theorem extracted_formal_statement_288.{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) (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))
  (this :
    (fun ω => (f ∘ Function.swap X) ω - ∫ (a : Ω), (f ∘ Function.swap X) a ∂μ ≤ -ε) = fun ω =>
      ((-f) ∘ Function.swap X) ω - ∫ (a : Ω), ((-f) ∘ Function.swap X) a ∂μ ≥ ε)
  (h_original_goal :
    (μ fun ω => (f ∘ Function.swap X) ω - ∫ (x : Ω), (f ∘ Function.swap X) x ∂μ ≤ -ε).toReal ≤
      Real.exp (-2 * ε ^ 2 * t)) :
  (μ fun ω => ((-f) ∘ Function.swap X) ω - ∫ (a : Ω), ((-f) ∘ Function.swap X) a ∂μ ≥ ε).toReal ≤
    Real.exp (-2 * ε ^ 2 * t) := 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)
this :
  (fun ω => (f ∘ Function.swap X) ω - ∫ (a : Ω), (f ∘ Function.swap X) a ∂μ ≤ -ε) = fun ω =>
    ((-f) ∘ Function.swap X) ω - ∫ (a : Ω), ((-f) ∘ Function.swap X) a ∂μ ≥ ε
⊢ (μ fun ω => (f ∘ Function.swap X) ω - ∫ (x : Ω), (f ∘ Function.swap X) x ∂μ ≤ -ε).toReal ≤ Real.exp (-2 * ε ^ 2 * t)

Tactic: rw [this]

State after:
[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)
 this :
   (fun ω => (f ∘ Function.swap X) ω - ∫ (a : Ω), (f ∘ Function.swap X) a ∂μ ≤ -ε) = fun ω =>
     ((-f) ∘ Function.swap X) ω - ∫ (a : Ω), ((-f) ∘ Function.swap X) a ∂μ ≥ ε
 ⊢ (μ fun ω => ((-f) ∘ Function.swap X) ω - ∫ (a : Ω), ((-f) ∘ Function.swap X) a ∂μ ≥ ε).toReal ≤
     Real.exp (-2 * ε ^ 2 * t)]

Relevant lemmas: