zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_20.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) →
        (∀ᵐ (ω : Ω) ∂μ, ∀ (i : Fin n), |X i ω| ≤ 1) → ∀ (ω : Ω), (∀ (i : Fin n), |X i ω| ≤ 1) → ∑ i, 1 = ↑n := sorry

Tactic state before:
Ω : Type u
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
hS_meas : Measurable S
f : ℝ → ℝ := fun x => rexp (τ * x)
f_meas : Measurable f
f_measω : Measurable fun ω => rexp (τ * S ω)
h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
h_all : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : Fin n), |X i ω| ≤ 1
ω : Ω
hω : ∀ (i : Fin n), |X i ω| ≤ 1
⊢ ∑ i, 1 = ↑n

Tactic:
simp [Finset.sum_const, nsmul_eq_mul, mul_one]

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_21.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) →
        (∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)) →
          HasFiniteIntegral (fun ω => rexp (τ * S ω)) μ → HasFiniteIntegral (fun x => rexp (↑n * τ)) μ := sorry

Tactic state before:
case right
Ω : Type u
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
hS_meas : Measurable S
f : ℝ → ℝ := fun x => rexp (τ * x)
f_meas : Measurable f
f_measω : Measurable fun ω => rexp (τ * S ω)
h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
h_sum_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)
⊢ HasFiniteIntegral (fun ω => rexp (τ * S ω)) μ

Tactic:
apply HasFiniteIntegral.mono' (g := fun _ ↦ Real.exp (↑n * τ))

Tactic state after:
[case right.hg
 Ω : Type u
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 S : Ω → ℝ := fun ω => ∑ i, X i ω
 hS_meas : Measurable S
 f : ℝ → ℝ := fun x => rexp (τ * x)
 f_meas : Measurable f
 f_measω : Measurable fun ω => rexp (τ * S ω)
 h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
 h_sum_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)
 ⊢ HasFiniteIntegral (fun x => rexp (↑n * τ)) μ,
 case right.h
 Ω : Type u
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 S : Ω → ℝ := fun ω => ∑ i, X i ω
 hS_meas : Measurable S
 f : ℝ → ℝ := fun x => rexp (τ * x)
 f_meas : Measurable f
 f_measω : Measurable fun ω => rexp (τ * S ω)
 h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
 h_sum_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)
 ⊢ ∀ᵐ (a : Ω) ∂μ, ‖rexp (τ * S a)‖ ≤ rexp (↑n * τ)]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_22.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) →
        (∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)) →
          HasFiniteIntegral (fun ω => rexp (τ * S ω)) μ → ∀ᵐ (a : Ω) ∂μ, ‖rexp (τ * S a)‖ ≤ rexp (↑n * τ) := sorry

Tactic state before:
case right
Ω : Type u
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
hS_meas : Measurable S
f : ℝ → ℝ := fun x => rexp (τ * x)
f_meas : Measurable f
f_measω : Measurable fun ω => rexp (τ * S ω)
h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
h_sum_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)
⊢ HasFiniteIntegral (fun ω => rexp (τ * S ω)) μ

Tactic:
apply HasFiniteIntegral.mono' (g := fun _ ↦ Real.exp (↑n * τ))

Tactic state after:
[case right.hg
 Ω : Type u
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 S : Ω → ℝ := fun ω => ∑ i, X i ω
 hS_meas : Measurable S
 f : ℝ → ℝ := fun x => rexp (τ * x)
 f_meas : Measurable f
 f_measω : Measurable fun ω => rexp (τ * S ω)
 h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
 h_sum_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)
 ⊢ HasFiniteIntegral (fun x => rexp (↑n * τ)) μ,
 case right.h
 Ω : Type u
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 S : Ω → ℝ := fun ω => ∑ i, X i ω
 hS_meas : Measurable S
 f : ℝ → ℝ := fun x => rexp (τ * x)
 f_meas : Measurable f
 f_measω : Measurable fun ω => rexp (τ * S ω)
 h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
 h_sum_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)
 ⊢ ∀ᵐ (a : Ω) ∂μ, ‖rexp (τ * S a)‖ ≤ rexp (↑n * τ)]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_23.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) →
        (∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)) →
          HasFiniteIntegral (fun x => rexp (↑n * τ)) μ → ∫⁻ (a : Ω), ‖rexp (↑n * τ)‖ₑ ∂μ < ⊤ := sorry

Tactic state before:
case right.hg
Ω : Type u
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
hS_meas : Measurable S
f : ℝ → ℝ := fun x => rexp (τ * x)
f_meas : Measurable f
f_measω : Measurable fun ω => rexp (τ * S ω)
h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
h_sum_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)
⊢ HasFiniteIntegral (fun x => rexp (↑n * τ)) μ

