awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
inst_1 : IsFiniteKernel ν
i j : ℕ
hij : i ≤ j
a : α
s : Set β
hs : MeasurableSet s
refine_1 : @Integrable ℝ _ _ _ mγ (fun (x : γ) ↦ κ.densityProcess ν j a x s) ((ν : (a : α) → Measure γ) a)
refine_2 :
  ∀ (s_1 : Set γ),
    MeasurableSet s_1 →
      LT.lt (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) s_1 : ℝ≥0∞) (∞ : ℝ≥0∞) →
        IntegrableOn (ε := ℝ) (mα := mγ) (fun (x : γ) ↦ κ.densityProcess ν i a x s) s_1 ((ν : (a : α) → Measure γ) a)
refine_3 :
  ∀ (s_1 : Set γ),
    MeasurableSet s_1 →
      LT.lt (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) s_1 : ℝ≥0∞) (∞ : ℝ≥0∞) →
        Eq (α := ℝ)
          (@integral _ _ _ _ mγ (Measure.restrict ((ν : (a : α) → Measure γ) a) s_1) fun (x : γ) ↦
            κ.densityProcess ν i a x s)
          (@integral _ _ _ _ mγ (Measure.restrict ((ν : (a : α) → Measure γ) a) s_1) fun (x : γ) ↦
            κ.densityProcess ν j a x s)
refine_4 :
  AEStronglyMeasurable (β := ℝ) (m₀ := mγ) (fun (x : γ) ↦ κ.densityProcess ν i a x s) ((ν : (a : α) → Measure γ) a)
⊢ (ae (F := Measure γ) ((ν : (a : α) → Measure γ) a : Measure γ)).EventuallyEq (β := ℝ)
    (condExp ((↑(countableFiltration γ) : ℕ → MeasurableSpace γ) i) (m₀ := mγ) ((ν : (a : α) → Measure γ) a)
      fun (x : γ) ↦ κ.densityProcess ν j a x s)
    fun (x : γ) ↦ κ.densityProcess ν i a x s

Tactic:
refine (ae_eq_condExp_of_forall_setIntegral_eq ?_ ?_ ?_ ?_ ?_).symm

Premises used: MeasureTheory.ae_eq_condExp_of_forall_setIntegral_eq, Filter.EventuallyEq.symm
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
inst_1 : IsFiniteKernel ν
i j : ℕ
hij : i ≤ j
a : α
s : Set β
hs : MeasurableSet s
⊢ @Integrable ℝ _ _ _ mγ (fun (x : γ) ↦ κ.densityProcess ν j a x s) ((ν : (a : α) → Measure γ) a)

Tactic:
integrable_densityProcess hκν j a hs

Premises used: ProbabilityTheory.Kernel.integrable_densityProcess
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
inst_1 : IsFiniteKernel ν
i j : ℕ
hij : i ≤ j
a : α
s : Set β
hs : MeasurableSet s
⊢ @Integrable ℝ _ _ _ mγ (fun (x : γ) ↦ κ.densityProcess ν j a x s) ((ν : (a : α) → Measure γ) a)

Tactic:
exact integrable_densityProcess hκν j a hs

Premises used: ProbabilityTheory.Kernel.integrable_densityProcess
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
inst_1 : IsFiniteKernel ν
i j : ℕ
hij : i ≤ j
a : α
s : Set β
hs : MeasurableSet s
⊢ ∀ (s_1 : Set γ),
    MeasurableSet s_1 →
      LT.lt (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) s_1 : ℝ≥0∞) (∞ : ℝ≥0∞) →
        IntegrableOn (ε := ℝ) (mα := mγ) (fun (x : γ) ↦ κ.densityProcess ν i a x s) s_1 ((ν : (a : α) → Measure γ) a)

Tactic:
fun _ _ _ ↦ (integrable_densityProcess hκν _ _ hs).integrableOn

Premises used: ProbabilityTheory.Kernel.integrable_densityProcess, MeasureTheory.Integrable.integrableOn
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
inst_1 : IsFiniteKernel ν
i j : ℕ
hij : i ≤ j
a : α
s : Set β
hs : MeasurableSet s
⊢ ∀ (s_1 : Set γ),
    MeasurableSet s_1 →
      LT.lt (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) s_1 : ℝ≥0∞) (∞ : ℝ≥0∞) →
        IntegrableOn (ε := ℝ) (mα := mγ) (fun (x : γ) ↦ κ.densityProcess ν i a x s) s_1 ((ν : (a : α) → Measure γ) a)

