-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
theorem lintegral_iSup_directed [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ b, AEMeasurable (f b) μ)
    (h_directed : Directed (· ≤ ·) f) : ∫⁻ a, ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ a, f b a ∂μ

Proof state:
case h.e'_3.e_s.h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 => x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
b : β
⊢ f b =ᶠ[ae μ] aeSeq hf p b

Tactic:
apply EventuallyEq.symm

Result:
case h.e'_3.e_s.h.H
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 => x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
b : β
⊢ aeSeq hf p b =ᶠ[ae μ] f b

Proof state:
case h.e'_3
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 => x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ⨆ b, ∫⁻ (a : α), f b a ∂μ = ⨆ b, ∫⁻ (a : α), aeSeq hf p b a ∂μ

Tactic:
congr 1

Result:
case h.e'_3.e_s
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 => x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ (fun b => ∫⁻ (a : α), f b a ∂μ) = fun b => ∫⁻ (a : α), aeSeq hf p b a ∂μ

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 => x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ

Tactic:
convert lintegral_iSup_directed_of_measurable (aeSeq.measurable hf p) h_ae_seq_directed using 1

Result:
case h.e'_2
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 => x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ∫⁻ (a : α), ⨆ b, aeSeq hf p b a ∂?m.363903

case h.e'_3
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 => x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ⨆ b, ∫⁻ (a : α), f b a ∂μ = ⨆ b, ∫⁻ (a : α), aeSeq hf p b a ∂?m.363903

α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 => x1 ≤ x2) f
p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ Measure α