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

Goal state before:
pp: case mpr
Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
s : ι → Set Ω
hπ : ∀ (i : ι), MeasurableSet (s i)
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ (∀ (S : Finset ι), μ[(⋂ i ∈ S, s i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ i ∈ S, μ[(s i).indicator fun ω ↦ 1 | m']) →
    ∀ (s_1 : Finset ι) {f : ι → Set Ω},
      (∀ i ∈ s_1, f i = s i) →
        μ[(⋂ i ∈ s_1, s i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ x ∈ s_1, μ[(s x).indicator fun ω ↦ 1 | m']

Tactic:
·

Goal state after:
case mpr
Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
s : ι → Set Ω
hπ : ∀ (i : ι), MeasurableSet (s i)
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ (∀ (S : Finset ι), μ[(⋂ i ∈ S, s i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ i ∈ S, μ[(s i).indicator fun ω ↦ 1 | m']) →
    ∀ (s_1 : Finset ι) {f : ι → Set Ω},
      (∀ i ∈ s_1, f i = s i) →
        μ[(⋂ i ∈ s_1, s i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ x ∈ s_1, μ[(s x).indicator fun ω ↦ 1 | m']
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: case hπ
Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
s : ι → Set Ω
hπ : ∀ (i : ι), MeasurableSet (s i)
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ ∀ (i : ι), ∀ s_1 ∈ {s i}, MeasurableSet s_1

Tactic:
·

Goal state after:
case hπ
Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
s : ι → Set Ω
hπ : ∀ (i : ι), MeasurableSet (s i)
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ ∀ (i : ι), ∀ s_1 ∈ {s i}, MeasurableSet s_1
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✝
⊢ ∫⁻ (x : Y₁✝.carrier),
      ∫⁻ (y : Y₂✝.carrier), (η.hom a✝.2) (Prod.mk (x, y) ⁻¹' ⇑prodAssoc ⁻¹' s✝) ∂κ₂.hom a✝.1.2 ∂κ₁.hom a✝.1.1 =
    ∫⁻ (x : Y₁✝.carrier),
      ((κ₂.hom (prodAssoc a✝).2.1).prod (η.hom (prodAssoc a✝).2.2)) (Prod.mk x ⁻¹' s✝) ∂κ₁.hom (prodAssoc a✝).1

pp: case hf
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✝
⊢ AEMeasurable (fun x ↦ (η.hom a✝.2) (Prod.mk x ⁻¹' ⇑prodAssoc ⁻¹' s✝)) ((κ₁.hom a✝.1.1).prod (κ₂.hom a✝.1.2))

Tactic:
·

Goal state after:
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✝
⊢ ∫⁻ (x : Y₁✝.carrier),
      ∫⁻ (y : Y₂✝.carrier), (η.hom a✝.2) (Prod.mk (x, y) ⁻¹' ⇑prodAssoc ⁻¹' s✝) ∂κ₂.hom a✝.1.2 ∂κ₁.hom a✝.1.1 =
    ∫⁻ (x : Y₁✝.carrier),
      ((κ₂.hom (prodAssoc a✝).2.1).prod (η.hom (prodAssoc a✝).2.2)) (Prod.mk x ⁻¹' s✝) ∂κ₁.hom (prodAssoc a✝).1
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: Ω : Type u_1
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
s t : Set Ω
hs : MeasurableSet s
ht : MeasurableSet t
⊢ (∀ (t1 t2 : Set Ω),
      t1 ∈ {s} →
        t2 ∈ {t} →
          μ[(t1 ∩ t2).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[t1.indicator fun ω ↦ 1 | m'] * μ[t2.indicator fun ω ↦ 1 | m']) ↔
    μ[(s ∩ t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m'] * μ[t.indicator fun ω ↦ 1 | m']

pp: Ω : Type u_1
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
s t : Set Ω
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ∀ s_1 ∈ {s}, MeasurableSet s_1

pp: Ω : Type u_1
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
s t : Set Ω
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ∀ s ∈ {t}, MeasurableSet s

Tactic:
·

Goal state after:
Ω : Type u_1
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
s t : Set Ω
hs : MeasurableSet s
ht : MeasurableSet t
⊢ (∀ (t1 t2 : Set Ω),
      t1 ∈ {s} →
        t2 ∈ {t} →
          μ[(t1 ∩ t2).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[t1.indicator fun ω ↦ 1 | m'] * μ[t2.indicator fun ω ↦ 1 | m']) ↔
    μ[(s ∩ t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m'] * μ[t.indicator fun ω ↦ 1 | m']
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: Ω : Type u_1
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
s t : Set Ω
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ∀ s_1 ∈ {s}, MeasurableSet s_1

pp: Ω : Type u_1
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
s t : Set Ω
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ∀ s ∈ {t}, MeasurableSet s

