awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
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
⊢ ∀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∞)
have 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∞)
Tactic: filter_upwards [h_indep {i, j} hf_m] with a h_indep'
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
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
⊢ IndepSets (s i) (s j) κ μ
have intro : ∀ (t₁ t2 : Set Ω),
  t₁ ∈ s i →
    t2 ∈ s j →
      ∀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: classical
  intro t₁ t₂ ht₁ ht₂
  have hf_m : ∀ x : ι, x ∈ ({i, j} : Finset ι) → ite (x = i) t₁ t₂ ∈ s x := by
    intro x hx
    rcases Finset.mem_insert.mp hx with hx | hx
    · simp [hx, ht₁]
    · simp [Finset.mem_singleton.mp hx, hij.symm, ht₂]
  have h_inter : ⋂ (t : ι) (_ : t ∈ ({i, j} : Finset ι)), ite (t = i) t₁ t₂ =
      ite (i = i) t₁ t₂ ∩ ite (j = i) t₁ t₂ := by
    simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert]
  filter_upwards [h_indep {i, j} hf_m] with a h_indep'
  grind
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
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
intro intro₁ intro₂ intro₃ :
  ∀ (t₁ t₂ : Set Ω) (ht₁ : t₁ ∈ s i) (ht₂ : t₂ ∈ s j),
    ∀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∞)
⊢ IndepSets (s i) (s j) κ μ
have intro₄ : ∀ (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),
  ∀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: classical
  intro t₁ t₂ ht₁ ht₂
  have hf_m : ∀ x : ι, x ∈ ({i, j} : Finset ι) → ite (x = i) t₁ t₂ ∈ s x := by
    intro x hx
    rcases Finset.mem_insert.mp hx with hx | hx
    · simp [hx, ht₁]
    · simp [Finset.mem_singleton.mp hx, hij.symm, ht₂]
  have h_inter : ⋂ (t : ι) (_ : t ∈ ({i, j} : Finset ι)), ite (t = i) t₁ t₂ =
      ite (i = i) t₁ t₂ ∩ ite (j = i) t₁ t₂ := by
    simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert]
  filter_upwards [h_indep {i, j} hf_m] with a h_indep'
  grind
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
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
intro intro₁ intro₂ intro₃ :
  ∀ (t₁ t₂ : Set Ω) (ht₁ : t₁ ∈ s i) (ht₂ : t₂ ∈ s j),
    ∀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∞)
intro₄ :
  ∀ (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),
    ∀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∞)
⊢ IndepSets (s i) (s j) κ μ
have intro₅ : ∀ (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),
  ∀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: classical
  intro t₁ t₂ ht₁ ht₂
  have hf_m : ∀ x : ι, x ∈ ({i, j} : Finset ι) → ite (x = i) t₁ t₂ ∈ s x := by
    intro x hx
    rcases Finset.mem_insert.mp hx with hx | hx
    · simp [hx, ht₁]
    · simp [Finset.mem_singleton.mp hx, hij.symm, ht₂]
  have h_inter : ⋂ (t : ι) (_ : t ∈ ({i, j} : Finset ι)), ite (t = i) t₁ t₂ =
      ite (i = i) t₁ t₂ ∩ ite (j = i) t₁ t₂ := by
    simp only [Finset.set_biInter_singleton, Finset.set_biInter_insert]
  filter_upwards [h_indep {i, j} hf_m] with a h_indep'
  grind
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
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
⊢ ∀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∞)
have 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∞)
Tactic: rcases eq_zero_or_isMarkovKernel κ with rfl | h
Cleared hypotheses: κ, inst, hyp
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
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∞)
⊢ ∀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∞)
have 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∞)
Tactic: rcases eq_zero_or_isMarkovKernel κ with rfl | h
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
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∞)
⊢ ∀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∞)
have 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∞)
Tactic: filter_upwards [ihu] with a ha
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
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∞)
⊢ (((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∞)
have 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∞)
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]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (hypothesis)
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,
    (((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∞)
have ihf₁ : ∀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
Cleared hypotheses: ihf
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
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∞)
⊢ ∀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∞)
have 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∞)
Tactic: filter_upwards [ihf] with a ha
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
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∞)
⊢ (((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∞)
have 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∞)
Tactic: rw [inter_iUnion, measure_iUnion, measure_iUnion hfd fun i ↦ h2 _ (hfm i)]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
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∞)
⊢ (((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∞)
have h₂ : Pairwise (α := ℕ) (Function.onFun.{1, u_2 + 1, 1} (β := Set Ω) Disjoint fun (i : ℕ) ↦ t1 ∩ f i)
Tactic: rw [inter_iUnion, measure_iUnion, measure_iUnion hfd fun i ↦ h2 _ (hfm i)]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
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)
⊢ (((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∞)
have h₃ : ∀ (i : ℕ), MeasurableSet (t1 ∩ f i)
Tactic: rw [inter_iUnion, measure_iUnion, measure_iUnion hfd fun i ↦ h2 _ (hfm i)]
-- Lean 4 proof step from Mathlib.Probability.Independence.Kernel.Indep (goal)
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
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∞)
have inr : ∀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: induction t2, ht2m using induction_on_inter hpm2 hp2 with
  | empty => simp
  | basic u hu => exact hyp t1 u ht1 hu
  | compl u hu ihu =>
    filter_upwards [ihu] with a ha
    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]
    exact fun _ _ ↦ measure_ne_top _ _
  | iUnion f hfd hfm ihf =>
    rw [← ae_all_iff] at ihf
    filter_upwards [ihf] with a ha
    rw [inter_iUnion, measure_iUnion, measure_iUnion hfd fun i ↦ h2 _ (hfm i)]
    · simp only [ENNReal.tsum_mul_left, ha]
    · exact hfd.mono fun i j h ↦ (h.inter_left' _).inter_right' _
    · exact fun i ↦ .inter ht1m (h2 _ <| hfm i)
Cleared hypotheses: ht2m