Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: case inr
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ × ℝ → E
f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E
a₂ b₂ : ℝ
s : Set (ℝ × ℝ)
hs : s.Countable
a₁ b₁ : ℝ
Hcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])
Hcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])
Hdf : ∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt f (f' x) x
Hdg : ∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt g (g' x) x
Hi : IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) volume
h₁ : a₁ ≤ b₁
this :
  ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →
    ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →
      (∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt f (f' x) x) →
        (∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt g (g' x) x) →
          IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) volume →
            b₂ ≤ a₂ →
              ∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in b₂..a₂, (f' (x, y)) (1, 0) + (g' (x, y)) (0, 1) =
                (((∫ (x : ℝ) in a₁..b₁, g (x, a₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, b₂)) + ∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -
                  ∫ (y : ℝ) in b₂..a₂, f (a₁, y)
h₂ : ¬a₂ ≤ b₂
⊢ -∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in b₂..a₂, (f' (x, y)) (1, 0) + (g' (x, y)) (0, 1) =
    (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + -∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -
      -∫ (y : ℝ) in b₂..a₂, f (a₁, y)

Tactic:
refine (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_ge h₂))).trans ?_

Goal state after:
case inr
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : ℝ × ℝ → E
f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E
a₂ b₂ : ℝ
s : Set (ℝ × ℝ)
hs : s.Countable
a₁ b₁ : ℝ
Hcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])
Hcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])
Hdf : ∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt f (f' x) x
Hdg : ∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt g (g' x) x
Hi : IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) volume
h₁ : a₁ ≤ b₁
this :
  ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →
    ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) →
      (∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt f (f' x) x) →
        (∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt g (g' x) x) →
          IntegrableOn (fun x ↦ (f' x) (1, 0) + (g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) volume →
            b₂ ≤ a₂ →
              ∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in b₂..a₂, (f' (x, y)) (1, 0) + (g' (x, y)) (0, 1) =
                (((∫ (x : ℝ) in a₁..b₁, g (x, a₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, b₂)) + ∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -
                  ∫ (y : ℝ) in b₂..a₂, f (a₁, y)
h₂ : ¬a₂ ≤ b₂
⊢ -((((∫ (x : ℝ) in a₁..b₁, g (x, a₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, b₂)) + ∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -
        ∫ (y : ℝ) in b₂..a₂, f (a₁, y)) =
    (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + -∫ (y : ℝ) in b₂..a₂, f (b₁, y)) -
      -∫ (y : ℝ) in b₂..a₂, f (a₁, y)