Tactic:
·

Goal state after:
Ω : Type u_1
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
s t : Set Ω
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ∀ s_1 ∈ {s}, MeasurableSet s_1
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: case hf
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✝
⊢ AEMeasurable (fun x ↦ (η.hom a✝.2) (Prod.mk x ⁻¹' ⇑prodAssoc ⁻¹' s✝)) ((κ₁.hom a✝.1.1).prod (κ₂.hom a✝.1.2))

Tactic:
·

Goal state after:
case hf
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✝
⊢ AEMeasurable (fun x ↦ (η.hom a✝.2) (Prod.mk x ⁻¹' ⇑prodAssoc ⁻¹' s✝)) ((κ₁.hom a✝.1.1).prod (κ₂.hom a✝.1.2))
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: Ω : Type u_1
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
s t : Set Ω
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ∀ s ∈ {t}, MeasurableSet s

Tactic:
·

Goal state after:
Ω : Type u_1
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
s t : Set Ω
hs : MeasurableSet s
ht : MeasurableSet t
⊢ ∀ s ∈ {t}, MeasurableSet s
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∞⦄,
    (∀ (n : ℕ), Measurable (f n)) →
      Monotone f →
        (∀ (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), B.indicator 1 x * (fun x ↦ ⨆ n, f n x) x ∂π x₀ =
            B.indicator 1 x₀ * ∫⁻ (x : X), (fun x ↦ ⨆ n, f n x) x ∂π x₀

Tactic:
·

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∞⦄,
    (∀ (n : ℕ), Measurable (f n)) →
      Monotone f →
        (∀ (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), B.indicator 1 x * (fun x ↦ ⨆ n, f n x) x ∂π x₀ =
            B.indicator 1 x₀ * ∫⁻ (x : X), (fun x ↦ ⨆ n, f n x) x ∂π x₀
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
m : ι → MeasurableSpace Ω
hm : ∀ (i : ι), m i ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ (∀ (s : Finset ι) {f : ι → Set Ω},
      (∀ i ∈ s, f i ∈ {s | MeasurableSet s}) →
        μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ i ∈ s, μ[(f i).indicator fun ω ↦ 1 | m']) ↔
    ∀ (s : Finset ι) {f : ι → Set Ω},
      (∀ i ∈ s, MeasurableSet (f i)) →
        μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ i ∈ s, μ[(f i).indicator fun ω ↦ 1 | m']

pp: case hπ
Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
m : ι → MeasurableSpace Ω
hm : ∀ (i : ι), m i ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ ∀ (i : ι), ∀ s ∈ {s | MeasurableSet s}, MeasurableSet s

Tactic:
·

Goal state after:
Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
m : ι → MeasurableSpace Ω
hm : ∀ (i : ι), m i ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ (∀ (s : Finset ι) {f : ι → Set Ω},
      (∀ i ∈ s, f i ∈ {s | MeasurableSet s}) →
        μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ i ∈ s, μ[(f i).indicator fun ω ↦ 1 | m']) ↔
    ∀ (s : Finset ι) {f : ι → Set Ω},
      (∀ i ∈ s, MeasurableSet (f i)) →
        μ[(⋂ i ∈ s, f i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ i ∈ s, μ[(f i).indicator fun ω ↦ 1 | m']
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
⊢ ⨆ n, ∫⁻ (a : X), B.indicator 1 a * f' n a ∂π x₀ = ⨆ i, B.indicator 1 x₀ * ∫⁻ (a : X), f' i a ∂π x₀

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
⊢ Monotone fun i x ↦ B.indicator 1 x * f' i x

Tactic:
·

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
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: case hπ
Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
m : ι → MeasurableSpace Ω
hm : ∀ (i : ι), m i ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ ∀ (i : ι), ∀ s ∈ {s | MeasurableSet s}, MeasurableSet s

Tactic:
·

Goal state after:
case hπ
Ω : Type u_1
ι : Type u_2
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
m : ι → MeasurableSpace Ω
hm : ∀ (i : ι), m i ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ ∀ (i : ι), ∀ s ∈ {s | MeasurableSet s}, MeasurableSet s
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
⊢ Monotone fun i x ↦ B.indicator 1 x * f' i x

Tactic:
·

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
⊢ Monotone fun i x ↦ B.indicator 1 x * f' i x
Module: Mathlib.Probability.Kernel.Proper

Goal state before:
pp: case refine_1
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
⊢ ∀ (c : ℝ≥0∞) ⦃s : Set X⦄,
    MeasurableSet s →
      ∫⁻ (x : X), s.indicator (fun x ↦ c) x * f x ∂π x₀ = s.indicator (fun x ↦ c) x₀ * ∫⁻ (x : X), f x ∂π x₀

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
⊢ ∀ ⦃f_1 g : X → ℝ≥0∞⦄,
    Disjoint (Function.support f_1) (Function.support g) →
      Measurable f_1 →
        Measurable g →
          ∫⁻ (x : X), f_1 x * f x ∂π x₀ = f_1 x₀ * ∫⁻ (x : X), f x ∂π x₀ →
            ∫⁻ (x : X), g x * f x ∂π x₀ = g x₀ * ∫⁻ (x : X), f x ∂π x₀ →
              ∫⁻ (x : X), (f_1 + g) x * f x ∂π x₀ = (f_1 + g) x₀ * ∫⁻ (x : X), f x ∂π x₀

pp: case refine_3
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
⊢ ∀ ⦃f_1 : ℕ → X → ℝ≥0∞⦄,
    (∀ (n : ℕ), Measurable (f_1 n)) →
      Monotone f_1 →
        (∀ (n : ℕ), ∫⁻ (x : X), f_1 n x * f x ∂π x₀ = f_1 n x₀ * ∫⁻ (x : X), f x ∂π x₀) →
          ∫⁻ (x : X), (fun x ↦ ⨆ n, f_1 n x) x * f x ∂π x₀ = (fun x ↦ ⨆ n, f_1 n x) x₀ * ∫⁻ (x : X), f x ∂π x₀

Tactic:
·

Goal state after:
case refine_1
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
⊢ ∀ (c : ℝ≥0∞) ⦃s : Set X⦄,
    MeasurableSet s →
      ∫⁻ (x : X), s.indicator (fun x ↦ c) x * f x ∂π x₀ = s.indicator (fun x ↦ c) x₀ * ∫⁻ (x : X), f x ∂π x₀
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: Ω : Type u_1
m' m₁ m₂ mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
hm₁ : m₁ ≤ mΩ
hm₂ : m₂ ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ (∀ (t1 t2 : Set Ω),
      t1 ∈ {s | MeasurableSet s} →
        t2 ∈ {s | MeasurableSet s} →
          μ[(t1 ∩ t2).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[t1.indicator fun ω ↦ 1 | m'] * μ[t2.indicator fun ω ↦ 1 | m']) ↔
    ∀ (t1 t2 : Set Ω),
      MeasurableSet t1 →
        MeasurableSet t2 →
          μ[(t1 ∩ t2).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[t1.indicator fun ω ↦ 1 | m'] * μ[t2.indicator fun ω ↦ 1 | m']

pp: case hs1
Ω : Type u_1
m' m₁ m₂ mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
hm₁ : m₁ ≤ mΩ
hm₂ : m₂ ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ ∀ s ∈ {s | MeasurableSet s}, MeasurableSet s

pp: case hs2
Ω : Type u_1
m' m₁ m₂ mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
hm₁ : m₁ ≤ mΩ
hm₂ : m₂ ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ ∀ s ∈ {s | MeasurableSet s}, MeasurableSet s

Tactic:
·

Goal state after:
Ω : Type u_1
m' m₁ m₂ mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
hm₁ : m₁ ≤ mΩ
hm₂ : m₂ ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ (∀ (t1 t2 : Set Ω),
      t1 ∈ {s | MeasurableSet s} →
        t2 ∈ {s | MeasurableSet s} →
          μ[(t1 ∩ t2).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[t1.indicator fun ω ↦ 1 | m'] * μ[t2.indicator fun ω ↦ 1 | m']) ↔
    ∀ (t1 t2 : Set Ω),
      MeasurableSet t1 →
        MeasurableSet t2 →
          μ[(t1 ∩ t2).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[t1.indicator fun ω ↦ 1 | m'] * μ[t2.indicator fun ω ↦ 1 | m']
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: case hs1
Ω : Type u_1
m' m₁ m₂ mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
hm₁ : m₁ ≤ mΩ
hm₂ : m₂ ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ ∀ s ∈ {s | MeasurableSet s}, MeasurableSet s

pp: case hs2
Ω : Type u_1
m' m₁ m₂ mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
hm₁ : m₁ ≤ mΩ
hm₂ : m₂ ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ ∀ s ∈ {s | MeasurableSet s}, MeasurableSet s

Tactic:
·

Goal state after:
case hs1
Ω : Type u_1
m' m₁ m₂ mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
hm₁ : m₁ ≤ mΩ
hm₂ : m₂ ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ ∀ s ∈ {s | MeasurableSet s}, MeasurableSet s
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: case hs2
Ω : Type u_1
m' m₁ m₂ mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
hm₁ : m₁ ≤ mΩ
hm₂ : m₂ ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ ∀ s ∈ {s | MeasurableSet s}, MeasurableSet s

Tactic:
·

Goal state after:
case hs2
Ω : Type u_1
m' m₁ m₂ mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
hm₁ : m₁ ≤ mΩ
hm₂ : m₂ ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ ∀ s ∈ {s | MeasurableSet s}, MeasurableSet s
Module: Mathlib.Probability.Kernel.Proper

Goal state before:
pp: case refine_1.hf
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
c : ℝ≥0∞
A : Set X
hA : MeasurableSet A
⊢ Measurable fun x ↦ A.indicator 1 x * f x

Tactic:
·

Goal state after:
case refine_1.hf
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
c : ℝ≥0∞
A : Set X
hA : MeasurableSet A
⊢ Measurable fun x ↦ A.indicator 1 x * f x
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: Ω : Type u_1
m' m₁ m₂ mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
hm₁ : m₁ ≤ mΩ
hm₂ : m₂ ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ CondIndep m' m₁ m₂ hm' μ ↔
    ∀ (t1 t2 : Set Ω),
      MeasurableSet t1 →
        MeasurableSet t2 →
          μ[(t1 ∩ t2).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[t1.indicator fun ω ↦ 1 | m'] * μ[t2.indicator fun ω ↦ 1 | m']

Tactic:
by
  rw [condIndep_iff_condIndepSets, condIndepSets_iff]
  · rfl
  · exact hm₁
  · exact hm₂

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
s : ι → Set Ω
hs : ∀ (i : ι), MeasurableSet (s i)
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ iCondIndepSet m' hm' s μ ↔
    ∀ (S : Finset ι), μ[(⋂ i ∈ S, s i).indicator fun ω ↦ 1 | m'] =ᵐ[μ] ∏ i ∈ S, μ[(s i).indicator fun ω ↦ 1 | m']

Tactic:
rw [iCondIndepSet_iff_iCondIndepSets_singleton _ _ _ hs, iCondIndepSets_singleton_iff _ _ _ hs]

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
⊢ ∀ ⦃f_1 g : X → ℝ≥0∞⦄,
    Disjoint (Function.support f_1) (Function.support g) →
      Measurable f_1 →
        Measurable g →
          ∫⁻ (x : X), f_1 x * f x ∂π x₀ = f_1 x₀ * ∫⁻ (x : X), f x ∂π x₀ →
            ∫⁻ (x : X), g x * f x ∂π x₀ = g x₀ * ∫⁻ (x : X), f x ∂π x₀ →
              ∫⁻ (x : X), (f_1 + g) x * f x ∂π x₀ = (f_1 + g) x₀ * ∫⁻ (x : X), f x ∂π x₀

pp: case refine_3
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
⊢ ∀ ⦃f_1 : ℕ → X → ℝ≥0∞⦄,
    (∀ (n : ℕ), Measurable (f_1 n)) →
      Monotone f_1 →
        (∀ (n : ℕ), ∫⁻ (x : X), f_1 n x * f x ∂π x₀ = f_1 n x₀ * ∫⁻ (x : X), f x ∂π x₀) →
          ∫⁻ (x : X), (fun x ↦ ⨆ n, f_1 n x) x * f x ∂π x₀ = (fun x ↦ ⨆ n, f_1 n x) x₀ * ∫⁻ (x : X), f x ∂π x₀

Tactic:
·

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
⊢ ∀ ⦃f_1 g : X → ℝ≥0∞⦄,
    Disjoint (Function.support f_1) (Function.support g) →
      Measurable f_1 →
        Measurable g →
          ∫⁻ (x : X), f_1 x * f x ∂π x₀ = f_1 x₀ * ∫⁻ (x : X), f x ∂π x₀ →
            ∫⁻ (x : X), g x * f x ∂π x₀ = g x₀ * ∫⁻ (x : X), f x ∂π x₀ →
              ∫⁻ (x : X), (f_1 + g) x * f x ∂π x₀ = (f_1 + g) x₀ * ∫⁻ (x : X), f x ∂π x₀
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: case refine_1
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
h :
  ∀ (t1 t2 : Set Ω),
    MeasurableSet t1 →
      MeasurableSet t2 →
        μ[(t1 ∩ t2).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[t1.indicator fun ω ↦ 1 | m'] * μ[t2.indicator fun ω ↦ 1 | m']
s : Set β
t : Set β'
hs : MeasurableSet s
ht : MeasurableSet t
⊢ μ[(f ⁻¹' s ∩ g ⁻¹' t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] fun ω ↦
    μ[(f ⁻¹' s).indicator fun ω ↦ 1 | m'] ω * μ[(g ⁻¹' t).indicator fun ω ↦ 1 | m'] ω

pp: case refine_2
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
h :
  ∀ (s : Set β) (t : Set β'),
    MeasurableSet s →
      MeasurableSet t →
        μ[(f ⁻¹' s ∩ g ⁻¹' t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] fun ω ↦
          μ[(f ⁻¹' s).indicator fun ω ↦ 1 | m'] ω * μ[(g ⁻¹' t).indicator fun ω ↦ 1 | m'] ω
s t : Set Ω
⊢ MeasurableSet s →
    MeasurableSet t →
      μ[(s ∩ t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m'] * μ[t.indicator fun ω ↦ 1 | m']

Tactic:
·

Goal state after:
case refine_1
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
h :
  ∀ (t1 t2 : Set Ω),
    MeasurableSet t1 →
      MeasurableSet t2 →
        μ[(t1 ∩ t2).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[t1.indicator fun ω ↦ 1 | m'] * μ[t2.indicator fun ω ↦ 1 | m']
s : Set β
t : Set β'
hs : MeasurableSet s
ht : MeasurableSet t
⊢ μ[(f ⁻¹' s ∩ g ⁻¹' t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] fun ω ↦
    μ[(f ⁻¹' s).indicator fun ω ↦ 1 | m'] ω * μ[(g ⁻¹' t).indicator fun ω ↦ 1 | m'] ω
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: case refine_2
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
h :
  ∀ (s : Set β) (t : Set β'),
    MeasurableSet s →
      MeasurableSet t →
        μ[(f ⁻¹' s ∩ g ⁻¹' t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] fun ω ↦
          μ[(f ⁻¹' s).indicator fun ω ↦ 1 | m'] ω * μ[(g ⁻¹' t).indicator fun ω ↦ 1 | m'] ω
s t : Set Ω
⊢ MeasurableSet s →
    MeasurableSet t →
      μ[(s ∩ t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m'] * μ[t.indicator fun ω ↦ 1 | m']

Tactic:
·

Goal state after:
case refine_2
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
h :
  ∀ (s : Set β) (t : Set β'),
    MeasurableSet s →
      MeasurableSet t →
        μ[(f ⁻¹' s ∩ g ⁻¹' t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] fun ω ↦
          μ[(f ⁻¹' s).indicator fun ω ↦ 1 | m'] ω * μ[(g ⁻¹' t).indicator fun ω ↦ 1 | m'] ω
s t : Set Ω
⊢ MeasurableSet s →
    MeasurableSet t →
      μ[(s ∩ t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[s.indicator fun ω ↦ 1 | m'] * μ[t.indicator fun ω ↦ 1 | m']
Module: Mathlib.Probability.Kernel.Category.SFinKer

Goal state before:
pp: case refine_1
X Y : SFinKer
⊢ { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ } ≫
      { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ } =
    𝟙 (X ⊗ Y)

pp: case refine_2
X Y : SFinKer
⊢ { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ } ≫
      { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ } =
    𝟙 (Y ⊗ X)

Tactic:
·

Goal state after:
case refine_1
X Y : SFinKer
⊢ { hom := Kernel.swap X.carrier Y.carrier, property := ⋯ } ≫
      { hom := Kernel.swap Y.carrier X.carrier, property := ⋯ } =
    𝟙 (X ⊗ Y)