/-- **Henstock-Sacks inequality**. Let `r₁ r₂ : ℝⁿ → (0, ∞)` be a function such that for any tagged
*partition* of `I` subordinate to `rₖ`, `k=1,2`, the integral sum of `f` over this partition differs
from the integral of `f` by at most `εₖ`. Then for any two tagged *prepartition* `π₁ π₂` subordinate
to `r₁` and `r₂` respectively and covering the same part of `I`, the integral sums of `f` over these
prepartitions differ from each other by at most `ε₁ + ε₂`.

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.

See also `BoxIntegral.Integrable.dist_integralSum_sum_integral_le_of_memBaseSet_of_iUnion_eq` and
`BoxIntegral.Integrable.dist_integralSum_sum_integral_le_of_memBaseSet`.
-/
theorem dist_integralSum_le_of_memBaseSet (h : Integrable I l f vol) (hpos₁ : 0 < ε₁)
    (hpos₂ : 0 < ε₂) (h₁ : l.MemBaseSet I c₁ (h.convergenceR ε₁ c₁) π₁)
    (h₂ : l.MemBaseSet I c₂ (h.convergenceR ε₂ c₂) π₂) (HU : π₁.iUnion = π₂.iUnion) :
    dist (integralSum f vol π₁) (integralSum f vol π₂) ≤ ε₁ + ε₂ := by
  rcases h₁.exists_common_compl h₂ HU with ⟨π, hπU, hπc₁, hπc₂⟩
  set r : ℝⁿ → Ioi (0 : ℝ) := fun x => min (h.convergenceR ε₁ c₁ x) (h.convergenceR ε₂ c₂ x)
  have hr : l.RCond r := (h.convergenceR_cond _ c₁).min (h.convergenceR_cond _ c₂)
  set πr := π.toSubordinate r
  have H₁ :
    dist (integralSum f vol (π₁.unionComplToSubordinate π hπU r)) (integral I l f vol) ≤ ε₁ :=
    h.dist_integralSum_integral_le_of_memBaseSet hpos₁
      (h₁.unionComplToSubordinate (fun _ _ => min_le_left _ _) hπU hπc₁)
      (isPartition_unionComplToSubordinate _ _ _ _)
  rw [HU] at hπU
  have H₂ :
    dist (integralSum f vol (π₂.unionComplToSubordinate π hπU r)) (integral I l f vol) ≤ ε₂ :=
    h.dist_integralSum_integral_le_of_memBaseSet hpos₂
      (h₂.unionComplToSubordinate (fun _ _ => min_le_right _ _) hπU hπc₂)
      (isPartition_unionComplToSubordinate _ _ _ _)
  

Proof state:
case intro.intro.intro
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
c c₁ c₂ : ℝ≥0
ε ε₁ ε₂ : ℝ
π₁ π₂ : TaggedPrepartition I
h : Integrable I l f vol
hpos₁ : 0 < ε₁
hpos₂ : 0 < ε₂
h₁ : MemBaseSet l I c₁ (convergenceR h ε₁ c₁) π₁
h₂ : MemBaseSet l I c₂ (convergenceR h ε₂ c₂) π₂
HU : TaggedPrepartition.iUnion π₁ = TaggedPrepartition.iUnion π₂
π : Prepartition I
hπU✝ : Prepartition.iUnion π = ↑I \ TaggedPrepartition.iUnion π₁
hπU : Prepartition.iUnion π = ↑I \ TaggedPrepartition.iUnion π₂
hπc₁ : l.bDistortion = true → Prepartition.distortion π ≤ c₁
hπc₂ : l.bDistortion = true → Prepartition.distortion π ≤ c₂
r : (ι → ℝ) → ↑(Set.Ioi 0) := fun x => min (convergenceR h ε₁ c₁ x) (convergenceR h ε₂ c₂ x)
hr : RCond l r
πr : TaggedPrepartition I := Prepartition.toSubordinate π r
H₁ : dist (integralSum f vol (unionComplToSubordinate π₁ π hπU✝ r)) (integral I l f vol) ≤ ε₁
H₂ : dist (integralSum f vol (unionComplToSubordinate π₂ π hπU r)) (integral I l f vol) ≤ ε₂
⊢ dist (integralSum f vol π₁) (integralSum f vol π₂) ≤ ε₁ + ε₂

Next tactic:
simpa [unionComplToSubordinate] using (dist_triangle_right _ _ _).trans (add_le_add H₁ H₂)