zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_245.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {n : ℕ}
  [inst_2 : Countable ι] [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ}
  (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) (ω : Fin n → Ω × Ω) (σ : Signs n) :
  let V := fun f => ∑ i, ↑↑(σ i) * (f (X (ω i).1) - f (X (ω i).2));
  (∀ (f : Z → ℝ), V (-f) = -V f) →
    ∀ ⦃x : ℝ⦄ (I : ι),
      (fun i => |V (f i)|) I = x → |∑ i, ↑↑(σ i) * (f I (X (ω i).1) - f I (X (ω i).2))| ≤ ↑n * (b + b) := sorry

Tactic state before:
case h
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
n : ℕ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
f' : Fin 2 × ι → Z → ℝ := fun i => if (↑i.1 == 0) = true then f i.2 else -f i.2
ω : Fin n → Ω × Ω
σ : Signs n
V : (Z → ℝ) → ℝ := fun f => ∑ i, ↑↑(σ i) * (f (X (ω i).1) - f (X (ω i).2))
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
x : ℝ
I : ι
eq : (fun i => |V (f i)|) I = x
⊢ |∑ i, ↑↑(σ i) * (f I (X (ω i).1) - f I (X (ω i).2))| ≤ ↑n * (b + b)

Tactic:
exact bound_lem' h𝓕' ω I σ

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_246.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {n : ℕ}
  [inst_2 : Countable ι] [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ}
  (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) :
  let f' := fun i => if (↑i.1 == 0) = true then f i.2 else -f i.2;
  ∀ (ω : Fin n → Ω × Ω) (σ : Signs n),
    let V := fun f => ∑ i, ↑↑(σ i) * (f (X (ω i).1) - f (X (ω i).2));
    (∀ (f : Z → ℝ), V (-f) = -V f) →
      BddAbove (Set.range fun i => |V (f i)|) →
        ⨆ I, ∑ i, ↑↑(σ i) * (f' I (X (ω i).1) - f' I (X (ω i).2)) =
          ⨆ I, |∑ i, ↑↑(σ i) * (f I (X (ω i).1) - f I (X (ω i).2))| := sorry

Tactic state before:
case e_f.h.e_a.e_f.h
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
n : ℕ
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
f' : Fin 2 × ι → Z → ℝ := fun i => if (↑i.1 == 0) = true then f i.2 else -f i.2
ω : Fin n → Ω × Ω
σ : Signs n
V : (Z → ℝ) → ℝ := fun f => ∑ i, ↑↑(σ i) * (f (X (ω i).1) - f (X (ω i).2))
hV₀ : ∀ (f : Z → ℝ), V (-f) = -V f
hV₁ : BddAbove (Set.range fun i => |V (f i)|)
⊢ ⨆ I, ∑ i, ↑↑(σ i) * (f' I (X (ω i).1) - f' I (X (ω i).2)) = ⨆ I, |∑ i, ↑↑(σ i) * (f I (X (ω i).1) - f I (X (ω i).2))|

Tactic:
exact (sup_abs_lemma hV₀ hV₁).symm

Tactic state after: