Theorem: ProbabilityTheory.kernel.tendsto_densityProcess_fst_atTop_univ_of_monotone
File: Mathlib/Probability/Kernel/Disintegration/Density.lean

tactic: simp_rw [densityProcess]
state_before: 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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ Tendsto (fun m => densityProcess κ (fst κ) n a x (seq m)) atTop (𝓝 (densityProcess κ (fst κ) n a x univ))
state_after: 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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ Tendsto (fun m => ((κ a) (countablePartitionSet n x ×ˢ seq m) / ((fst κ) a) (countablePartitionSet n x)).toReal) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ) / ((fst κ) a) (countablePartitionSet n x)).toReal)

tactic: refine (ENNReal.tendsto_toReal ?_).comp ?_
state_before: 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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ Tendsto (fun m => ((κ a) (countablePartitionSet n x ×ˢ seq m) / ((fst κ) a) (countablePartitionSet n x)).toReal) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ) / ((fst κ) a) (countablePartitionSet n x)).toReal)
state_after: case refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ (κ a) (countablePartitionSet n x ×ˢ univ) / ((fst κ) a) (countablePartitionSet n x) ≠ ⊤

case refine_2
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m) / ((fst κ) a) (countablePartitionSet n x)) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ) / ((fst κ) a) (countablePartitionSet n x)))

tactic: by_cases h0 : fst κ a (countablePartitionSet n x) = 0
state_before: case refine_2
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m) / ((fst κ) a) (countablePartitionSet n x)) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ) / ((fst κ) a) (countablePartitionSet n x)))
state_after: case pos
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ((fst κ) a) (countablePartitionSet n x) = 0
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m) / ((fst κ) a) (countablePartitionSet n x)) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ) / ((fst κ) a) (countablePartitionSet n x)))

case neg
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m) / ((fst κ) a) (countablePartitionSet n x)) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ) / ((fst κ) a) (countablePartitionSet n x)))

tactic: refine ENNReal.Tendsto.div_const ?_ ?_
state_before: case neg
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m) / ((fst κ) a) (countablePartitionSet n x)) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ) / ((fst κ) a) (countablePartitionSet n x)))
state_after: case neg.refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m)) atTop (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ)))

case neg.refine_2
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
⊢ (κ a) (countablePartitionSet n x ×ˢ univ) ≠ 0 ∨ ((fst κ) a) (countablePartitionSet n x) ≠ 0

tactic: rw [ne_eq, ENNReal.div_eq_top]
state_before: case refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ (κ a) (countablePartitionSet n x ×ˢ univ) / ((fst κ) a) (countablePartitionSet n x) ≠ ⊤
state_after: case refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ ¬((κ a) (countablePartitionSet n x ×ˢ univ) ≠ 0 ∧ ((fst κ) a) (countablePartitionSet n x) = 0 ∨
      (κ a) (countablePartitionSet n x ×ˢ univ) = ⊤ ∧ ((fst κ) a) (countablePartitionSet n x) ≠ ⊤)

tactic: push_neg
state_before: case refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ ¬((κ a) (countablePartitionSet n x ×ˢ univ) ≠ 0 ∧ ((fst κ) a) (countablePartitionSet n x) = 0 ∨
      (κ a) (countablePartitionSet n x ×ˢ univ) = ⊤ ∧ ((fst κ) a) (countablePartitionSet n x) ≠ ⊤)
state_after: case refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ ((κ a) (countablePartitionSet n x ×ˢ univ) ≠ 0 → ((fst κ) a) (countablePartitionSet n x) ≠ 0) ∧
    ((κ a) (countablePartitionSet n x ×ˢ univ) = ⊤ → ((fst κ) a) (countablePartitionSet n x) = ⊤)

