theorem extracted_formal_statement_287.{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))
  (ω : Ω)
  (this :
    ((-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)) :
  (f ∘ Function.swap X) ω - ∫ (a : Ω), (f ∘ Function.swap X) a ∂μ ≤ -ε ↔
    -((f ∘ Function.swap X) ω - ∫ (a : Ω), (f ∘ Function.swap X) a ∂μ) ≥ ε := sorry

State before:
case h.a
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
this :
  ((-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)
⊢ (f ∘ Function.swap X) ω - ∫ (a : Ω), (f ∘ Function.swap X) a ∂μ ≤ -ε ↔
    -((f ∘ Function.swap X) ω - ∫ (a : Ω), (f ∘ Function.swap X) a ∂μ) ≥ ε

Tactic: exact le_neg

State after:

Relevant lemmas:
le_neg : ∀ {α : Type u} [inst : AddGroup α] [inst_4 : LE α] [inst_5 : AddLeftMono α] [inst_6 : AddRightMono α]
  {a b : α}, a ≤ -b ↔ b ≤ -a