ Measurable fun x ↦ (PUnit.unit, x)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: X : SFinKer
⊢ Measurable fun x ↦ (x, PUnit.unit)

Tactic:
by fun_prop

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
G : Type u_6
H : Type u_7
inst✝⁸ : TopologicalSpace G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : HasOuterApproxClosed G
inst✝⁴ : TopologicalSpace H
inst✝³ : MeasurableSpace H
inst✝² : BorelSpace H
inst✝¹ : HasOuterApproxClosed H
Z : Ω → G
U : Ω → H
inst✝ : IsFiniteMeasure P
mZ : AEMeasurable Z P
mU : AEMeasurable U P
h :
  ∀ (f : G →ᵇ ℝ) (g : H →ᵇ ℝ), ∫ (x : Ω), (⇑f ∘ Z * ⇑g ∘ U) x ∂P = (∫ (x : Ω), (⇑f ∘ Z) x ∂P) * ∫ (x : Ω), (⇑g ∘ U) x ∂P
f : G →ᵇ ℝ
g : H →ᵇ ℝ
⊢ Measurable fun p ↦ f p.1 * g p.2

Tactic:
by fun_prop

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

Goal state before:
pp: Ω : Type u_1
β : Type u_3
β' : Type u_4
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
hf : Measurable f
hg : Measurable g
⊢ Measurable fun ω ↦ (f ω, g ω)

Tactic:
by fun_prop

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

Goal state before:
pp: Ω : Type u_1
β : Type u_3
β' : Type u_4
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
hf : Measurable f
hg : Measurable g
⊢ Measurable (Prod.map id fun ω ↦ (f ω, g ω))

Tactic:
by fun_prop

Goal state after: