-- Mathlib/Probability/Independence/ZeroOne.lean
theorem Kernel.indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
    (hf : ∀ t, p t → tᶜ ∈ f) {t : Set ι} (ht : p t) :
    Indep (⨆ n ∈ t, s n) (limsup s f) κ μα

Proof state:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
p : Set ι → Prop
f : Filter ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
hf : ∀ (t : Set ι), p t → tᶜ ∈ f
t : Set ι
ht : p t
⊢ ∀ᶠ (n : MeasurableSpace Ω) in Filter.map s f, n ≤ ⨆ n ∈ tᶜ, s n

Tactic:
simp only [Set.mem_compl_iff, eventually_map]

Result:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
p : Set ι → Prop
f : Filter ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
hf : ∀ (t : Set ι), p t → tᶜ ∈ f
t : Set ι
ht : p t
⊢ ∀ᶠ (a : ι) in f, s a ≤ ⨆ n, ⨆ (_ : n ∉ t), s n

Proof state:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
p : Set ι → Prop
f : Filter ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
hf : ∀ (t : Set ι), p t → tᶜ ∈ f
t : Set ι
ht : p t
⊢ Indep (⨆ n ∈ t, s n) (limsup s f) κ μα

Tactic:
refine indep_of_indep_of_le_right (indep_biSup_compl h_le h_indep t) ?_

Result:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
p : Set ι → Prop
f : Filter ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
hf : ∀ (t : Set ι), p t → tᶜ ∈ f
t : Set ι
ht : p t
⊢ limsup s f ≤ ⨆ n ∈ tᶜ, s n

Proof state:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
p : Set ι → Prop
f : Filter ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
hf : ∀ (t : Set ι), p t → tᶜ ∈ f
t : Set ι
ht : p t
⊢ limsup s f ≤ ⨆ n ∈ tᶜ, s n

Tactic:
refine limsSup_le_of_le (by isBoundedDefault) ?_

Result:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
p : Set ι → Prop
f : Filter ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
hf : ∀ (t : Set ι), p t → tᶜ ∈ f
t : Set ι
ht : p t
⊢ ∀ᶠ (n : MeasurableSpace Ω) in Filter.map s f, n ≤ ⨆ n ∈ tᶜ, s n

Proof state:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
p : Set ι → Prop
f : Filter ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
hf : ∀ (t : Set ι), p t → tᶜ ∈ f
t : Set ι
ht : p t
⊢ ∀ᶠ (a : ι) in f, s a ≤ ⨆ n, ⨆ (_ : n ∉ t), s n

Tactic:
exact eventually_of_mem (hf t ht) le_iSup₂

Result:
no goals

Proof state:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → MeasurableSpace Ω
m0 : MeasurableSpace Ω
κ : Kernel α Ω
μα : Measure α
p : Set ι → Prop
f : Filter ι
h_le : ∀ (n : ι), s n ≤ m0
h_indep : iIndep s κ μα
hf : ∀ (t : Set ι), p t → tᶜ ∈ f
t : Set ι
ht : p t
⊢ IsCobounded (fun x1 x2 => x1 ≤ x2) (Filter.map s f)

Tactic:
isBoundedDefault

Result:
no goals