/-- **Henstock-Sacks inequality**. Let `r : ℝⁿ → (0, ∞)` be a function such that for any tagged
*partition* of `I` subordinate to `r`, the integral sum of `f` over this partition differs from the
integral of `f` by at most `ε`. Then for any tagged *prepartition* `π` subordinate to `r`, the
integral sum of `f` over this prepartition differs from the integral of `f` over the part of `I`
covered by `π` by at most `ε`.

The actual statement

- uses `BoxIntegral.Integrable.convergenceR` instead of a predicate assumption on `r`;
- uses `BoxIntegral.IntegrationParams.MemBaseSet` instead of “subordinate to `r`” to
  account for additional requirements like being a Henstock partition or having a bounded
  distortion;
- takes an extra argument `π₀ : prepartition I` and an assumption `π.Union = π₀.Union` instead of
  using `π.to_prepartition`.
-/
theorem dist_integralSum_sum_integral_le_of_memBaseSet_of_iUnion_eq (h : Integrable I l f vol)
    (h0 : 0 < ε) (hπ : l.MemBaseSet I c (h.convergenceR ε c) π) {π₀ : Prepartition I}
    (hU : π.iUnion = π₀.iUnion) :
    dist (integralSum f vol π) (∑ J in π₀.boxes, integral J l f vol) ≤ ε := by
  -- Let us prove that the distance is less than or equal to `ε + δ` for all positive `δ`.
  refine' le_of_forall_pos_le_add fun δ δ0 => _
  -- First we choose some constants.
  set δ' : ℝ := δ / (π₀.boxes.card + 1)
  have H0 : 0 < (π₀.boxes.card + 1 : ℝ) := Nat.cast_add_one_pos _
  have δ'0 : 0 < δ' := div_pos δ0 H0
  set C := max π₀.distortion π₀.compl.distortion
  /- Next we choose a tagged partition of each `J ∈ π₀` such that the integral sum of `f` over this
    partition is `δ'`-close to the integral of `f` over `J`. -/
  have : ∀ J ∈ π₀, ∃ πi : TaggedPrepartition J,
      πi.IsPartition ∧ dist (integralSum f vol πi) (integral J l f vol) ≤ δ' ∧
        l.MemBaseSet J C (h.convergenceR δ' C) πi := by
    intro J hJ
    have Hle : J ≤ I := π₀.le_of_mem hJ
    have HJi : Integrable J l f vol := h.to_subbox Hle
    set r := fun x => min (h.convergenceR δ' C x) (HJi.convergenceR δ' C x)
    have hJd : J.distortion ≤ C := le_trans (Finset.le_sup hJ) (le_max_left _ _)
    rcases l.exists_memBaseSet_isPartition J hJd r with ⟨πJ, hC, hp⟩
    have hC₁ : l.MemBaseSet J C (HJi.convergenceR δ' C) πJ := by
      refine' hC.mono J le_rfl le_rfl fun x _ => _; exact min_le_right _ _
    have hC₂ : l.MemBaseSet J C (h.convergenceR δ' C) πJ := by
      refine' hC.mono J le_rfl le_rfl fun x _ => _; exact min_le_left _ _
    exact ⟨πJ, hp, HJi.dist_integralSum_integral_le_of_memBaseSet δ'0 hC₁ hp, hC₂⟩
  /- Now we combine these tagged partitions into a tagged prepartition of `I` that covers the
    same part of `I` as `π₀` and apply `BoxIntegral.dist_integralSum_le_of_memBaseSet` to
    `π` and this prepartition. -/
  choose! πi hπip hπiδ' hπiC using this
  have : l.MemBaseSet I C (h.convergenceR δ' C) (π₀.biUnionTagged πi) :=
    biUnionTagged_memBaseSet hπiC hπip fun _ => le_max_right _ _
  have hU' : π.iUnion = (π₀.biUnionTagged πi).iUnion :=
    hU.trans (Prepartition.iUnion_biUnion_partition _ hπip).symm
  have := h.dist_integralSum_le_of_memBaseSet h0 δ'0 hπ this hU'
  rw [integralSum_biUnionTagged] at this
  calc
    dist (integralSum f vol π) (∑ J in π₀.boxes, integral J l f vol) ≤
        dist (integralSum f vol π) (∑ J in π₀.boxes, integralSum f vol (πi J)) +
          dist (∑ J in π₀.boxes, integralSum f vol (πi J)) (∑ J in π₀.boxes, integral J l f vol) :=
      dist_triangle _ _ _
    _ ≤ ε + δ' + ∑ _J in π₀.boxes, δ' := (add_le_add this (dist_sum_sum_le_of_le _ hπiδ'))
    _ = ε + δ := by field_simp; 

Proof state:
ι : Type u
E : Type v
F : Type w
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
I J : Box ι
π : TaggedPrepartition I
inst✝¹ : Fintype ι
l : IntegrationParams
f g : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y y' : F
c c₁ c₂ : ℝ≥0
ε ε₁ ε₂ : ℝ
π₁ π₂ : TaggedPrepartition I
inst✝ : CompleteSpace F
h : Integrable I l f vol
h0 : 0 < ε
hπ : MemBaseSet l I c (convergenceR h ε c) π
π₀ : Prepartition I
hU : TaggedPrepartition.iUnion π = Prepartition.iUnion π₀
δ : ℝ
δ0 : 0 < δ
δ' : ℝ := δ / (↑(card π₀.boxes) + 1)
H0 : 0 < ↑(card π₀.boxes) + 1
δ'0 : 0 < δ'
C : ℝ≥0 := max (Prepartition.distortion π₀) (Prepartition.distortion (Prepartition.compl π₀))
πi : (J : Box ι) → TaggedPrepartition J
hπip : ∀ J ∈ π₀, IsPartition (πi J)
hπiδ' : ∀ J ∈ π₀, dist (integralSum f vol (πi J)) (integral J l f vol) ≤ δ'
hπiC : ∀ J ∈ π₀, MemBaseSet l J C (convergenceR h δ' C) (πi J)
this✝ : MemBaseSet l I C (convergenceR h δ' C) (Prepartition.biUnionTagged π₀ πi)
hU' : TaggedPrepartition.iUnion π = TaggedPrepartition.iUnion (Prepartition.biUnionTagged π₀ πi)
this : dist (integralSum f vol π) (∑ J in π₀.boxes, integralSum f vol (πi J)) ≤ ε + δ'
⊢ ε * (↑(card π₀.boxes) + 1) + δ + ↑(card π₀.boxes) * δ = (ε + δ) * (↑(card π₀.boxes) + 1)

Next tactic:
ring