/-- This is an auxiliary lemma used to prove two statements at once. Use one of the next two
lemmas instead. -/
theorem HasIntegral.of_bRiemann_eq_false_of_forall_isLittleO (hl : l.bRiemann = false)
    (B : ι →ᵇᵃ[I] ℝ) (hB0 : ∀ J, 0 ≤ B J) (g : ι →ᵇᵃ[I] F) (s : Set ℝⁿ) (hs : s.Countable)
    (hlH : s.Nonempty → l.bHenstock = true)
    (H₁ : ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I ∩ s, ∀ ε > (0 : ℝ),
      ∃ δ > 0, ∀ J ≤ I, Box.Icc J ⊆ Metric.closedBall x δ → x ∈ Box.Icc J →
        (l.bDistortion → J.distortion ≤ c) → dist (vol J (f x)) (g J) ≤ ε)
    (H₂ : ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I \ s, ∀ ε > (0 : ℝ),
      ∃ δ > 0, ∀ J ≤ I, Box.Icc J ⊆ Metric.closedBall x δ → (l.bHenstock → x ∈ Box.Icc J) →
        (l.bDistortion → J.distortion ≤ c) → dist (vol J (f x)) (g J) ≤ ε * B J) :
    HasIntegral I l f vol (g I) := by
  /- We choose `r x` differently for `x ∈ s` and `x ∉ s`.

    For `x ∈ s`, we choose `εs` such that `∑' x : s, εs x < ε / 2 / 2 ^ #ι`, then choose `r x` so
    that `dist (vol J (f x)) (g J) ≤ εs x` for `J` in the `r x`-neighborhood of `x`. This guarantees
    that the sum of these distances over boxes `J` such that `π.tag J ∈ s` is less than `ε / 2`. We
    need an additional multiplier `2 ^ #ι` because different boxes can have the same tag.

    For `x ∉ s`, we choose `r x` so that `dist (vol (J (f x))) (g J) ≤ (ε / 2 / B I) * B J` for a
    box `J` in the `δ`-neighborhood of `x`. -/
  refine' ((l.hasBasis_toFilteriUnion_top _).tendsto_iff Metric.nhds_basis_closedBall).2 _
  intro ε ε0
  simp only [← exists_prop, gt_iff_lt, Subtype.exists'] at H₁ H₂
  choose! δ₁ Hδ₁ using H₁
  choose! δ₂ Hδ₂ using H₂
  have ε0' := half_pos ε0; have H0 : 0 < (2 : ℝ) ^ Fintype.card ι := pow_pos zero_lt_two _
  rcases hs.exists_pos_forall_sum_le (div_pos ε0' H0) with ⟨εs, hεs0, hεs⟩
  simp only [le_div_iff' H0, mul_sum] at hεs
  rcases exists_pos_mul_lt ε0' (B I) with ⟨ε', ε'0, hεI⟩
  set δ : ℝ≥0 → ℝⁿ → Ioi (0 : ℝ) := fun c x => if x ∈ s then δ₁ c x (εs x) else (δ₂ c) x ε'
  refine' ⟨δ, fun c => l.rCond_of_bRiemann_eq_false hl, _⟩
  simp only [Set.mem_iUnion, mem_inter_iff, mem_setOf_eq]
  rintro π ⟨c, hπδ, hπp⟩
  -- Now we split the sum into two parts based on whether `π.tag J` belongs to `s` or not.
  rw [← g.sum_partition_boxes le_rfl hπp, mem_closedBall, integralSum,
    ← sum_filter_add_sum_filter_not π.boxes fun J => π.tag J ∈ s,
    ← sum_filter_add_sum_filter_not π.boxes fun J => π.tag J ∈ s, ← add_halves ε]
  refine' dist_add_add_le_of_le _ _
  · rcases s.eq_empty_or_nonempty with (rfl | hsne); · simp [ε0'.le]
    /- For the boxes such that `π.tag J ∈ s`, we use the fact that at most `2 ^ #ι` boxes have the
        same tag. -/
    specialize hlH hsne
    have : ∀ J ∈ π.boxes.filter fun J => π.tag J ∈ s,
        dist (vol J (f <| π.tag J)) (g J) ≤ εs (π.tag J) := fun J hJ ↦ by
      rw [Finset.mem_filter] at hJ; cases' hJ with hJ hJs
      refine' Hδ₁ c _ ⟨π.tag_mem_Icc _, hJs⟩ _ (hεs0 _) _ (π.le_of_mem' _ hJ) _
        (hπδ.2 hlH J hJ) fun hD => (Finset.le_sup hJ).trans (hπδ.3 hD)
      convert hπδ.1 J hJ using 3; exact (if_pos hJs).symm
    

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_1.inr
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I J : Box ι
π✝ : TaggedPrepartition I
inst✝ : Fintype ι
l : IntegrationParams
f g✝ : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y y' : F
hl : l.bRiemann = false
B : ι →ᵇᵃ[↑I] ℝ
hB0 : ∀ (J : Box ι), 0 ≤ B J
g : ι →ᵇᵃ[↑I] F
s : Set (ι → ℝ)
hs : Set.Countable s
ε : ℝ
ε0 : 0 < ε
δ₁ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₁ :
  ∀ (c : ℝ≥0),
    ∀ x ∈ Box.Icc I ∩ s,
      ∀ (ε : ℝ),
        0 < ε →
          ∀ J ≤ I,
            Box.Icc J ⊆ closedBall x ↑(δ₁ c x ε) →
              x ∈ Box.Icc J → (l.bDistortion = true → Box.distortion J ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε
δ₂ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₂ :
  ∀ (c : ℝ≥0),
    ∀ x ∈ Box.Icc I \ s,
      ∀ (ε : ℝ),
        0 < ε →
          ∀ J ≤ I,
            Box.Icc J ⊆ closedBall x ↑(δ₂ c x ε) →
              (l.bHenstock = true → x ∈ Box.Icc J) →
                (l.bDistortion = true → Box.distortion J ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε * B J
ε0' : 0 < ε / 2
H0 : 0 < 2 ^ Fintype.card ι
εs : (ι → ℝ) → ℝ
hεs0 : ∀ (i : ι → ℝ), 0 < εs i
hεs : ∀ (t : Finset (ι → ℝ)), ↑t ⊆ s → ∑ x in t, 2 ^ Fintype.card ι * εs x ≤ ε / 2
ε' : ℝ
ε'0 : 0 < ε'
hεI : B I * ε' < ε / 2
δ : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x => if x ∈ s then δ₁ c x (εs x) else δ₂ c x ε'
π : TaggedPrepartition I
c : ℝ≥0
hπδ : MemBaseSet l I c (fun x => if x ∈ s then δ₁ c x (εs x) else δ₂ c x ε') π
hπp : IsPartition π
hsne : Set.Nonempty s
hlH : l.bHenstock = true
this : ∀ J ∈ Finset.filter (fun J => tag π J ∈ s) π.boxes, dist ((vol J) (f (tag π J))) (g J) ≤ εs (tag π J)
⊢ dist (∑ x in Finset.filter (fun J => tag π J ∈ s) π.boxes, (vol x) (f (tag π x)))
      (∑ x in Finset.filter (fun J => tag π J ∈ s) π.boxes, g x) ≤
    ε / 2

Next tactic:
refine' (dist_sum_sum_le_of_le _ this).trans _