awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Probability.Kernel.Disintegration.Density (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst : CountablyGenerated γ
κ : Kernel α (γ × β)
ν ν' : Kernel α γ
hνν' : ν ≤ ν'
hκν : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
h_le :
  LE.le (α := ℝ≥0∞)
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
        (countablePartitionSet n x ×ˢ s) :
      ℝ≥0∞)
    ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
h0 :
  ¬Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
      (0 : ℝ≥0∞)
⊢ (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
            (countablePartitionSet n x ×ˢ s) :
          ℝ≥0∞)
        ((((ν' : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal ≤
    (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
            (countablePartitionSet n x ×ˢ s) :
          ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal
have neg : HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
        (countablePartitionSet n x ×ˢ s) :
      ℝ≥0∞)
    ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞) ≠
  ∞
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Probability.Kernel.Disintegration.Density (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst : CountablyGenerated γ
κ : Kernel α (γ × β)
ν ν' : Kernel α γ
hνν' : ν ≤ ν'
hκν : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
h_le :
  LE.le (α := ℝ≥0∞)
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
        (countablePartitionSet n x ×ˢ s) :
      ℝ≥0∞)
    ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
h0 :
  ¬Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
      (0 : ℝ≥0∞)
neg :
  HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (countablePartitionSet n x ×ˢ s) :
        ℝ≥0∞)
      ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞) ≠
    ∞
⊢ (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
            (countablePartitionSet n x ×ˢ s) :
          ℝ≥0∞)
        ((((ν' : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal ≤
    (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
            (countablePartitionSet n x ×ˢ s) :
          ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal
have neg₁ : LE.le.{u_3} (α := Measure γ) ((ν : (a : α) → Measure γ) a : Measure γ) ((ν' : (a : α) → Measure γ) a : Measure γ)
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Probability.Kernel.Disintegration.Density (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst : CountablyGenerated γ
κ : Kernel α (γ × β)
ν ν' : Kernel α γ
hνν' : ν ≤ ν'
hκν : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
h_le :
  LE.le (α := ℝ≥0∞)
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
        (countablePartitionSet n x ×ˢ s) :
      ℝ≥0∞)
    ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
h0 :
  ¬Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
      (0 : ℝ≥0∞)
⊢ HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (countablePartitionSet n x ×ˢ s) :
        ℝ≥0∞)
      ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞) ≠
    ∞
have neg : Eq (α := ℝ≥0∞)
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
        (countablePartitionSet n x ×ˢ s) :
      ℝ≥0∞)
    (∞ : ℝ≥0∞) →
  Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
    (∞ : ℝ≥0∞)
Tactic: simp only [ne_eq, ENNReal.div_eq_top, h0, and_false, false_or, not_and, not_not]
-- Lean 4 proof step from Mathlib.Probability.Kernel.Disintegration.Density (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst_1 : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : Eq.{u_2 + 1} (α := Set β) (⋂ (i : ℕ), seq i) ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
⊢ Tendsto
    (fun (m : ℕ) ↦
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (countablePartitionSet n x ×ˢ seq m) :
            ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal)
    atTop
    (𝓝
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
            ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal)
have pos : ∀
  (h0 :
    Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
      (0 : ℝ≥0∞)),
  Tendsto
    (fun (m : ℕ) ↦
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (countablePartitionSet n x ×ˢ seq m) :
            ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal)
    atTop
    (𝓝
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
            ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal)
Tactic: by_cases h0 : ν a (countablePartitionSet n x) = 0
-- Lean 4 proof step from Mathlib.Probability.Kernel.Disintegration.Density (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst_1 : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : Eq.{u_2 + 1} (α := Set β) (⋂ (i : ℕ), seq i) ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
pos :
  ∀
    (h0 :
      Eq (α := ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
        (0 : ℝ≥0∞)),
    Tendsto
      (fun (m : ℕ) ↦
        (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
            ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
