-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
theorem lintegral_finset_sum' (s : Finset β) {f : β → α → ℝ≥0∞}
    (hf : ∀ b ∈ s, AEMeasurable (f b) μ) :
    ∫⁻ a, ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ a, f b a ∂μ

Proof state:
case insert
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
f : β → α → ℝ≥0∞
a : β
s : Finset β
has : a ∉ s
ih : (∀ b ∈ s, AEMeasurable (f b) μ) → ∫⁻ (a : α), ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ
hf : AEMeasurable (f a) μ ∧ ∀ x ∈ s, AEMeasurable (f x) μ
⊢ ∫⁻ (a_1 : α), f a a_1 + ∑ b ∈ s, f b a_1 ∂μ = ∫⁻ (a_1 : α), f a a_1 ∂μ + ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ

Tactic:
rw [lintegral_add_left' hf.1, ih hf.2]

Result:
no goals

Proof state:
case empty
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
f : β → α → ℝ≥0∞
hf : ∀ b ∈ ∅, AEMeasurable (f b) μ
⊢ ∫⁻ (a : α), ∑ b ∈ ∅, f b a ∂μ = ∑ b ∈ ∅, ∫⁻ (a : α), f b a ∂μ

Tactic:
simp

Result:
no goals

Proof state:
case insert
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
f : β → α → ℝ≥0∞
a : β
s : Finset β
has : a ∉ s
ih : (∀ b ∈ s, AEMeasurable (f b) μ) → ∫⁻ (a : α), ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ
hf : ∀ b ∈ insert a s, AEMeasurable (f b) μ
⊢ ∫⁻ (a_1 : α), ∑ b ∈ insert a s, f b a_1 ∂μ = ∑ b ∈ insert a s, ∫⁻ (a : α), f b a ∂μ

Tactic:
simp only [Finset.sum_insert has]

Result:
case insert
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
f : β → α → ℝ≥0∞
a : β
s : Finset β
has : a ∉ s
ih : (∀ b ∈ s, AEMeasurable (f b) μ) → ∫⁻ (a : α), ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ
hf : ∀ b ∈ insert a s, AEMeasurable (f b) μ
⊢ ∫⁻ (a_1 : α), f a a_1 + ∑ b ∈ s, f b a_1 ∂μ = ∫⁻ (a_1 : α), f a a_1 ∂μ + ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ

Proof state:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
s : Finset β
f : β → α → ℝ≥0∞
hf : ∀ b ∈ s, AEMeasurable (f b) μ
⊢ ∫⁻ (a : α), ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ

Tactic:
induction' s using Finset.induction_on with a s has ih

Result:
case empty
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
f : β → α → ℝ≥0∞
hf : ∀ b ∈ ∅, AEMeasurable (f b) μ
⊢ ∫⁻ (a : α), ∑ b ∈ ∅, f b a ∂μ = ∑ b ∈ ∅, ∫⁻ (a : α), f b a ∂μ

case insert
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
f : β → α → ℝ≥0∞
a : β
s : Finset β
has : a ∉ s
ih : (∀ b ∈ s, AEMeasurable (f b) μ) → ∫⁻ (a : α), ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ
hf : ∀ b ∈ insert a s, AEMeasurable (f b) μ
⊢ ∫⁻ (a_1 : α), ∑ b ∈ insert a s, f b a_1 ∂μ = ∑ b ∈ insert a s, ∫⁻ (a : α), f b a ∂μ

Proof state:
case insert
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
f : β → α → ℝ≥0∞
a : β
s : Finset β
has : a ∉ s
ih : (∀ b ∈ s, AEMeasurable (f b) μ) → ∫⁻ (a : α), ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ
hf : ∀ b ∈ insert a s, AEMeasurable (f b) μ
⊢ ∫⁻ (a_1 : α), f a a_1 + ∑ b ∈ s, f b a_1 ∂μ = ∫⁻ (a_1 : α), f a a_1 ∂μ + ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ

Tactic:
rw [Finset.forall_mem_insert] at hf

Result:
case insert
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
f : β → α → ℝ≥0∞
a : β
s : Finset β
has : a ∉ s
ih : (∀ b ∈ s, AEMeasurable (f b) μ) → ∫⁻ (a : α), ∑ b ∈ s, f b a ∂μ = ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ
hf : AEMeasurable (f a) μ ∧ ∀ x ∈ s, AEMeasurable (f x) μ
⊢ ∫⁻ (a_1 : α), f a a_1 + ∑ b ∈ s, f b a_1 ∂μ = ∫⁻ (a_1 : α), f a a_1 ∂μ + ∑ b ∈ s, ∫⁻ (a : α), f b a ∂μ