Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem

Goal state before:
pp: case refine_3.refine_1
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (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 : 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)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
c : ℝ≥0
x : Fin (n + 1) → ℝ
hx : x ∈ Box.Icc I ∩ s
ε : ℝ
ε0 : ε > 0
this :
  ∀ᶠ (δ : ℝ) in 𝓝[>] 0,
    δ ∈ Set.Ioc 0 (1 / 2) ∧
      (∀ y₁ ∈ closedBall x δ ∩ Box.Icc I, ∀ y₂ ∈ closedBall x δ ∩ Box.Icc I, ‖f y₁ - f y₂‖ ≤ ε / 2) ∧
        (2 * δ) ^ (n + 1) * ‖(f' x) (Pi.single i 1)‖ ≤ ε / 2
δ : ℝ
hδ0 : 0 < δ
hδ12 : δ ≤ 1 / 2
hdfδ : ∀ y₁ ∈ closedBall x δ ∩ Box.Icc I, ∀ y₂ ∈ closedBall x δ ∩ Box.Icc I, ‖f y₁ - f y₂‖ ≤ ε / 2
hδ : (2 * δ) ^ (n + 1) * ‖(f' x) (Pi.single i 1)‖ ≤ ε / 2
J : Box (Fin (n + 1))
hJI : J ≤ I
hJδ : Box.Icc J ⊆ closedBall x δ
x✝¹ : x ∈ Box.Icc J
x✝ : GP.bDistortion = true → J.distortion ≤ c
Hl : J.lower i ∈ Set.Icc (J.lower i) (J.upper i)
Hu : J.upper i ∈ Set.Icc (J.lower i) (J.upper i)
Hi : ∀ x ∈ Set.Icc (J.lower i) (J.upper i), Integrable (J.face i) GP (fun y ↦ f (i.insertNth x y)) BoxAdditiveMap.volume
hJδ' : Box.Icc J ⊆ closedBall x δ ∩ Box.Icc I
Hmaps : ∀ z ∈ Set.Icc (J.lower i) (J.upper i), MapsTo (i.insertNth z) (Box.Icc (J.face i)) (closedBall x δ ∩ Box.Icc I)
⊢ ‖(BoxAdditiveMap.volume J) ((f' x) (Pi.single i 1))‖ ≤ ε / 2

pp: case refine_3.refine_2
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (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 : 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)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
c : ℝ≥0
x : Fin (n + 1) → ℝ
hx : x ∈ Box.Icc I ∩ s
ε : ℝ
ε0 : ε > 0
this :
  ∀ᶠ (δ : ℝ) in 𝓝[>] 0,
    δ ∈ Set.Ioc 0 (1 / 2) ∧
      (∀ y₁ ∈ closedBall x δ ∩ Box.Icc I, ∀ y₂ ∈ closedBall x δ ∩ Box.Icc I, ‖f y₁ - f y₂‖ ≤ ε / 2) ∧
        (2 * δ) ^ (n + 1) * ‖(f' x) (Pi.single i 1)‖ ≤ ε / 2
δ : ℝ
hδ0 : 0 < δ
hδ12 : δ ≤ 1 / 2
hdfδ : ∀ y₁ ∈ closedBall x δ ∩ Box.Icc I, ∀ y₂ ∈ closedBall x δ ∩ Box.Icc I, ‖f y₁ - f y₂‖ ≤ ε / 2
hδ : (2 * δ) ^ (n + 1) * ‖(f' x) (Pi.single i 1)‖ ≤ ε / 2
J : Box (Fin (n + 1))
hJI : J ≤ I
hJδ : Box.Icc J ⊆ closedBall x δ
x✝¹ : x ∈ Box.Icc J
x✝ : GP.bDistortion = true → J.distortion ≤ c
Hl : J.lower i ∈ Set.Icc (J.lower i) (J.upper i)
Hu : J.upper i ∈ Set.Icc (J.lower i) (J.upper i)
Hi : ∀ x ∈ Set.Icc (J.lower i) (J.upper i), Integrable (J.face i) GP (fun y ↦ f (i.insertNth x y)) BoxAdditiveMap.volume
hJδ' : Box.Icc J ⊆ closedBall x δ ∩ Box.Icc I
Hmaps : ∀ z ∈ Set.Icc (J.lower i) (J.upper i), MapsTo (i.insertNth z) (Box.Icc (J.face i)) (closedBall x δ ∩ Box.Icc I)
⊢ ‖integral (J.face i) GP ((fun y ↦ f (i.insertNth (J.upper i) y)) - fun y ↦ f (i.insertNth (J.lower i) y))
        BoxAdditiveMap.volume‖ ≤
    ε / 2

Tactic:
· simp_rw [BoxAdditiveMap.volume_apply, norm_smul, Real.norm_eq_abs, abs_prod]
      refine (mul_le_mul_of_nonneg_right ?_ <| norm_nonneg _).trans hδ
      have : ∀ j, |J.upper j - J.lower j| ≤ 2 * δ := fun j ↦
        calc
          dist (J.upper j) (J.lower j) ≤ dist J.upper J.lower := dist_le_pi_dist _ _ _
          _ ≤ dist J.upper x + dist J.lower x := dist_triangle_right _ _ _
          _ ≤ δ + δ := add_le_add (hJδ J.upper_mem_Icc) (hJδ J.lower_mem_Icc)
          _ = 2 * δ := (two_mul δ).symm
      calc
        ∏ j, |J.upper j - J.lower j| ≤ ∏ j : Fin (n + 1), 2 * δ := by
          gcongr with j _; exact this j
        _ = (2 * δ) ^ (n + 1) := by simp

Goal state after:
case refine_3.refine_2
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (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 : 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)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
F : Fin (n + 1) →ᵇᵃ[↑I] E := BoxAdditiveMap.upperSubLower I i fI fb ⋯
c : ℝ≥0
x : Fin (n + 1) → ℝ
hx : x ∈ Box.Icc I ∩ s
ε : ℝ
ε0 : ε > 0
this :
  ∀ᶠ (δ : ℝ) in 𝓝[>] 0,
    δ ∈ Set.Ioc 0 (1 / 2) ∧
      (∀ y₁ ∈ closedBall x δ ∩ Box.Icc I, ∀ y₂ ∈ closedBall x δ ∩ Box.Icc I, ‖f y₁ - f y₂‖ ≤ ε / 2) ∧
        (2 * δ) ^ (n + 1) * ‖(f' x) (Pi.single i 1)‖ ≤ ε / 2
δ : ℝ
hδ0 : 0 < δ
hδ12 : δ ≤ 1 / 2
hdfδ : ∀ y₁ ∈ closedBall x δ ∩ Box.Icc I, ∀ y₂ ∈ closedBall x δ ∩ Box.Icc I, ‖f y₁ - f y₂‖ ≤ ε / 2
hδ : (2 * δ) ^ (n + 1) * ‖(f' x) (Pi.single i 1)‖ ≤ ε / 2
J : Box (Fin (n + 1))
hJI : J ≤ I
hJδ : Box.Icc J ⊆ closedBall x δ
x✝¹ : x ∈ Box.Icc J
x✝ : GP.bDistortion = true → J.distortion ≤ c
Hl : J.lower i ∈ Set.Icc (J.lower i) (J.upper i)
Hu : J.upper i ∈ Set.Icc (J.lower i) (J.upper i)
Hi : ∀ x ∈ Set.Icc (J.lower i) (J.upper i), Integrable (J.face i) GP (fun y ↦ f (i.insertNth x y)) BoxAdditiveMap.volume
hJδ' : Box.Icc J ⊆ closedBall x δ ∩ Box.Icc I
Hmaps : ∀ z ∈ Set.Icc (J.lower i) (J.upper i), MapsTo (i.insertNth z) (Box.Icc (J.face i)) (closedBall x δ ∩ Box.Icc I)
⊢ ‖integral (J.face i) GP ((fun y ↦ f (i.insertNth (J.upper i) y)) - fun y ↦ f (i.insertNth (J.lower i) y))
        BoxAdditiveMap.volume‖ ≤
    ε / 2