theorem extracted_formal_statement_57.{u_3, u_2, u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : Measure Ω}
  [inst_1 : IsProbabilityMeasure μ] {𝓧 : Type u_2} {ι : Type u_3} [inst_2 : DecidableEq ι] {m𝓧 : MeasurableSpace 𝓧}
  [hnonempty𝓧 : Nonempty 𝓧] [inst_3 : Fintype ι] (X : ι → Ω → 𝓧) (hX : ∀ (i : ι), Measurable (X i))
  (hX' : iIndepFun X μ) (f : (ι → 𝓧) → ℝ) (c : ι → ℝ)
  (hf : ∀ (i : ι) (x : ι → 𝓧) (x' : 𝓧), |f x - f (Function.update x i x')| ≤ c i) (hf' : Measurable f) (t : ℝ)
  (ht' : t * ∑ i, c i ^ 2 ≤ 1) (hmf : ∀ (i : ι) (x : ι → 𝓧) (x' : 𝓧), |(-f) x - (-f) (Function.update x i x')| ≤ c i)
  (hmf' : Measurable (-f)) (ω : Ω) :
  ((-f) ∘ Function.swap X) ω - ∫ (a : Ω), ((-f) ∘ Function.swap X) a ∂μ =
    -((f ∘ Function.swap X) ω - ∫ (a : Ω), (f ∘ Function.swap X) a ∂μ) := sorry

State before:
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
𝓧 : Type u_2
ι : Type u_3
inst✝¹ : DecidableEq ι
m𝓧 : MeasurableSpace 𝓧
hnonempty𝓧 : Nonempty 𝓧
inst✝ : Fintype ι
X : ι → Ω → 𝓧
hX : ∀ (i : ι), Measurable (X i)
hX' : iIndepFun X μ
f : (ι → 𝓧) → ℝ
c : ι → ℝ
hf : ∀ (i : ι) (x : ι → 𝓧) (x' : 𝓧), |f x - f (Function.update x i x')| ≤ c i
hf' : Measurable f
ε : ℝ
hε : ε > 0
t : ℝ
ht' : t * ∑ i, c i ^ 2 ≤ 1
hmf : ∀ (i : ι) (x : ι → 𝓧) (x' : 𝓧), |(-f) x - (-f) (Function.update x i x')| ≤ c i
hmf' : Measurable (-f)
ω : Ω
⊢ ((-f) ∘ Function.swap X) ω - ∫ (a : Ω), ((-f) ∘ Function.swap X) a ∂μ =
    -((f ∘ Function.swap X) ω - ∫ (a : Ω), (f ∘ Function.swap X) a ∂μ)

Tactic: simp only [Function.comp_apply, Pi.neg_apply, neg_sub]
    rw [integral_neg]
    ring_nf

State after:

Relevant lemmas:
neg_sub : ∀ {α : Type u_1} [inst : SubtractionMonoid α] (a b : α), -(a - b) = b - a
Function.comp_apply : ∀ {β : Sort u_1} {δ : Sort u_2} {α : Sort u_3} {f : β → δ} {g : α → β} {x : α},
  (f ∘ g) x = f (g x)
Pi.neg_apply : ∀ {I : Type u} {f : I → Type v₁} (x : (i : I) → f i) (i : I) [inst : (i : I) → Neg (f i)], (-x) i = -x i
MeasureTheory.integral_neg : ∀ {α : Type u_1} {G : Type u_5} [inst : NormedAddCommGroup G] [inst_4 : NormedSpace ℝ G]
  {m : MeasurableSpace α} {μ : Measure α} (f : α → G), ∫ (a : α), -f a ∂μ = -∫ (a : α), f a ∂μ