awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndepSets s κ μ
i j : ι
hij : i ≠ j
t₁ t₂ : Set Ω
ht₁ : t₁ ∈ s i
ht₂ : t₂ ∈ s j
hf_m : ∀ (x : ι), Membership.mem (γ := Finset ι) {i, j} x → (if x = i then t₁ else t₂) ∈ s x
h_inter :
  (⋂ (t : ι), ⋂ (_ : Membership.mem (γ := Finset ι) {i, j} t), if t = i then t₁ else t₂) =
    (if i = i then t₁ else t₂) ∩ if j = i then t₁ else t₂
h :
  ∀ (a : α)
    (h_indep' :
      Eq (α := ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞)
            (⋂ (i_1 : ι), ⋂ (_ : Membership.mem (γ := Finset ι) {i, j} i_1), if i_1 = i then t₁ else t₂) :
          ℝ≥0∞)
        (∏ i_1 ∈ {i, j}, (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (if i_1 = i then t₁ else t₂) :
          ℝ≥0∞)),
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t₁ ∩ t₂) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t₁ : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t₂ : ℝ≥0∞)
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t₁ ∩ t₂) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t₁ : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t₂ : ℝ≥0∞)

Tactic:
filter_upwards [h_indep {i, j} hf_m] with a h_indep'

Premises used: Insert.insert, Singleton.singleton
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndepSets s κ μ
i j : ι
hij : i ≠ j
t₁ t₂ : Set Ω
ht₁ : t₁ ∈ s i
ht₂ : t₂ ∈ s j
hf_m : ∀ (x : ι), Membership.mem (γ := Finset ι) {i, j} x → (if x = i then t₁ else t₂) ∈ s x
h_inter :
  (⋂ (t : ι), ⋂ (_ : Membership.mem (γ := Finset ι) {i, j} t), if t = i then t₁ else t₂) =
    (if i = i then t₁ else t₂) ∩ if j = i then t₁ else t₂
a : α
h_indep' :
  Eq (α := ℝ≥0∞)
    ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞)
        (⋂ (i_1 : ι), ⋂ (_ : Membership.mem (γ := Finset ι) {i, j} i_1), if i_1 = i then t₁ else t₂) :
      ℝ≥0∞)
    (∏ i_1 ∈ {i, j}, (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (if i_1 = i then t₁ else t₂) :
      ℝ≥0∞)
⊢ (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t₁ ∩ t₂) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t₁ : ℝ≥0∞)
      ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t₂ : ℝ≥0∞)

Tactic:
grind
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
m : ι → MeasurableSpace Ω
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndep m κ μ
i j : ι
hij : i ≠ j
⊢ Indep (m i) (m j) κ μ

Tactic:
iIndepSets.indepSets h_indep hij

Premises used: ProbabilityTheory.Kernel.iIndepSets.indepSets, ProbabilityTheory.Kernel.Indep
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
m : ι → MeasurableSpace Ω
s : ι → Set (Set Ω)
hms : ∀ (n : ι), m n = generateFrom (s n)
h_indep : iIndep m κ μ
⊢ Kernel.iIndepSets s κ μ

Tactic:
fun S f hfs =>
  h_indep S fun x hxS =>
    ((hms x).symm ▸ measurableSet_generateFrom (hfs x hxS) : MeasurableSet[m x] (f x))

Premises used: MeasurableSpace.measurableSet_generateFrom, Eq.rec, ProbabilityTheory.Kernel.iIndepSets, Eq.symm
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s1 s2 : Set (Set Ω)
h_indep : Indep (generateFrom s1) (generateFrom s2) κ μ
⊢ IndepSets s1 s2 κ μ

Tactic:
fun t1 t2 ht1 ht2 =>
  h_indep t1 t2 (measurableSet_generateFrom ht1) (measurableSet_generateFrom ht2)

Premises used: MeasurableSpace.measurableSet_generateFrom, ProbabilityTheory.Kernel.IndepSets
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
ht2m : MeasurableSet t2
inl :
  ∀ [IsZeroOrMarkovKernel.{u_1, u_2} (α := α) (β := Ω) (mα := _mα) (mβ := m) 0] (hyp : IndepSets (_mΩ := m) p1 p2 0 μ),
    ∀ᵐ (a : α) ∂μ,
      ((((0 : Kernel α Ω) : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
        HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞)
          (((((0 : Kernel α Ω) : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          (((((0 : Kernel α Ω) : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)
inr :
  ∀ (h : IsMarkovKernel κ),
    ∀ᵐ (a : α) ∂μ,
      (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
        HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)

Tactic:
rcases eq_zero_or_isMarkovKernel κ with rfl | h

Premises used: ProbabilityTheory.eq_zero_or_isMarkovKernel, rfl
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
μ : Measure α
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
ht2m : MeasurableSet t2
inst : IsZeroOrMarkovKernel.{u_1, u_2} (α := α) (β := Ω) (mα := _mα) (mβ := m) 0
hyp : IndepSets (_mΩ := m) p1 p2 0 μ
⊢ ∀ᵐ (a : α) ∂μ,
    ((((0 : Kernel α Ω) : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ t2) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞)
        (((((0 : Kernel α Ω) : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        (((((0 : Kernel α Ω) : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t2 : ℝ≥0∞)

Tactic:
simp
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ ∅) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) ∅ : ℝ≥0∞)

Tactic:
simp
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
u : Set Ω
hu : u ∈ p2
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ u) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞)

Tactic:
hyp t1 u ht1 hu
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
u : Set Ω
hu : u ∈ p2
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ u) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞)

Tactic:
exact hyp t1 u ht1 hu
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
u : Set Ω
hu : MeasurableSet u
ihu :
  ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ u) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞)
h₁ :
  ∀ (a : α)
    (ha :
      (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ u) =
        HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞)),
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ uᶜ) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) uᶜ : ℝ≥0∞)
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ uᶜ) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) uᶜ : ℝ≥0∞)

Tactic:
filter_upwards [ihu] with a ha
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
u : Set Ω
hu : MeasurableSet u
ihu :
  ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ u) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞)
a : α
ha :
  (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ u) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞)
h₁ :
  LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞) →
    LT.lt (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞) (1 : ℝ≥0∞) →
      Ne (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ uᶜ) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) uᶜ : ℝ≥0∞)

Tactic:
rw [← Set.diff_eq, ← Set.diff_self_inter,
      measure_diff inter_subset_left (ht1m.inter (h2 _ hu)).nullMeasurableSet (measure_ne_top _ _),
      ha, measure_compl (h2 _ hu) (measure_ne_top _ _), measure_univ, ENNReal.mul_sub, mul_one]

Premises used: Set.diff_self_inter, MeasureTheory.measure_compl, MeasurableSet.nullMeasurableSet, MeasureTheory.measure_ne_top, ENNReal.mul_sub, MeasureTheory.IsProbabilityMeasure.measure_univ, Set.diff_eq, Set.inter_subset_left, MeasureTheory.measure_diff, mul_one, MeasurableSet.inter
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
u : Set Ω
hu : MeasurableSet u
ihu :
  ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ u) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞)
a : α
ha :
  (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ u) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞)
⊢ LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞) →
    LT.lt (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞) (1 : ℝ≥0∞) →
      Ne (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
fun _ _ ↦ measure_ne_top _ _

Premises used: MeasureTheory.measure_ne_top, DFunLike.coe
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
u : Set Ω
hu : MeasurableSet u
ihu :
  ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ u) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞)
a : α
ha :
  (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ u) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞)
⊢ LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞) →
    LT.lt (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) u : ℝ≥0∞) (1 : ℝ≥0∞) →
      Ne (α := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
exact fun _ _ ↦ measure_ne_top _ _

Premises used: MeasureTheory.measure_ne_top
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
f : ℕ → Set Ω
hfd : Pairwise (Function.onFun Disjoint f)
hfm : ∀ (i : ℕ), MeasurableSet (f i)
ihf :
  ∀ (i : ℕ),
    ∀ᵐ (a : α) ∂μ,
      (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ f i) =
        HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f i) : ℝ≥0∞)
⊢ ∀ᵐ (a : α) ∂μ,
    ∀ (i : ℕ),
      (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ f i) =
        HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f i) : ℝ≥0∞)

