/-- If `f : ℝⁿ⁺¹ → E` is differentiable on a closed rectangular box `I` with derivative `f'`, then
the partial derivative `fun x ↦ f' x (Pi.single i 1)` is Henstock-Kurzweil integrable with integral
equal to the difference of integrals of `f` over the faces `x i = I.upper i` and `x i = I.lower i`.

More precisely, we use a non-standard generalization of the Henstock-Kurzweil integral and
we allow `f` to be non-differentiable (but still continuous) at a countable set of points.

TODO: If `n > 0`, then the condition at `x ∈ s` can be replaced by a much weaker estimate but this
requires either better integrability theorems, or usage of a filter depending on the countable set
`s` (we need to ensure that none of the faces of a partition contain a point from `s`). -/
theorem hasIntegral_GP_pderiv (f : (Fin (n + 1) → ℝ) → E)
    (f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E) (s : Set (Fin (n + 1) → ℝ))
    (hs : s.Countable) (Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x)
    (Hd : ∀ x ∈ (Box.Icc I) \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x) (i : Fin (n + 1)) :
    HasIntegral.{0, u, u} I GP (fun x => f' x (Pi.single i 1)) BoxAdditiveMap.volume
      (integral.{0, u, u} (I.face i) GP (fun x => f (i.insertNth (I.upper i) x))
          BoxAdditiveMap.volume -
        integral.{0, u, u} (I.face i) GP (fun x => f (i.insertNth (I.lower i) x))
          BoxAdditiveMap.volume) := by
  /- Note that `f` is continuous on `I.Icc`, hence it is integrable on the faces of all boxes
    `J ≤ I`, thus the difference of integrals over `x i = J.upper i` and `x i = J.lower i` is a
    box-additive function of `J ≤ I`. -/
  have Hc : ContinuousOn f (Box.Icc I) := fun x hx ↦ by
    by_cases hxs : x ∈ s
    exacts [Hs x hxs, (Hd x ⟨hx, hxs⟩).continuousWithinAt]
  set fI : ℝ → Box (Fin n) → E := fun y J =>
    integral.{0, u, u} J GP (fun x => f (i.insertNth y x)) BoxAdditiveMap.volume
  set fb : Icc (I.lower i) (I.upper i) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x =>
    (integrable_of_continuousOn GP (Box.continuousOn_face_Icc Hc x.2) volume).toBoxAdditive
  set F : Fin (n + 1) →ᵇᵃ[I] E := BoxAdditiveMap.upperSubLower I i fI fb fun x _ J => rfl
  -- Thus our statement follows from some local estimates.
  change HasIntegral I GP (fun x => f' x (Pi.single i 1)) _ (F I)
  refine' HasIntegral.of_le_Henstock_of_forall_isLittleO gp_le _ _ _ s hs _ _
  ·-- We use the volume as an upper estimate.
    exact (volume : Measure (Fin (n + 1) → ℝ)).toBoxAdditive.restrict _ le_top
  · exact fun J => ENNReal.toReal_nonneg
  · 

Proof state:
case refine'_3
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i✝ : Fin (n + 1)
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : Set.Countable s
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J => integral J GP (fun x => f (Fin.insertNth i y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (Box.lower I i) (Box.upper I i)) → Fin n →ᵇᵃ[↑(Box.face I i)] E :=
  fun x =>
    Integrable.toBoxAdditive
      (_ : Integrable (Box.face I i) GP (f ∘ Fin.insertNth i ↑x) (BoxAdditiveMap.toSMul (Measure.toBoxAdditive volume)))
F : Fin (n + 1) →ᵇᵃ[↑I] E :=
  BoxAdditiveMap.upperSubLower I i fI fb
    (_ : ∀ x ∈ Set.Icc (Box.lower I i) (Box.upper I i), ∀ (J : Box (Fin n)), fI x J = fI x J)
⊢ ∀ (c : ℝ≥0),
    ∀ x ∈ Box.Icc I ∩ s,
      ∀ ε > 0,
        ∃ δ > 0,
          ∀ J ≤ I,
            Box.Icc J ⊆ closedBall x δ →
              x ∈ Box.Icc J →
                (GP.bDistortion = true → Box.distortion J ≤ c) →
                  dist ((BoxAdditiveMap.volume J) ((f' x) (Pi.single i 1))) (F J) ≤ ε

Next tactic:
intro c x hx ε ε0