Tactic:
simp only [HasFiniteIntegral]

Tactic state after:
[case right.hg
 Ω : Type u
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 S : Ω → ℝ := fun ω => ∑ i, X i ω
 hS_meas : Measurable S
 f : ℝ → ℝ := fun x => rexp (τ * x)
 f_meas : Measurable f
 f_measω : Measurable fun ω => rexp (τ * S ω)
 h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
 h_sum_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)
 ⊢ ∫⁻ (a : Ω), ‖rexp (↑n * τ)‖ₑ ∂μ < ⊤]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_24.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) →
        (∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)) →
          ∫⁻ (a : Ω), ‖rexp (↑n * τ)‖ₑ ∂μ < ⊤ → ‖rexp (↑n * τ)‖ₑ * μ univ < ⊤ := sorry

Tactic state before:
case right.hg
Ω : Type u
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
hS_meas : Measurable S
f : ℝ → ℝ := fun x => rexp (τ * x)
f_meas : Measurable f
f_measω : Measurable fun ω => rexp (τ * S ω)
h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
h_sum_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)
⊢ ∫⁻ (a : Ω), ‖rexp (↑n * τ)‖ₑ ∂μ < ⊤

Tactic:
rw [lintegral_const]

Tactic state after:
[case right.hg
 Ω : Type u
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 S : Ω → ℝ := fun ω => ∑ i, X i ω
 hS_meas : Measurable S
 f : ℝ → ℝ := fun x => rexp (τ * x)
 f_meas : Measurable f
 f_measω : Measurable fun ω => rexp (τ * S ω)
 h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
 h_sum_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)
 ⊢ ‖rexp (↑n * τ)‖ₑ * μ univ < ⊤]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_25.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) →
        (∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)) → ‖rexp (↑n * τ)‖ₑ * μ univ < ⊤ := sorry

Tactic state before:
case right.hg
Ω : Type u
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
hS_meas : Measurable S
f : ℝ → ℝ := fun x => rexp (τ * x)
f_meas : Measurable f
f_measω : Measurable fun ω => rexp (τ * S ω)
h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
h_sum_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)
⊢ ‖rexp (↑n * τ)‖ₑ * μ univ < ⊤

Tactic:
simp [mul_eq_top, ENNReal.ofReal_lt_top]

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_26.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) →
        (∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)) → ∀ᵐ (a : Ω) ∂μ, ‖rexp (τ * S a)‖ ≤ rexp (↑n * τ) := sorry

Tactic state before:
case right.h
Ω : Type u
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
hS_meas : Measurable S
f : ℝ → ℝ := fun x => rexp (τ * x)
f_meas : Measurable f
f_measω : Measurable fun ω => rexp (τ * S ω)
h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
h_sum_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ)
⊢ ∀ᵐ (a : Ω) ∂μ, ‖rexp (τ * S a)‖ ≤ rexp (↑n * τ)

Tactic:
exact h_sum_bound

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_27.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S → Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ → Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ := sorry

Tactic state before:
Ω : Type u
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
hS_meas : Measurable S
h_exp_prod : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ
⊢ Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ

Tactic:
exact h_exp_prod

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_28.{u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (i : Fin n) (ω : Ω) (hω : |X i ω| ≤ 1)
  (h_original_goal : X i ω ∈ Icc (-1) 1) : -1 ≤ X i ω ∧ X i ω ≤ 1 := sorry

Tactic state before:
Ω : Type u_1
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
i : Fin n
ω : Ω
hω : |X i ω| ≤ 1
⊢ X i ω ∈ Icc (-1) 1

Tactic:
rw [Set.mem_Icc]

Tactic state after:
[Ω : Type u_1
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 i : Fin n
 ω : Ω
 hω : |X i ω| ≤ 1
 ⊢ -1 ≤ X i ω ∧ X i ω ≤ 1]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_29.{u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (i : Fin n) (ω : Ω) (hω : |X i ω| ≤ 1) :
  -1 ≤ X i ω ∧ X i ω ≤ 1 := sorry

Tactic state before:
Ω : Type u_1
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
i : Fin n
ω : Ω
hω : |X i ω| ≤ 1
⊢ -1 ≤ X i ω ∧ X i ω ≤ 1

Tactic:
exact abs_le.mp hω

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_30.{u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) (i : Fin n) (ω : Ω) (hω : |X i ω| ≤ 1)
  (this : -1 ≤ X i ω ∧ X i ω ≤ 1) (h_original_goal : rexp (-τ) ≤ rexp (τ * X i ω)) : -τ ≤ τ * X i ω := sorry