Tactic:
exact fun _ _ _ ↦ (integrable_densityProcess hκν _ _ hs).integrableOn

Premises used: ProbabilityTheory.Kernel.integrable_densityProcess, MeasureTheory.Integrable.integrableOn
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
inst_1 : IsFiniteKernel ν
i j : ℕ
hij : i ≤ j
a : α
s : Set β
hs : MeasurableSet s
x : Set γ
hx : MeasurableSet x
refine_3 :
  LT.lt (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) x : ℝ≥0∞) (∞ : ℝ≥0∞) →
    Eq (α := ℝ)
      (@integral _ _ _ _ mγ (Measure.restrict ((ν : (a : α) → Measure γ) a) x) fun (x : γ) ↦ κ.densityProcess ν i a x s)
      (@integral _ _ _ _ mγ (Measure.restrict ((ν : (a : α) → Measure γ) a) x) fun (x : γ) ↦ κ.densityProcess ν j a x s)
⊢ LT.lt (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) x : ℝ≥0∞) (∞ : ℝ≥0∞) →
    Eq (α := ℝ)
      (@integral _ _ _ _ mγ (Measure.restrict ((ν : (a : α) → Measure γ) a) x) fun (x : γ) ↦ κ.densityProcess ν i a x s)
      (@integral _ _ _ _ mγ (Measure.restrict ((ν : (a : α) → Measure γ) a) x) fun (x : γ) ↦ κ.densityProcess ν j a x s)

Tactic:
_
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
inst_1 : IsFiniteKernel ν
i j : ℕ
hij : i ≤ j
a : α
s : Set β
hs : MeasurableSet s
x : Set γ
hx : MeasurableSet x
h : LT.lt (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) x : ℝ≥0∞) (∞ : ℝ≥0∞)
refine_3 :
  Measure.real (m := Prod.instMeasurableSpace) ((κ : (a : α) → Measure (γ × β)) a) (x ×ˢ s) =
    @integral _ _ _ _ mγ (Measure.restrict ((ν : (a : α) → Measure γ) a) x) fun (x : γ) ↦ κ.densityProcess ν j a x s
refine_3₁ refine_3₂ :
  Measure.real (m := Prod.instMeasurableSpace) ((κ : (a : α) → Measure (γ × β)) a) (x ×ˢ s) =
    Measure.real (m := Prod.instMeasurableSpace) ((κ : (a : α) → Measure (γ × β)) a) (x ×ˢ s)
⊢ Eq (α := ℝ)
    (@integral _ _ _ _ mγ (Measure.restrict ((ν : (a : α) → Measure γ) a) x) fun (x : γ) ↦ κ.densityProcess ν i a x s)
    (@integral _ _ _ _ mγ (Measure.restrict ((ν : (a : α) → Measure γ) a) x) fun (x : γ) ↦ κ.densityProcess ν j a x s)

Tactic:
rw [setIntegral_densityProcess hκν i a hs hx,
      setIntegral_densityProcess_of_le hκν hij a hs hx]

Premises used: ProbabilityTheory.Kernel.setIntegral_densityProcess_of_le, ProbabilityTheory.Kernel.setIntegral_densityProcess
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
inst_1 : IsFiniteKernel ν
i j : ℕ
hij : i ≤ j
a : α
s : Set β
hs : MeasurableSet s
⊢ AEStronglyMeasurable (β := ℝ) (m₀ := mγ) (fun (x : γ) ↦ κ.densityProcess ν i a x s) ((ν : (a : α) → Measure γ) a)

Tactic:
StronglyMeasurable.aestronglyMeasurable
      (stronglyMeasurable_countableFiltration_densityProcess κ ν i a hs)