                (countablePartitionSet n x ×ˢ seq m) :
              ℝ≥0∞)
            ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) :
              ℝ≥0∞)).toReal)
      atTop
      (𝓝
        (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
            ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
                (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
              ℝ≥0∞)
            ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) :
              ℝ≥0∞)).toReal)
⊢ Tendsto
    (fun (m : ℕ) ↦
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (countablePartitionSet n x ×ˢ seq m) :
            ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal)
    atTop
    (𝓝
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
            ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal)
have neg : ∀
  (h0 :
    ¬Eq (α := ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
        (0 : ℝ≥0∞)),
  Tendsto
    (fun (m : ℕ) ↦
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (countablePartitionSet n x ×ˢ seq m) :
            ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal)
    atTop
    (𝓝
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
            ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal)
Tactic: by_cases h0 : ν a (countablePartitionSet n x) = 0
-- Lean 4 proof step from Mathlib.Probability.Kernel.Disintegration.Density (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst_1 : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : Eq.{u_2 + 1} (α := Set β) (⋂ (i : ℕ), seq i) ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 :
  Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
    (0 : ℝ≥0∞)
⊢ Tendsto
    (fun (m : ℕ) ↦
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (countablePartitionSet n x ×ˢ seq m) :
            ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal)
    atTop
    (𝓝
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
            ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal)
have pos : Tendsto
  (fun (m : ℕ) ↦
    ENNReal.toReal
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (countablePartitionSet n x ×ˢ seq m)) /
      ENNReal.toReal 0)
  atTop
  (𝓝
    (ENNReal.toReal
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅)) /
      ENNReal.toReal 0))
Tactic: simp_rw [h0, ENNReal.toReal_div]
-- Lean 4 proof step from Mathlib.Probability.Kernel.Disintegration.Density (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst_1 : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : Eq.{u_2 + 1} (α := Set β) (⋂ (i : ℕ), seq i) ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 :
  ¬Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
      (0 : ℝ≥0∞)
⊢ Tendsto
    (fun (m : ℕ) ↦
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (countablePartitionSet n x ×ˢ seq m) :
            ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal)
    atTop
    (𝓝
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
            ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal)
have neg : HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
        (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
      ℝ≥0∞)
    ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞) ≠
  ∞
Tactic: refine (ENNReal.tendsto_toReal ?_).comp ?_
-- Lean 4 proof step from Mathlib.Probability.Kernel.Disintegration.Density (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst_1 : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : Eq.{u_2 + 1} (α := Set β) (⋂ (i : ℕ), seq i) ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 :
  ¬Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
      (0 : ℝ≥0∞)
neg :
  HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
        ℝ≥0∞)
      ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞) ≠
    ∞
⊢ Tendsto
    (fun (m : ℕ) ↦
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (countablePartitionSet n x ×ˢ seq m) :
            ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal)
    atTop
    (𝓝
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
            ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)).toReal)
have neg₁ : Tendsto
  (fun (m : ℕ) ↦
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (countablePartitionSet n x ×ˢ seq m) :
        ℝ≥0∞)
      ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞))
  atTop
  (𝓝
    (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
        ℝ≥0∞)
      ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)))
Tactic: refine (ENNReal.tendsto_toReal ?_).comp ?_
-- Lean 4 proof step from Mathlib.Probability.Kernel.Disintegration.Density (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst_1 : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : Eq.{u_2 + 1} (α := Set β) (⋂ (i : ℕ), seq i) ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 :
  ¬Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
      (0 : ℝ≥0∞)
⊢ HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
      ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
        ℝ≥0∞)
      ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞) ≠
    ∞
have neg : ¬(Ne (α := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
            (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
          ℝ≥0∞)
        (0 : ℝ≥0∞) ∧
      Eq (α := ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
        (0 : ℝ≥0∞) ∨
    Eq (α := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
            (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
          ℝ≥0∞)
        (∞ : ℝ≥0∞) ∧
      Ne (α := ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
        (∞ : ℝ≥0∞))
Tactic: rw [ne_eq, ENNReal.div_eq_top]
-- Lean 4 proof step from Mathlib.Probability.Kernel.Disintegration.Density (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst_1 : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : Eq.{u_2 + 1} (α := Set β) (⋂ (i : ℕ), seq i) ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 :
  ¬Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
      (0 : ℝ≥0∞)
⊢ ¬(Ne (α := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
            ℝ≥0∞)
          (0 : ℝ≥0∞) ∧
        Eq (α := ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
          (0 : ℝ≥0∞) ∨
      Eq (α := ℝ≥0∞)
          ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
              (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
            ℝ≥0∞)
          (∞ : ℝ≥0∞) ∧
        Ne (α := ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
          (∞ : ℝ≥0∞))
have neg : (∀
    (_ :
      Ne (α := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
            (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
          ℝ≥0∞)
        (0 : ℝ≥0∞)),
    Ne (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
      (0 : ℝ≥0∞)) ∧
  ∀
    (_ :
      Eq (α := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
            (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
          ℝ≥0∞)
        (∞ : ℝ≥0∞)),
    Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
      (∞ : ℝ≥0∞)
Tactic: push Not
-- Lean 4 proof step from Mathlib.Probability.Kernel.Disintegration.Density (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst_1 : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : Eq.{u_2 + 1} (α := Set β) (⋂ (i : ℕ), seq i) ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 :
  ¬Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
      (0 : ℝ≥0∞)
⊢ Tendsto
    (fun (m : ℕ) ↦
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
            (countablePartitionSet n x ×ˢ seq m) :
          ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞))
    atTop
    (𝓝
      (HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
            (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅) :
          ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)))
have neg : Tendsto (β := ℝ≥0∞)
  (fun (m : ℕ) ↦
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
        (countablePartitionSet n x ×ˢ seq m) :
      ℝ≥0∞))
  atTop
  (𝓝
      ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
        (SProd.sprod (β := Set β) (countablePartitionSet n x) ∅)) :
    Filter ℝ≥0∞)
Tactic: refine ENNReal.Tendsto.div_const ?_ (.inr h0)
-- Lean 4 proof step from Mathlib.Probability.Kernel.Disintegration.Density (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst_1 : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : Eq.{u_2 + 1} (α := Set β) (⋂ (i : ℕ), seq i) ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 :
  ¬Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
      (0 : ℝ≥0∞)
⊢ Tendsto (β := ℝ≥0∞)
    (fun (m : ℕ) ↦
      ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (countablePartitionSet n x ×ˢ seq m) :
        ℝ≥0∞))
    atTop
    (𝓝
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (⋂ (n_1 : ℕ), countablePartitionSet n x ×ˢ seq n_1)) :
      Filter ℝ≥0∞)
have hs : ∀ (i : ℕ), NullMeasurableSet (countablePartitionSet n x ×ˢ seq i) ((κ : (a : α) → Measure (γ × β)) a)
Tactic: apply tendsto_measure_iInter_atTop
-- Lean 4 proof step from Mathlib.Probability.Kernel.Disintegration.Density (goal)
Goal:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst_1 : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : Eq.{u_2 + 1} (α := Set β) (⋂ (i : ℕ), seq i) ∅
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
h0 :
  ¬Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) (countablePartitionSet n x) : ℝ≥0∞)
      (0 : ℝ≥0∞)
hs : ∀ (i : ℕ), NullMeasurableSet (countablePartitionSet n x ×ˢ seq i) ((κ : (a : α) → Measure (γ × β)) a)
⊢ Tendsto (β := ℝ≥0∞)
    (fun (m : ℕ) ↦
      ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (countablePartitionSet n x ×ˢ seq m) :
        ℝ≥0∞))
    atTop
    (𝓝
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (⋂ (n_1 : ℕ), countablePartitionSet n x ×ˢ seq n_1)) :
      Filter ℝ≥0∞)
have hm : Antitone (β := Set (γ × β)) fun (m : ℕ) ↦ countablePartitionSet n x ×ˢ seq m
Tactic: apply tendsto_measure_iInter_atTop