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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this_1 : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u,
      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∞) ∂(ν : (a : α) → Measure γ) a =
    ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a
e_a :
  HMul.hMul (β := 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∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞))
      ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) =
    (((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s)
⊢ ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a =
    (((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s)

Tactic:
simp only [MeasureTheory.lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]

Premises used: MeasureTheory.lintegral_const, MeasureTheory.Measure.restrict_apply, Set.univ_inter, MeasurableSet.univ
-- 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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this_1 : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u,
      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∞) ∂(ν : (a : α) → Measure γ) a =
    ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a
pos :
  ∀ (h0 : Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)),
    HMul.hMul (β := 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∞) (u ×ˢ s) : ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞))
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) =
      (((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s)
neg :
  ∀ (h0 : ¬Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)),
    HMul.hMul (β := 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∞) (u ×ˢ s) : ℝ≥0∞)
          ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞))
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) =
      (((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s)
⊢ HMul.hMul (β := 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∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞))
      ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) =
    (((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s)

Tactic:
by_cases h0 : ν a u = 0

Premises used: 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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this_1 : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u,
      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∞) ∂(ν : (a : α) → Measure γ) a =
    ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a
h0 : Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
pos :
  Eq (α := ℝ≥0∞) (0 : ℝ≥0∞)
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
⊢ HMul.hMul (β := 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∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞))
      ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) =
    (((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s)

Tactic:
simp only [h0, mul_zero]

Premises used: MulZeroClass.mul_zero
-- 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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this_1 : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u,
      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∞) ∂(ν : (a : α) → Measure γ) a =
    ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a
h0 : Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((((κ.fst : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
simpa using (hκν a _).trans h0.le

Premises used: Eq.le, LE.le.trans
-- 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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this_1 : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u,
      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∞) ∂(ν : (a : α) → Measure γ) a =
    ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a
h0 : Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((((κ.fst : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
have h0' : fst κ a u = 0 := by simpa using (hκν a _).trans h0.le
-- 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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this_1 : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u,
      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∞) ∂(ν : (a : α) → Measure γ) a =
    ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a
h0 : Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
h0' : Eq (α := ℝ≥0∞) ((((κ.fst : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : 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∞) {p : γ × β | p.1 ∈ u} : ℝ≥0∞)
    (0 : ℝ≥0∞)

Tactic:
rw [fst_apply' _ _ hu_meas] at h0'

Premises used: ProbabilityTheory.Kernel.fst_apply'
-- 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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this_1 : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u,
      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∞) ∂(ν : (a : α) → Measure γ) a =
    ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a
h0 : Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
h0' :
  Eq (α := ℝ≥0∞)
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) {p : γ × β | p.1 ∈ u} : ℝ≥0∞)
    (0 : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) (0 : ℝ≥0∞)
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)

Tactic:
(measure_mono_null ?_ h0').symm

Premises used: MeasureTheory.measure_mono_null, Eq.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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this_1 : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u,
      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∞) ∂(ν : (a : α) → Measure γ) a =
    ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a
h0 : Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
h0' :
  Eq (α := ℝ≥0∞)
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) {p : γ × β | p.1 ∈ u} : ℝ≥0∞)
    (0 : ℝ≥0∞)
pos : u ×ˢ s ⊆ {p : γ × β | p.1 ∈ u}
⊢ Eq (α := ℝ≥0∞) (0 : ℝ≥0∞)
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)

Tactic:
refine (measure_mono_null ?_ h0').symm

Premises used: MeasureTheory.measure_mono_null, Eq.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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this_1 : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u,
      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∞) ∂(ν : (a : α) → Measure γ) a =
    ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a
h0 : Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
h0' :
  Eq (α := ℝ≥0∞)
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) {p : γ × β | p.1 ∈ u} : ℝ≥0∞)
    (0 : ℝ≥0∞)
p : γ × β
pos : p.1 ∈ u → p.2 ∈ s → p.1 ∈ u
⊢ p ∈ u ×ˢ s → Membership.mem (γ := Set (γ × β)) {p : γ × β | p.1 ∈ u} p

Tactic:
simp only [mem_prod, mem_setOf_eq, and_imp]

Premises used: and_imp, Set.mem_setOf_eq, Set.mem_prod
-- 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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this_1 : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u,
      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∞) ∂(ν : (a : α) → Measure γ) a =
    ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a
h0 : Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
h0' :
  Eq (α := ℝ≥0∞)
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) {p : γ × β | p.1 ∈ u} : ℝ≥0∞)
    (0 : ℝ≥0∞)
p : γ × β
⊢ p.1 ∈ u → p.2 ∈ s → p.1 ∈ u

Tactic:
fun h _ ↦ h
-- 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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this_1 : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u,
      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∞) ∂(ν : (a : α) → Measure γ) a =
    ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a
h0 : Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
h0' :
  Eq (α := ℝ≥0∞)
    ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) {p : γ × β | p.1 ∈ u} : ℝ≥0∞)
    (0 : ℝ≥0∞)
p : γ × β
⊢ p.1 ∈ u → p.2 ∈ s → p.1 ∈ u

Tactic:
exact fun h _ ↦ h
-- 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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this_1 : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u,
      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∞) ∂(ν : (a : α) → Measure γ) a =
    ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a
h0 : ¬Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
neg : Ne (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ HMul.hMul (β := 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∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞))
      ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) =
    (((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s)

Tactic:
rw [div_eq_mul_inv, mul_assoc, ENNReal.inv_mul_cancel h0, mul_one]

Premises used: mul_assoc, div_eq_mul_inv, ENNReal.inv_mul_cancel, mul_one
-- 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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this_1 : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u,
      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∞) ∂(ν : (a : α) → Measure γ) a =
    ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a
h0 : ¬Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ Ne (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
measure_ne_top _ _

Premises used: MeasureTheory.measure_ne_top
-- 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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this_1 : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u,
      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∞) ∂(ν : (a : α) → Measure γ) a =
    ∫⁻ (x : γ) in u,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞)
        ((((κ : (a : α) → Measure (γ × β)) a : Measure (γ × β)) : (a : Set (γ × β)) → ℝ≥0∞) (u ×ˢ s) : ℝ≥0∞)
        ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) ∂(ν : (a : α) → Measure γ) a
h0 : ¬Eq (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ Ne (α := ℝ≥0∞) ((((ν : (a : α) → Measure γ) a : Measure γ) : (a : Set γ) → ℝ≥0∞) u : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
exact measure_ne_top _ _

Premises used: MeasureTheory.measure_ne_top
-- 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
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
⊢ IsFiniteKernel κ

Tactic:
have : IsFiniteKernel κ := isFiniteKernel_of_isFiniteKernel_fst (h := isFiniteKernel_of_le hκν)