Tactic:
rw [← ae_all_iff] at ihf

Premises used: MeasureTheory.ae_all_iff
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
f : ℕ → Set Ω
hfd : Pairwise (Function.onFun Disjoint f)
hfm : ∀ (i : ℕ), MeasurableSet (f i)
ihf :
  ∀ᵐ (a : α) ∂μ,
    ∀ (i : ℕ),
      (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ f i) =
        HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f i) : ℝ≥0∞)
h₁ :
  ∀ (a : α)
    (ha :
      ∀ (i : ℕ),
        (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ f i) =
          HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
            ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f i) : ℝ≥0∞)),
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ ⋃ (i : ℕ), f i) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)
⊢ ∀ᵐ (a : α) ∂μ,
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ ⋃ (i : ℕ), f i) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)

Tactic:
filter_upwards [ihf] with a ha
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
f : ℕ → Set Ω
hfd : Pairwise (Function.onFun Disjoint f)
hfm : ∀ (i : ℕ), MeasurableSet (f i)
ihf :
  ∀ᵐ (a : α) ∂μ,
    ∀ (i : ℕ),
      (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ f i) =
        HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f i) : ℝ≥0∞)
a : α
ha :
  ∀ (i : ℕ),
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ f i) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f i) : ℝ≥0∞)
h₁ :
  ∑' (i : ℕ), (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ f i) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      (∑' (i : ℕ), (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f i) : ℝ≥0∞)
h₂ : Pairwise (α := ℕ) (Function.onFun.{1, u_2 + 1, 1} (β := Set Ω) Disjoint fun (i : ℕ) ↦ t1 ∩ f i)
h₃ : ∀ (i : ℕ), MeasurableSet (t1 ∩ f i)
⊢ (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ ⋃ (i : ℕ), f i) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), f i) : ℝ≥0∞)

Tactic:
rw [inter_iUnion, measure_iUnion, measure_iUnion hfd fun i ↦ h2 _ (hfm i)]

Premises used: MeasureTheory.measure_iUnion, Set.inter_iUnion
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
f : ℕ → Set Ω
hfd : Pairwise (Function.onFun Disjoint f)
hfm : ∀ (i : ℕ), MeasurableSet (f i)
ihf :
  ∀ᵐ (a : α) ∂μ,
    ∀ (i : ℕ),
      (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ f i) =
        HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f i) : ℝ≥0∞)
a : α
ha :
  ∀ (i : ℕ),
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ f i) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f i) : ℝ≥0∞)
⊢ ∑' (i : ℕ), (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ f i) =
    HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
      (∑' (i : ℕ), (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f i) : ℝ≥0∞)

Tactic:
simp only [ENNReal.tsum_mul_left, ha]

Premises used: ENNReal.tsum_mul_left
-- Module: Mathlib.Probability.Independence.Kernel.Indep
Goal:
α : Type u_1
Ω : Type u_2
_mα : MeasurableSpace α
m₂ m : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst : IsZeroOrMarkovKernel κ
p1 p2 : Set (Set Ω)
h2 : m₂ ≤ m
hp2 : IsPiSystem p2
hpm2 : m₂ = generateFrom p2
hyp : IndepSets p1 p2 κ μ
t1 t2 : Set Ω
ht1 : t1 ∈ p1
ht1m : MeasurableSet t1
h : IsMarkovKernel κ
f : ℕ → Set Ω
hfd : Pairwise (Function.onFun Disjoint f)
hfm : ∀ (i : ℕ), MeasurableSet (f i)
ihf :
  ∀ᵐ (a : α) ∂μ,
    ∀ (i : ℕ),
      (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ f i) =
        HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
          ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f i) : ℝ≥0∞)
a : α
ha :
  ∀ (i : ℕ),
    (((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (t1 ∩ f i) =
      HMul.hMul (α := ℝ≥0∞) (β := ℝ≥0∞) ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) t1 : ℝ≥0∞)
        ((((κ : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (f i) : ℝ≥0∞)
⊢ Pairwise (α := ℕ) (Function.onFun.{1, u_2 + 1, 1} (β := Set Ω) Disjoint fun (i : ℕ) ↦ t1 ∩ f i)

Tactic:
hfd.mono fun i j h ↦ (h.inter_left' _).inter_right' _

Premises used: Disjoint.inter_left', Disjoint.inter_right', Pairwise.mono