Tactic state before:
case h.left
Ω : Type u_1
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
i : Fin n
ω : Ω
hω : |X i ω| ≤ 1
this : -1 ≤ X i ω ∧ X i ω ≤ 1
⊢ rexp (-τ) ≤ rexp (τ * X i ω)

Tactic:
apply Real.exp_le_exp.mpr

Tactic state after:
[case h.left
 Ω : Type u_1
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 i : Fin n
 ω : Ω
 hω : |X i ω| ≤ 1
 this : -1 ≤ X i ω ∧ X i ω ≤ 1
 ⊢ -τ ≤ τ * X i ω]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_31.{u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) (i : Fin n) (ω : Ω) (hω : |X i ω| ≤ 1)
  (this : -1 ≤ X i ω ∧ X i ω ≤ 1) (h_original_goal : -τ ≤ τ * X i ω) : -τ ≤ X i ω * τ := sorry

Tactic state before:
case h.left
Ω : Type u_1
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
i : Fin n
ω : Ω
hω : |X i ω| ≤ 1
this : -1 ≤ X i ω ∧ X i ω ≤ 1
⊢ -τ ≤ τ * X i ω

Tactic:
rw [mul_comm]

Tactic state after:
[case h.left
 Ω : Type u_1
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 i : Fin n
 ω : Ω
 hω : |X i ω| ≤ 1
 this : -1 ≤ X i ω ∧ X i ω ≤ 1
 ⊢ -τ ≤ X i ω * τ]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_32.{u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) (i : Fin n) (ω : Ω) (hω : |X i ω| ≤ 1)
  (this : -1 ≤ X i ω ∧ X i ω ≤ 1) (h_original_goal : rexp (τ * X i ω) ≤ rexp τ) : τ * X i ω ≤ τ := sorry

Tactic state before:
case h.right
Ω : Type u_1
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
i : Fin n
ω : Ω
hω : |X i ω| ≤ 1
this : -1 ≤ X i ω ∧ X i ω ≤ 1
⊢ rexp (τ * X i ω) ≤ rexp τ

Tactic:
apply Real.exp_le_exp.mpr

Tactic state after:
[case h.right
 Ω : Type u_1
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 i : Fin n
 ω : Ω
 hω : |X i ω| ≤ 1
 this : -1 ≤ X i ω ∧ X i ω ≤ 1
 ⊢ τ * X i ω ≤ τ]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_33.{u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) (i : Fin n) (ω : Ω) (hω : |X i ω| ≤ 1)
  (this : -1 ≤ X i ω ∧ X i ω ≤ 1) (h_original_goal : τ * X i ω ≤ τ) : X i ω * τ ≤ τ := sorry

Tactic state before:
case h.right
Ω : Type u_1
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
i : Fin n
ω : Ω
hω : |X i ω| ≤ 1
this : -1 ≤ X i ω ∧ X i ω ≤ 1
⊢ τ * X i ω ≤ τ

Tactic:
rw [mul_comm]

Tactic state after:
[case h.right
 Ω : Type u_1
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 i : Fin n
 ω : Ω
 hω : |X i ω| ≤ 1
 this : -1 ≤ X i ω ∧ X i ω ≤ 1
 ⊢ X i ω * τ ≤ τ]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_34.{u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) (i : Fin n)
  (hbdd : ∀ᵐ (ω : Ω) ∂μ, rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)) (ω : Ω)
  (hω : rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)) (h_original_goal : ‖rexp (τ * X i ω)‖ ≤ rexp τ) :
  rexp (τ * X i ω) ≤ rexp τ := sorry

Tactic state before:
case h
Ω : Type u_1
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
i : Fin n
hbdd : ∀ᵐ (ω : Ω) ∂μ, rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)
ω : Ω
hω : rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)
⊢ ‖rexp (τ * X i ω)‖ ≤ rexp τ

Tactic:
rw [norm_eq_abs, abs_of_nonneg (exp_pos _).le]

Tactic state after:
[case h
 Ω : Type u_1
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 i : Fin n
 hbdd : ∀ᵐ (ω : Ω) ∂μ, rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)
 ω : Ω
 hω : rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)
 ⊢ rexp (τ * X i ω) ≤ rexp τ]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_35.{u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) (i : Fin n)
  (hbdd : ∀ᵐ (ω : Ω) ∂μ, rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)) (ω : Ω)
  (hω : rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)) : rexp (τ * X i ω) ≤ rexp τ := sorry

Tactic state before:
case h
Ω : Type u_1
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
i : Fin n
hbdd : ∀ᵐ (ω : Ω) ∂μ, rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)
ω : Ω
hω : rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)
⊢ rexp (τ * X i ω) ≤ rexp τ

Tactic:
exact hω.2

Tactic state after: