-- Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : (a : α) → ℝ≥0∞ := fun (a : α) ↦ ⨆ (n : ℕ), f n a
s : α →ₛ ℝ≥0
hsf : ∀ (x : α), (↑((s : (a : α) → ℝ≥0) x) : ℝ≥0∞) ≤ ⨆ (n : ℕ), f n x
r : ℝ≥0
right ha_1 : (↑r : ℝ≥0∞) < 1
ha : r < 1
rs : α →ₛ ℝ≥0 := SimpleFunc.map (fun (a : ℝ≥0) ↦ r * a) s
eq_rs : SimpleFunc.map c rs = const α (↑r : ℝ≥0∞) * SimpleFunc.map c s
eq :
  ∀ (p : ℝ≥0∞),
    Eq (α := Set α) (preimage (β := ℝ≥0∞) (⇑(SimpleFunc.map c rs) : (a : α) → ℝ≥0∞) {p})
      (⋃ (n : ℕ), preimage (β := ℝ≥0∞) (⇑(SimpleFunc.map c rs) : (a : α) → ℝ≥0∞) {p} ∩ {a : α | p ≤ f n a})
mono :
  ∀ (r : ℝ≥0∞),
    Monotone.{0, u_1} (β := Set α) fun (n : ℕ) ↦
      preimage (β := ℝ≥0∞) (⇑(SimpleFunc.map c rs) : (a : α) → ℝ≥0∞) {r} ∩ {a : α | r ≤ f n a}
h_meas : ∀ (n : ℕ), MeasurableSet.{u_1} (α := α) {a : α | (SimpleFunc.map c rs : (a : α) → ℝ≥0∞) a ≤ f n a}
μ₁ :
  (↑r : ℝ≥0∞) * (SimpleFunc.map c s).lintegral μ =
    ∑ r ∈ (SimpleFunc.map c rs).range,
      HMul.hMul (β := ℝ≥0∞) r
        ((μ : (a : Set α) → ℝ≥0∞) (preimage (β := ℝ≥0∞) (⇑(SimpleFunc.map c rs) : (a : α) → ℝ≥0∞) {r}) : ℝ≥0∞)
μ₂ :
  Eq (α := ℝ≥0∞)
    (∑ r ∈ (SimpleFunc.map c rs).range,
      HMul.hMul (β := ℝ≥0∞) r
        ((μ : (a : Set α) → ℝ≥0∞) (preimage (β := ℝ≥0∞) (⇑(SimpleFunc.map c rs) : (a : α) → ℝ≥0∞) {r}) : ℝ≥0∞))
    (∑ r ∈ (SimpleFunc.map c rs).range,
      HMul.hMul (β := ℝ≥0∞) r
        ((μ : (a : Set α) → ℝ≥0∞)
            (⋃ (n : ℕ), preimage (β := ℝ≥0∞) (⇑(SimpleFunc.map c rs) : (a : α) → ℝ≥0∞) {r} ∩ {a : α | r ≤ f n a}) :
          ℝ≥0∞))
μ₃ :
  Eq (α := ℝ≥0∞)
    (∑ r ∈ (SimpleFunc.map c rs).range,
      ⨆ (n : ℕ),
        HMul.hMul (β := ℝ≥0∞) r
          ((μ : (a : Set α) → ℝ≥0∞)
              (preimage (β := ℝ≥0∞) (⇑(SimpleFunc.map c rs) : (a : α) → ℝ≥0∞) {r} ∩ {a : α | r ≤ f n a}) :
            ℝ≥0∞))
    (⨆ (n : ℕ),
      ∑ r ∈ (SimpleFunc.map c rs).range,
        HMul.hMul (β := ℝ≥0∞) r
          ((μ : (a : Set α) → ℝ≥0∞)
              (preimage (β := ℝ≥0∞) (⇑(SimpleFunc.map c rs) : (a : α) → ℝ≥0∞) {r} ∩ {a : α | r ≤ f n a}) :
            ℝ≥0∞))
μ₄ :
  LE.le (α := ℝ≥0∞)
    (⨆ (n : ℕ),
      ∑ r ∈ (SimpleFunc.map c rs).range,
        HMul.hMul (β := ℝ≥0∞) r
          ((μ : (a : Set α) → ℝ≥0∞)
              (preimage (β := ℝ≥0∞) (⇑(SimpleFunc.map c rs) : (a : α) → ℝ≥0∞) {r} ∩ {a : α | r ≤ f n a}) :
            ℝ≥0∞))
    (⨆ (n : ℕ), ((SimpleFunc.map c rs).restrict {a : α | (SimpleFunc.map c rs : (a : α) → ℝ≥0∞) a ≤ f n a}).lintegral μ)
μ₅ :
  LE.le (α := ℝ≥0∞)
    (⨆ (n : ℕ), ((SimpleFunc.map c rs).restrict {a : α | (SimpleFunc.map c rs : (a : α) → ℝ≥0∞) a ≤ f n a}).lintegral μ)
    (⨆ (n : ℕ), ∫⁻ (a : α), f n a ∂μ)
⊢ (↑r : ℝ≥0∞) * (SimpleFunc.map ofNNReal s).lintegral μ ≤ ⨆ (n : ℕ), ∫⁻ (a : α), f n a ∂μ

Tactic:
calc
    (r : ℝ≥0∞) * (s.map c).lintegral μ = ∑ r ∈ (rs.map c).range, r * μ (rs.map c ⁻¹' {r}) := by
      rw [← const_mul_lintegral, eq_rs, SimpleFunc.lintegral]
    _ = ∑ r ∈ (rs.map c).range, r * μ (⋃ n, rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) := by
      simp only [(eq _).symm]
    _ = ∑ r ∈ (rs.map c).range, ⨆ n, r * μ (rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) :=
      Finset.sum_congr rfl fun x _ => by rw [(mono x).measure_iUnion, ENNReal.mul_iSup]
    _ = ⨆ n, ∑ r ∈ (rs.map c).range, r * μ (rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) := by
      refine ENNReal.finsetSum_iSup_of_monotone fun p i j h ↦ ?_
      gcongr _ * μ ?_
      exact mono p h
    _ ≤ ⨆ n : ℕ, ((rs.map c).restrict { a | (rs.map c) a ≤ f n a }).lintegral μ := by
      gcongr with n
      rw [restrict_lintegral _ (h_meas n)]
      refine le_of_eq (Finset.sum_congr rfl fun r _ => ?_)
      congr 2 with a
      refine and_congr_right ?_
      simp +contextual
    _ ≤ ⨆ n, ∫⁻ a, f n a ∂μ := by
      simp only [← SimpleFunc.lintegral_eq_lintegral]
      gcongr with n a
      simp only [map_apply] at h_meas
      simp only [coe_map, restrict_apply _ (h_meas _), (· ∘ ·)]
      exact indicator_apply_le id

Premises used: MeasureTheory.SimpleFunc.lintegral, ENNReal.ofNNReal, MeasureTheory.SimpleFunc.range, Finset.sum_congr, rfl, MeasureTheory.SimpleFunc.map