tactic: simp_rw [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)]
state_before: case refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ ((κ a) (countablePartitionSet n x ×ˢ univ) ≠ 0 → ((fst κ) a) (countablePartitionSet n x) ≠ 0) ∧
    ((κ a) (countablePartitionSet n x ×ˢ univ) = ⊤ → ((fst κ) a) (countablePartitionSet n x) = ⊤)
state_after: case refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ ((κ a) (countablePartitionSet n x ×ˢ univ) ≠ 0 → (κ a) {p | p.1 ∈ countablePartitionSet n x} ≠ 0) ∧
    ((κ a) (countablePartitionSet n x ×ˢ univ) = ⊤ → (κ a) {p | p.1 ∈ countablePartitionSet n x} = ⊤)

tactic: constructor
state_before: case refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ ((κ a) (countablePartitionSet n x ×ˢ univ) ≠ 0 → (κ a) {p | p.1 ∈ countablePartitionSet n x} ≠ 0) ∧
    ((κ a) (countablePartitionSet n x ×ˢ univ) = ⊤ → (κ a) {p | p.1 ∈ countablePartitionSet n x} = ⊤)
state_after: case refine_1.left
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ (κ a) (countablePartitionSet n x ×ˢ univ) ≠ 0 → (κ a) {p | p.1 ∈ countablePartitionSet n x} ≠ 0

case refine_1.right
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ (κ a) (countablePartitionSet n x ×ˢ univ) = ⊤ → (κ a) {p | p.1 ∈ countablePartitionSet n x} = ⊤

tactic: refine fun h h0 ↦ h (measure_mono_null (fun x ↦ ?_) h0)
state_before: case refine_1.left
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ (κ a) (countablePartitionSet n x ×ˢ univ) ≠ 0 → (κ a) {p | p.1 ∈ countablePartitionSet n x} ≠ 0
state_after: case refine_1.left
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))
n : ℕ
a : α
x✝ : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h : (κ a) (countablePartitionSet n x✝ ×ˢ univ) ≠ 0
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x✝} = 0
x : γ × β
⊢ x ∈ countablePartitionSet n x✝ ×ˢ univ → x ∈ {p | p.1 ∈ countablePartitionSet n x✝}

tactic: simp only [mem_prod, mem_setOf_eq, and_imp]
state_before: case refine_1.left
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))
n : ℕ
a : α
x✝ : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h : (κ a) (countablePartitionSet n x✝ ×ˢ univ) ≠ 0
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x✝} = 0
x : γ × β
⊢ x ∈ countablePartitionSet n x✝ ×ˢ univ → x ∈ {p | p.1 ∈ countablePartitionSet n x✝}
state_after: case refine_1.left
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))
n : ℕ
a : α
x✝ : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h : (κ a) (countablePartitionSet n x✝ ×ˢ univ) ≠ 0
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x✝} = 0
x : γ × β
⊢ x.1 ∈ countablePartitionSet n x✝ → x.2 ∈ univ → x.1 ∈ countablePartitionSet n x✝

tactic: exact fun h _ ↦ h
state_before: case refine_1.left
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))
n : ℕ
a : α
x✝ : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h : (κ a) (countablePartitionSet n x✝ ×ˢ univ) ≠ 0
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x✝} = 0
x : γ × β
⊢ x.1 ∈ countablePartitionSet n x✝ → x.2 ∈ univ → x.1 ∈ countablePartitionSet n x✝
state_after: no goals

tactic: refine fun h_top ↦ eq_top_mono (measure_mono (fun x ↦ ?_)) h_top
state_before: case refine_1.right
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
⊢ (κ a) (countablePartitionSet n x ×ˢ univ) = ⊤ → (κ a) {p | p.1 ∈ countablePartitionSet n x} = ⊤
state_after: case refine_1.right
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))
n : ℕ
a : α
x✝ : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h_top : (κ a) (countablePartitionSet n x✝ ×ˢ univ) = ⊤
x : γ × β
⊢ x ∈ countablePartitionSet n x✝ ×ˢ univ → x ∈ {p | p.1 ∈ countablePartitionSet n x✝}

tactic: simp only [mem_prod, mem_setOf_eq, and_imp]
state_before: case refine_1.right
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))
n : ℕ
a : α
x✝ : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h_top : (κ a) (countablePartitionSet n x✝ ×ˢ univ) = ⊤
x : γ × β
⊢ x ∈ countablePartitionSet n x✝ ×ˢ univ → x ∈ {p | p.1 ∈ countablePartitionSet n x✝}
state_after: case refine_1.right
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))
n : ℕ
a : α
x✝ : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h_top : (κ a) (countablePartitionSet n x✝ ×ˢ univ) = ⊤
x : γ × β
⊢ x.1 ∈ countablePartitionSet n x✝ → x.2 ∈ univ → x.1 ∈ countablePartitionSet n x✝

tactic: exact fun h _ ↦ h
state_before: case refine_1.right
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))
n : ℕ
a : α
x✝ : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h_top : (κ a) (countablePartitionSet n x✝ ×ˢ univ) = ⊤
x : γ × β
⊢ x.1 ∈ countablePartitionSet n x✝ → x.2 ∈ univ → x.1 ∈ countablePartitionSet n x✝
state_after: no goals

tactic: rw [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)] at h0 ⊢
state_before: case pos
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ((fst κ) a) (countablePartitionSet n x) = 0
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m) / ((fst κ) a) (countablePartitionSet n x)) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ) / ((fst κ) a) (countablePartitionSet n x)))
state_after: case pos
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x} = 0
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m) / (κ a) {p | p.1 ∈ countablePartitionSet n x}) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ) / (κ a) {p | p.1 ∈ countablePartitionSet n x}))

tactic: suffices ∀ m, κ a (countablePartitionSet n x ×ˢ seq m) = 0 by
  simp only [this, h0, ENNReal.zero_div, tendsto_const_nhds_iff]
  suffices κ a (countablePartitionSet n x ×ˢ univ) = 0 by
    simp only [this, ENNReal.zero_div]
  convert h0
  ext x
  simp only [mem_prod, mem_univ, and_true, mem_setOf_eq]
state_before: case pos
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x} = 0
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m) / (κ a) {p | p.1 ∈ countablePartitionSet n x}) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ) / (κ a) {p | p.1 ∈ countablePartitionSet n x}))
state_after: case pos
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x} = 0
⊢ ∀ (m : ℕ), (κ a) (countablePartitionSet n x ×ˢ seq m) = 0

tactic: refine fun m ↦ measure_mono_null (fun x ↦ ?_) h0
state_before: case pos
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x} = 0
⊢ ∀ (m : ℕ), (κ a) (countablePartitionSet n x ×ˢ seq m) = 0
state_after: case pos
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))
n : ℕ
a : α
x✝ : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x✝} = 0
m : ℕ
x : γ × β
⊢ x ∈ countablePartitionSet n x✝ ×ˢ seq m → x ∈ {p | p.1 ∈ countablePartitionSet n x✝}

tactic: simp only [mem_prod, mem_setOf_eq, and_imp]
state_before: case pos
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))
n : ℕ
a : α
x✝ : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x✝} = 0
m : ℕ
x : γ × β
⊢ x ∈ countablePartitionSet n x✝ ×ˢ seq m → x ∈ {p | p.1 ∈ countablePartitionSet n x✝}
state_after: case pos
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))
n : ℕ
a : α
x✝ : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x✝} = 0
m : ℕ
x : γ × β
⊢ x.1 ∈ countablePartitionSet n x✝ → x.2 ∈ seq m → x.1 ∈ countablePartitionSet n x✝

tactic: exact fun h _ ↦ h
state_before: case pos
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))
n : ℕ
a : α
x✝ : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x✝} = 0
m : ℕ
x : γ × β
⊢ x.1 ∈ countablePartitionSet n x✝ → x.2 ∈ seq m → x.1 ∈ countablePartitionSet n x✝
state_after: no goals

tactic: simp only [this, h0, ENNReal.zero_div, tendsto_const_nhds_iff]
state_before: 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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x} = 0
this : ∀ (m : ℕ), (κ a) (countablePartitionSet n x ×ˢ seq m) = 0
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m) / (κ a) {p | p.1 ∈ countablePartitionSet n x}) atTop
    (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ) / (κ a) {p | p.1 ∈ countablePartitionSet n x}))
state_after: 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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x} = 0
this : ∀ (m : ℕ), (κ a) (countablePartitionSet n x ×ˢ seq m) = 0
⊢ 0 = (κ a) (countablePartitionSet n x ×ˢ univ) / 0

tactic: suffices κ a (countablePartitionSet n x ×ˢ univ) = 0 by
  simp only [this, ENNReal.zero_div]
state_before: 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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x} = 0
this : ∀ (m : ℕ), (κ a) (countablePartitionSet n x ×ˢ seq m) = 0
⊢ 0 = (κ a) (countablePartitionSet n x ×ˢ univ) / 0
state_after: 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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x} = 0
this : ∀ (m : ℕ), (κ a) (countablePartitionSet n x ×ˢ seq m) = 0
⊢ (κ a) (countablePartitionSet n x ×ˢ univ) = 0

tactic: convert h0
state_before: 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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x} = 0
this : ∀ (m : ℕ), (κ a) (countablePartitionSet n x ×ˢ seq m) = 0
⊢ (κ a) (countablePartitionSet n x ×ˢ univ) = 0
state_after: case h.e'_2.h.e'_6
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x} = 0
this : ∀ (m : ℕ), (κ a) (countablePartitionSet n x ×ˢ seq m) = 0
⊢ countablePartitionSet n x ×ˢ univ = {p | p.1 ∈ countablePartitionSet n x}

tactic: ext x
state_before: case h.e'_2.h.e'_6
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x} = 0
this : ∀ (m : ℕ), (κ a) (countablePartitionSet n x ×ˢ seq m) = 0
⊢ countablePartitionSet n x ×ˢ univ = {p | p.1 ∈ countablePartitionSet n x}
state_after: case h.e'_2.h.e'_6.h
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))
n : ℕ
a : α
x✝ : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x✝} = 0
this : ∀ (m : ℕ), (κ a) (countablePartitionSet n x✝ ×ˢ seq m) = 0
x : γ × β
⊢ x ∈ countablePartitionSet n x✝ ×ˢ univ ↔ x ∈ {p | p.1 ∈ countablePartitionSet n x✝}

tactic: simp only [mem_prod, mem_univ, and_true, mem_setOf_eq]
state_before: case h.e'_2.h.e'_6.h
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))
n : ℕ
a : α
x✝ : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x✝} = 0
this : ∀ (m : ℕ), (κ a) (countablePartitionSet n x✝ ×ˢ seq m) = 0
x : γ × β
⊢ x ∈ countablePartitionSet n x✝ ×ˢ univ ↔ x ∈ {p | p.1 ∈ countablePartitionSet n x✝}
state_after: no goals

tactic: simp only [this, ENNReal.zero_div]
state_before: 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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : (κ a) {p | p.1 ∈ countablePartitionSet n x} = 0
this✝ : ∀ (m : ℕ), (κ a) (countablePartitionSet n x ×ˢ seq m) = 0
this : (κ a) (countablePartitionSet n x ×ˢ univ) = 0
⊢ 0 = (κ a) (countablePartitionSet n x ×ˢ univ) / 0
state_after: no goals

tactic: have h := tendsto_measure_iUnion (μ := κ a)
  (s := fun m ↦ countablePartitionSet n x ×ˢ seq m) ?_
state_before: case neg.refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m)) atTop (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ)))
state_after: case neg.refine_1.refine_2
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
h :
  Tendsto (⇑(κ a) ∘ fun m => countablePartitionSet n x ×ˢ seq m) atTop
    (𝓝 ((κ a) (⋃ n_1, countablePartitionSet n x ×ˢ seq n_1)))
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m)) atTop (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ)))

case neg.refine_1.refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
⊢ Monotone fun m => countablePartitionSet n x ×ˢ seq m

tactic: swap
state_before: case neg.refine_1.refine_2
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
h :
  Tendsto (⇑(κ a) ∘ fun m => countablePartitionSet n x ×ˢ seq m) atTop
    (𝓝 ((κ a) (⋃ n_1, countablePartitionSet n x ×ˢ seq n_1)))
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m)) atTop (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ)))

case neg.refine_1.refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
⊢ Monotone fun m => countablePartitionSet n x ×ˢ seq m
state_after: case neg.refine_1.refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
⊢ Monotone fun m => countablePartitionSet n x ×ˢ seq m

case neg.refine_1.refine_2
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
h :
  Tendsto (⇑(κ a) ∘ fun m => countablePartitionSet n x ×ˢ seq m) atTop
    (𝓝 ((κ a) (⋃ n_1, countablePartitionSet n x ×ˢ seq n_1)))
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m)) atTop (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ)))

tactic: convert h
state_before: case neg.refine_1.refine_2
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
h :
  Tendsto (⇑(κ a) ∘ fun m => countablePartitionSet n x ×ˢ seq m) atTop
    (𝓝 ((κ a) (⋃ n_1, countablePartitionSet n x ×ˢ seq n_1)))
⊢ Tendsto (fun m => (κ a) (countablePartitionSet n x ×ˢ seq m)) atTop (𝓝 ((κ a) (countablePartitionSet n x ×ˢ univ)))
state_after: case h.e'_5.h.e'_3.h.e'_6
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
h :
  Tendsto (⇑(κ a) ∘ fun m => countablePartitionSet n x ×ˢ seq m) atTop
    (𝓝 ((κ a) (⋃ n_1, countablePartitionSet n x ×ˢ seq n_1)))
⊢ countablePartitionSet n x ×ˢ univ = ⋃ n_1, countablePartitionSet n x ×ˢ seq n_1

tactic: rw [← prod_iUnion, hseq_iUnion]
state_before: case h.e'_5.h.e'_3.h.e'_6
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
h :
  Tendsto (⇑(κ a) ∘ fun m => countablePartitionSet n x ×ˢ seq m) atTop
    (𝓝 ((κ a) (⋃ n_1, countablePartitionSet n x ×ˢ seq n_1)))
⊢ countablePartitionSet n x ×ˢ univ = ⋃ n_1, countablePartitionSet n x ×ˢ seq n_1
state_after: no goals

tactic: intro m m' hmm'
state_before: case neg.refine_1.refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
⊢ Monotone fun m => countablePartitionSet n x ×ˢ seq m
state_after: case neg.refine_1.refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
m m' : ℕ
hmm' : m ≤ m'
⊢ (fun m => countablePartitionSet n x ×ˢ seq m) m ≤ (fun m => countablePartitionSet n x ×ˢ seq m) m'

tactic: simp only [le_eq_subset, prod_subset_prod_iff, subset_rfl, true_and]
state_before: case neg.refine_1.refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
m m' : ℕ
hmm' : m ≤ m'
⊢ (fun m => countablePartitionSet n x ×ˢ seq m) m ≤ (fun m => countablePartitionSet n x ×ˢ seq m) m'
state_after: case neg.refine_1.refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
m m' : ℕ
hmm' : m ≤ m'
⊢ seq m ⊆ seq m' ∨ countablePartitionSet n x = ∅ ∨ seq m = ∅

tactic: exact Or.inl <| hseq hmm'
state_before: case neg.refine_1.refine_1
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
m m' : ℕ
hmm' : m ≤ m'
⊢ seq m ⊆ seq m' ∨ countablePartitionSet n x = ∅ ∨ seq m = ∅
state_after: no goals

tactic: exact Or.inr h0
state_before: case neg.refine_2
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))
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
h0 : ¬((fst κ) a) (countablePartitionSet n x) = 0
⊢ (κ a) (countablePartitionSet n x ×ˢ univ) ≠ 0 ∨ ((fst κ) a) (countablePartitionSet n x) ≠ 0
state_after: no goals