Premises used: MeasureTheory.StronglyMeasurable.aestronglyMeasurable, ProbabilityTheory.Kernel.stronglyMeasurable_countableFiltration_densityProcess
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
inst_1 : IsFiniteKernel ν
i j : ℕ
hij : i ≤ j
a : α
s : Set β
hs : MeasurableSet s
⊢ AEStronglyMeasurable (β := ℝ) (m₀ := mγ) (fun (x : γ) ↦ κ.densityProcess ν i a x s) ((ν : (a : α) → Measure γ) a)

Tactic:
exact StronglyMeasurable.aestronglyMeasurable
      (stronglyMeasurable_countableFiltration_densityProcess κ ν i a hs)

Premises used: MeasureTheory.StronglyMeasurable.aestronglyMeasurable, ProbabilityTheory.Kernel.stronglyMeasurable_countableFiltration_densityProcess
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
inst_1 : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
⊢ Martingale (E := ℝ) (m0 := mγ) (fun (n : ℕ) (x : γ) ↦ κ.densityProcess ν n a x s) (countableFiltration γ)
    ((ν : (a : α) → Measure γ) a)

Tactic:
⟨stronglyAdapted_densityProcess κ ν a hs, fun _ _ h ↦ condExp_densityProcess hκν h a hs⟩

Premises used: MeasureTheory.Martingale, ProbabilityTheory.Kernel.condExp_densityProcess, And.intro, ProbabilityTheory.Kernel.stronglyAdapted_densityProcess
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
inl :
  ∀
    (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
inr :
  ∀
    (h₀ :
      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∞)),
    (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
⊢ (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

Tactic:
obtain h₀ | h₀ := eq_or_ne (ν a (countablePartitionSet n x)) 0

Premises used: MeasurableSpace.countablePartitionSet, eq_or_ne, OfNat.ofNat, DFunLike.coe
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
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

Tactic:
simp [h₀]
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
h₀ :
  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∞)
inr :
  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

Tactic:
gcongr
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
h₀ :
  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∞)
inr :
  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∞)
⊢ 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:
simp only [ne_eq, ENNReal.div_eq_top, h₀, and_false, false_or, not_and, not_not]

Premises used: Classical.not_not, false_or, ne_eq, ENNReal.div_eq_top, and_false, not_and
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
h₀ :
  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∞)
          (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:
eq_top_mono (meas_countablePartitionSet_le_of_fst_le hκν n a x s')

Premises used: ProbabilityTheory.Kernel.meas_countablePartitionSet_le_of_fst_le, eq_top_mono
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
hκν : κ.fst ≤ ν
n : ℕ
a : α
x : γ
s s' : Set β
h : s ⊆ s'
h₀ :
  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∞)
          (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:
exact eq_top_mono (meas_countablePartitionSet_le_of_fst_le hκν n a x s')

Premises used: ProbabilityTheory.Kernel.meas_countablePartitionSet_le_of_fst_le, eq_top_mono
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
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∞)),
    (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
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∞)),
    (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
⊢ (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

Tactic:
by_cases h0 : ν a (countablePartitionSet n x) = 0

Premises used: MeasurableSpace.countablePartitionSet, DFunLike.coe
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
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∞)
pos :
  ENNReal.toReal
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (countablePartitionSet n x ×ˢ s)) /
      ENNReal.toReal 0 ≤
    ENNReal.toReal
        ((((κ' : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (countablePartitionSet n x ×ˢ s)) /
      ENNReal.toReal 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

Tactic:
rw [h0, ENNReal.toReal_div, ENNReal.toReal_div]

Premises used: ENNReal.toReal_div
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
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∞)
⊢ ENNReal.toReal
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (countablePartitionSet n x ×ˢ s)) /
      ENNReal.toReal 0 ≤
    ENNReal.toReal
        ((((κ' : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞)
          (countablePartitionSet n x ×ˢ s)) /
      ENNReal.toReal 0

Tactic:
simp
-- Module: Mathlib.Probability.Kernel.Disintegration.Density
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
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
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∞)
⊢ 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∞)

Tactic:
have h_le : κ' a (countablePartitionSet n x ×ˢ s) ≤ ν a (countablePartitionSet n x) :=
    meas_countablePartitionSet_le_of_fst_le hκ'ν n a x s