-- File: FoML/FoML/Main.lean
Theorem statement:
theorem extracted_formal_statement_5.{w, v, u} {n : ℕ} {Ω : Type u} [inst : MeasurableSpace Ω] {ι : Type v} {𝒳 : Type w}
  {μ : Measure Ω} [inst_1 : MeasurableSpace 𝒳] [inst_2 : Nonempty 𝒳] [inst_3 : Nonempty ι] [inst_4 : Countable ι]
  [inst_5 : IsProbabilityMeasure μ] (f : ι → 𝒳 → ℝ) (hf : ∀ (i : ι), Measurable (f i)) (X : Ω → 𝒳) (hX : Measurable X)
  {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b) {t : ℝ} (ht : 0 ≤ t) (ht' : t * b ^ 2 ≤ 1 / 2) {ε : ℝ}
  (hε : 0 ≤ ε) (hn_1 : ¬n = 0) (hn : 0 < n)
  (h_original_goal :
    ((Measure.pi fun x => μ) fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)).toReal ≤
      rexp (-ε ^ 2 * t * ↑n)) :
  ((Measure.pi fun x => μ) fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)).toReal ≤
    ((Measure.pi fun x => μ) fun ω =>
        (uniformDeviation n f μ X (X ∘ ω) -
            ∫ (x : Fin n → Ω), (fun ω => uniformDeviation n f μ X (X ∘ ω)) x ∂Measure.pi fun x => μ) ≥
          ε).toReal := sorry

Tactic state before:
case neg
n : ℕ
Ω : Type u
inst✝⁵ : MeasurableSpace Ω
ι : Type v
𝒳 : Type w
μ : Measure Ω
inst✝⁴ : MeasurableSpace 𝒳
inst✝³ : Nonempty 𝒳
inst✝² : Nonempty ι
inst✝¹ : Countable ι
inst✝ : IsProbabilityMeasure μ
f : ι → 𝒳 → ℝ
hf : ∀ (i : ι), Measurable (f i)
X : Ω → 𝒳
hX : Measurable X
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
t : ℝ
ht : 0 ≤ t
ht' : t * b ^ 2 ≤ 1 / 2
ε : ℝ
hε : 0 ≤ ε
hn✝ : ¬n = 0
hn : 0 < n
⊢ ((Measure.pi fun x => μ) fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)).toReal ≤
    rexp (-ε ^ 2 * t * ↑n)

Tactic: apply le_trans _ (uniformDeviation_mcdiarmid (μ := μ) hX hf hb hf' ht ht' hε)

Tactic state after:
[n : ℕ
 Ω : Type u
 inst✝⁵ : MeasurableSpace Ω
 ι : Type v
 𝒳 : Type w
 μ : Measure Ω
 inst✝⁴ : MeasurableSpace 𝒳
 inst✝³ : Nonempty 𝒳
 inst✝² : Nonempty ι
 inst✝¹ : Countable ι
 inst✝ : IsProbabilityMeasure μ
 f : ι → 𝒳 → ℝ
 hf : ∀ (i : ι), Measurable (f i)
 X : Ω → 𝒳
 hX : Measurable X
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
 t : ℝ
 ht : 0 ≤ t
 ht' : t * b ^ 2 ≤ 1 / 2
 ε : ℝ
 hε : 0 ≤ ε
 hn✝ : ¬n = 0
 hn : 0 < n
 ⊢ ((Measure.pi fun x => μ) fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)).toReal ≤
     ((Measure.pi fun x => μ) fun ω =>
         (uniformDeviation n f μ X (X ∘ ω) -
             ∫ (x : Fin n → Ω), (fun ω => uniformDeviation n f μ X (X ∘ ω)) x ∂Measure.pi fun x => μ) ≥
           ε).toReal]

Premises used:
le_trans : ∀ {α : Type u_1} [inst : Preorder α] {a b c : α}, a ≤ b → b ≤ c → a ≤ c
uniformDeviation_mcdiarmid : ∀ {n : ℕ} {Ω : Type u} [inst : MeasurableSpace Ω] {ι : Type v} {𝒳 : Type w} {μ : Measure Ω}
  {f : ι → 𝒳 → ℝ} [inst_6 : MeasurableSpace 𝒳] [inst_7 : Nonempty 𝒳] [inst_8 : Nonempty ι] [inst_9 : Countable ι]
  [inst_10 : IsProbabilityMeasure μ] {X : Ω → 𝒳},
  Measurable X →
    (∀ (i : ι), Measurable (f i)) →
      ∀ {b : ℝ},
        0 ≤ b →
          (∀ (i : ι) (x : 𝒳), |f i x| ≤ b) →
            ∀ {t : ℝ},
              0 ≤ t →
                t * b ^ 2 ≤ 1 / 2 →
                  ∀ {ε : ℝ},
                    0 ≤ ε →
                      ((Measure.pi fun x => μ) fun ω =>
                            (uniformDeviation n f μ X (X ∘ ω) -
                                ∫ (x : Fin n → Ω),
                                  (fun ω => uniformDeviation n f μ X (X ∘ ω)) x ∂Measure.pi fun x => μ) ≥
                              ε).toReal ≤
                        rexp (-ε ^ 2 * t * ↑n)