zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_0.{w, v, u} {n : ℕ} {Ω : Type u} [inst : MeasurableSpace Ω] {ι : Type v} {𝒳 : Type w}
  {μ : Measure Ω} {f : ι → 𝒳 → ℝ} [inst_1 : Nonempty ι] [inst_2 : Countable ι] [inst_3 : IsProbabilityMeasure μ]
  (hn : 0 < n) (X : Ω → 𝒳) (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b)
  (hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b)
  (h_original_goal :
    (∫ (x : Fin n → Ω), (fun ω => uniformDeviation n f μ X (X ∘ ω)) x ∂Measure.pi fun x => μ) ≤
      2 • rademacherComplexity n f μ X) :
  (↑n * ∫ (x : Fin n → Ω), (fun ω => uniformDeviation n f μ X (X ∘ ω)) x ∂Measure.pi fun x => μ) ≤
    ↑n * 2 • rademacherComplexity n f μ X := sorry

Tactic state before:
n : ℕ
Ω : Type u
inst✝³ : MeasurableSpace Ω
ι : Type v
𝒳 : Type w
μ : Measure Ω
f : ι → 𝒳 → ℝ
inst✝² : Nonempty ι
inst✝¹ : Countable ι
inst✝ : IsProbabilityMeasure μ
hn : 0 < n
X : Ω → 𝒳
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
⊢ (∫ (x : Fin n → Ω), (fun ω => uniformDeviation n f μ X (X ∘ ω)) x ∂Measure.pi fun x => μ) ≤
    2 • rademacherComplexity n f μ X

Tactic:
apply le_of_mul_le_mul_left _ (Nat.cast_pos.mpr hn)

Tactic state after:
[n : ℕ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 ι : Type v
 𝒳 : Type w
 μ : Measure Ω
 f : ι → 𝒳 → ℝ
 inst✝² : Nonempty ι
 inst✝¹ : Countable ι
 inst✝ : IsProbabilityMeasure μ
 hn : 0 < n
 X : Ω → 𝒳
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
 ⊢ (↑n * ∫ (x : Fin n → Ω), (fun ω => uniformDeviation n f μ X (X ∘ ω)) x ∂Measure.pi fun x => μ) ≤
     ↑n * 2 • rademacherComplexity n f μ X]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_1.{w, v, u} {n : ℕ} {Ω : Type u} [inst : MeasurableSpace Ω] {ι : Type v} {𝒳 : Type w}
  {μ : Measure Ω} {f : ι → 𝒳 → ℝ} [inst_1 : Nonempty ι] [inst_2 : Countable ι] [inst_3 : IsProbabilityMeasure μ]
  (hn : 0 < n) (X : Ω → 𝒳) (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b)
  (hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b)
  (h_original_goal :
    (↑n * ∫ (x : Fin n → Ω), (fun ω => uniformDeviation n f μ X (X ∘ ω)) x ∂Measure.pi fun x => μ) =
      ∫ (x : Fin n → Ω),
        (fun ω => ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ|) x ∂Measure.pi fun x => μ) :
  (∫ (a : Fin n → Ω), ↑n * uniformDeviation n f μ X (X ∘ a) ∂Measure.pi fun x => μ) =
    ∫ (x : Fin n → Ω),
      (fun ω => ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ|)
        x ∂Measure.pi fun x => μ := sorry

Tactic state before:
case h.e'_3
n : ℕ
Ω : Type u
inst✝³ : MeasurableSpace Ω
ι : Type v
𝒳 : Type w
μ : Measure Ω
f : ι → 𝒳 → ℝ
inst✝² : Nonempty ι
inst✝¹ : Countable ι
inst✝ : IsProbabilityMeasure μ
hn : 0 < n
X : Ω → 𝒳
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
⊢ (↑n * ∫ (x : Fin n → Ω), (fun ω => uniformDeviation n f μ X (X ∘ ω)) x ∂Measure.pi fun x => μ) =
    ∫ (x : Fin n → Ω),
      (fun ω => ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ|) x ∂Measure.pi fun x => μ

Tactic:
rw [← integral_mul_left]

Tactic state after:
[case h.e'_3
 n : ℕ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 ι : Type v
 𝒳 : Type w
 μ : Measure Ω
 f : ι → 𝒳 → ℝ
 inst✝² : Nonempty ι
 inst✝¹ : Countable ι
 inst✝ : IsProbabilityMeasure μ
 hn : 0 < n
 X : Ω → 𝒳
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
 ⊢ (∫ (a : Fin n → Ω), ↑n * uniformDeviation n f μ X (X ∘ a) ∂Measure.pi fun x => μ) =
     ∫ (x : Fin n → Ω),
       (fun ω => ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ|) x ∂Measure.pi fun x => μ]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_2.{w, v, u} {n : ℕ} {Ω : Type u} [inst : MeasurableSpace Ω] {ι : Type v} {𝒳 : Type w}
  {μ : Measure Ω} {f : ι → 𝒳 → ℝ} [inst_1 : Nonempty ι] [inst_2 : Countable ι] [inst_3 : IsProbabilityMeasure μ]
  (hn : 0 < n) (X : Ω → 𝒳) (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b)
  (hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b)
  (h_original_goal :
    (∫ (a : Fin n → Ω), ↑n * uniformDeviation n f μ X (X ∘ a) ∂Measure.pi fun x => μ) =
      ∫ (x : Fin n → Ω),
        (fun ω => ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ|) x ∂Measure.pi fun x => μ) :
  (fun a => ↑n * uniformDeviation n f μ X (X ∘ a)) = fun ω =>
    ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ| := sorry

Tactic state before:
case h.e'_3
n : ℕ
Ω : Type u
inst✝³ : MeasurableSpace Ω
ι : Type v
𝒳 : Type w
μ : Measure Ω
f : ι → 𝒳 → ℝ
inst✝² : Nonempty ι
inst✝¹ : Countable ι
inst✝ : IsProbabilityMeasure μ
hn : 0 < n
X : Ω → 𝒳
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
⊢ (∫ (a : Fin n → Ω), ↑n * uniformDeviation n f μ X (X ∘ a) ∂Measure.pi fun x => μ) =
    ∫ (x : Fin n → Ω),
      (fun ω => ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ|) x ∂Measure.pi fun x => μ

Tactic:
apply integral_congr_ae (Filter.EventuallyEq.of_eq _)

Tactic state after:
[n : ℕ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 ι : Type v
 𝒳 : Type w
 μ : Measure Ω
 f : ι → 𝒳 → ℝ
 inst✝² : Nonempty ι
 inst✝¹ : Countable ι
 inst✝ : IsProbabilityMeasure μ
 hn : 0 < n
 X : Ω → 𝒳
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
 ⊢ (fun a => ↑n * uniformDeviation n f μ X (X ∘ a)) = fun ω =>
     ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_3.{w, v, u} {n : ℕ} {Ω : Type u} [inst : MeasurableSpace Ω] {ι : Type v} {𝒳 : Type w}
  {μ : Measure Ω} {f : ι → 𝒳 → ℝ} [inst_1 : Nonempty ι] [inst_2 : Countable ι] [inst_3 : IsProbabilityMeasure μ]
  (hn : 0 < n) (X : Ω → 𝒳) (hf : ∀ (i : ι), Measurable (f i ∘ X)) {b : ℝ} (hb : 0 ≤ b)
  (hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b) (ω : Fin n → Ω)
  (h_original_goal :
    ↑n * uniformDeviation n f μ X (X ∘ ω) = ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ|) :
  ⨆ i, ↑n * |(↑n)⁻¹ * ∑ k, f i ((X ∘ ω) k) - ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ| =
    ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ| := sorry

Tactic state before:
case h
n : ℕ
Ω : Type u
inst✝³ : MeasurableSpace Ω
ι : Type v
𝒳 : Type w
μ : Measure Ω
f : ι → 𝒳 → ℝ
inst✝² : Nonempty ι
inst✝¹ : Countable ι
inst✝ : IsProbabilityMeasure μ
hn : 0 < n
X : Ω → 𝒳
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hb : 0 ≤ b
hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
ω : Fin n → Ω
⊢ ↑n * uniformDeviation n f μ X (X ∘ ω) = ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ|

Tactic:
rw [uniformDeviation, Real.mul_iSup_of_nonneg (by norm_num)]

Tactic state after:
[case h
 n : ℕ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 ι : Type v
 𝒳 : Type w
 μ : Measure Ω
 f : ι → 𝒳 → ℝ
 inst✝² : Nonempty ι
 inst✝¹ : Countable ι
 inst✝ : IsProbabilityMeasure μ
 hn : 0 < n
 X : Ω → 𝒳
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hb : 0 ≤ b
 hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b
 ω : Fin n → Ω
 ⊢ ⨆ i, ↑n * |(↑n)⁻¹ * ∑ k, f i ((X ∘ ω) k) - ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ| =
     ⨆ i, |∑ k, f i (X (ω k)) - n • ∫ (x : Ω), (fun ω' => f i (X ω')) x ∂μ|]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_4.{w, v, u} {n : ℕ} {Ω : Type u} [inst : MeasurableSpace Ω] {ι : Type v} {𝒳 : Type w}
  {μ : Measure Ω} {f : ι → 𝒳 → ℝ} [inst_1 : MeasurableSpace 𝒳] [inst_2 : Nonempty 𝒳] [inst_3 : Nonempty ι]
  [inst_4 : Countable ι] [inst_5 : IsProbabilityMeasure μ] {X : Ω → 𝒳} (hX : Measurable X)
  (hf : ∀ (i : ι), Measurable (f i)) {b : ℝ} (hb : 0 ≤ b) (hf' : ∀ (i : ι) (x : 𝒳), |f i x| ≤ b) {t : ℝ} (ht_1 : 0 ≤ t)
  (ht' : t * b ^ 2 ≤ 1 / 2) (hn_1 : ¬n = 0) (hn : 0 < n) (hn' : 0 < ↑n) :
  let c := fun i => (↑n)⁻¹ * 2 * b;
  ↑n * t / 2 ≥ 0 → ↑n * t / 2 * ∑ i, c i ^ 2 ≤ 1 → 1 / 2 * (↑n * t / 2 * ∑ i, c i ^ 2) ≤ 1 / 2 * 1 := sorry

Tactic state before:
n : ℕ
Ω : Type u
inst✝⁵ : MeasurableSpace Ω
ι : Type v
𝒳 : Type w
μ : Measure Ω
f : ι → 𝒳 → ℝ
inst✝⁴ : MeasurableSpace 𝒳
inst✝³ : Nonempty 𝒳
inst✝² : Nonempty ι
inst✝¹ : Countable ι
inst✝ : IsProbabilityMeasure μ
X : Ω → 𝒳
hX : Measurable X
hf : ∀ (i : ι), Measurable (f i)
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
hn' : 0 < ↑n
c : Fin n → ℝ := fun i => (↑n)⁻¹ * 2 * b
ht : ↑n * t / 2 ≥ 0
⊢ ↑n * t / 2 * ∑ i, c i ^ 2 ≤ 1

Tactic:
apply le_of_mul_le_mul_left _ (show (0 : ℝ) < 1 / 2 from by linarith)

Tactic state after:
[n : ℕ
 Ω : Type u
 inst✝⁵ : MeasurableSpace Ω
 ι : Type v
 𝒳 : Type w
 μ : Measure Ω
 f : ι → 𝒳 → ℝ
 inst✝⁴ : MeasurableSpace 𝒳
 inst✝³ : Nonempty 𝒳
 inst✝² : Nonempty ι
 inst✝¹ : Countable ι
 inst✝ : IsProbabilityMeasure μ
 X : Ω → 𝒳
 hX : Measurable X
 hf : ∀ (i : ι), Measurable (f i)
 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
 hn' : 0 < ↑n
 c : Fin n → ℝ := fun i => (↑n)⁻¹ * 2 * b
 ht : ↑n * t / 2 ≥ 0
 ⊢ 1 / 2 * (↑n * t / 2 * ∑ i, c i ^ 2) ≤ 1 / 2 * 1]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

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]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_6.{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 ≤
      ((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) :
  ((Measure.pi fun x => μ) fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)) ≤
    (Measure.pi fun x => μ) fun ω =>
      ε ≤
        uniformDeviation n f μ X (X ∘ ω) -
          ∫ (x : Fin n → Ω), uniformDeviation n f μ X (X ∘ x) ∂Measure.pi fun x => μ := sorry

Tactic state before:
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

Tactic:
simp only [ge_iff_le, ne_eq, measure_ne_top, not_false_eq_true, ENNReal.toReal_le_toReal]

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)) ≤
     (Measure.pi fun x => μ) fun ω =>
       ε ≤
         uniformDeviation n f μ X (X ∘ ω) - ∫ (x : Fin n → Ω), uniformDeviation n f μ X (X ∘ x) ∂Measure.pi fun x => μ]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

open scoped ENNReal

theorem extracted_formal_statement_7.{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)) ≤
      (Measure.pi fun x => μ) fun ω =>
        ε ≤
          uniformDeviation n f μ X (X ∘ ω) -
            ∫ (x : Fin n → Ω), uniformDeviation n f μ X (X ∘ x) ∂Measure.pi fun x => μ) :
  (fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)) ⊆ fun ω =>
    ε ≤
      uniformDeviation n f μ X (X ∘ ω) -
        ∫ (x : Fin n → Ω), uniformDeviation n f μ X (X ∘ x) ∂Measure.pi fun x => μ := sorry

Tactic state before:
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)) ≤
    (Measure.pi fun x => μ) fun ω =>
      ε ≤ uniformDeviation n f μ X (X ∘ ω) - ∫ (x : Fin n → Ω), uniformDeviation n f μ X (X ∘ x) ∂Measure.pi fun x => μ

Tactic:
apply measure_mono

Tactic state after:
[case h
 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
 ⊢ (fun ω => 2 • rademacherComplexity n f μ X + ε ≤ uniformDeviation n f μ X (X ∘ ω)) ⊆ fun ω =>
     ε ≤ uniformDeviation n f μ X (X ∘ ω) - ∫ (x : Fin n → Ω), uniformDeviation n f μ X (X ∘ x) ∂Measure.pi fun x => μ]