zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_36.{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))
  (h_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) : HasFiniteIntegral (fun ω => rexp (τ * X i ω)) μ := sorry

Tactic state before:
case 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
hbdd : ∀ᵐ (ω : Ω) ∂μ, rexp (τ * X i ω) ∈ Icc (rexp (-τ)) (rexp τ)
h_bound : ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
⊢ HasFiniteIntegral (fun ω => rexp (τ * X i ω)) μ

Tactic:
exact hasFiniteIntegral_of_bounded h_bound

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_37.{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) (i : Fin n) (h1 : AEStronglyMeasurable (fun x => 1) μ)
  (h2 : AEStronglyMeasurable (fun ω => τ * X i ω) μ) (h3 : AEStronglyMeasurable (fun ω => X i ω ^ 2) μ)
  (h4 : AEStronglyMeasurable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ)
  (h5 : AEStronglyMeasurable ((fun x => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ)
  (ω : Ω) :
  1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ) =
    ((fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) ω := 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
i : Fin n
h1 : AEStronglyMeasurable (fun x => 1) μ
h2 : AEStronglyMeasurable (fun ω => τ * X i ω) μ
h3 : AEStronglyMeasurable (fun ω => X i ω ^ 2) μ
h4 : AEStronglyMeasurable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ
h5 : AEStronglyMeasurable ((fun x => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ
ω : Ω
⊢ 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ) =
    ((fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) ω

Tactic:
simp only [Pi.add_apply, add_assoc]

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_38.{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) (i : Fin n) (h1 : AEStronglyMeasurable (fun x => 1) μ)
  (h2 : AEStronglyMeasurable (fun ω => τ * X i ω) μ) (h3 : AEStronglyMeasurable (fun ω => X i ω ^ 2) μ)
  (h4 : AEStronglyMeasurable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ)
  (h5 : AEStronglyMeasurable ((fun x => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ)
  (this :
    (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) =
      (fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)))
  (h_original_goal : AEStronglyMeasurable (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) μ) :
  AEStronglyMeasurable ((fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ := sorry

Tactic state before:
case 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
i : Fin n
h1 : AEStronglyMeasurable (fun x => 1) μ
h2 : AEStronglyMeasurable (fun ω => τ * X i ω) μ
h3 : AEStronglyMeasurable (fun ω => X i ω ^ 2) μ
h4 : AEStronglyMeasurable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ
h5 : AEStronglyMeasurable ((fun x => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ
this :
  (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) =
    (fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))
⊢ AEStronglyMeasurable (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) μ

Tactic:
rw [this]

Tactic state after:
[case 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
 i : Fin n
 h1 : AEStronglyMeasurable (fun x => 1) μ
 h2 : AEStronglyMeasurable (fun ω => τ * X i ω) μ
 h3 : AEStronglyMeasurable (fun ω => X i ω ^ 2) μ
 h4 : AEStronglyMeasurable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ
 h5 : AEStronglyMeasurable ((fun x => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ
 this :
   (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) =
     (fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))
 ⊢ AEStronglyMeasurable ((fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_39.{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) (i : Fin n) (h1 : AEStronglyMeasurable (fun x => 1) μ)
  (h2 : AEStronglyMeasurable (fun ω => τ * X i ω) μ) (h3 : AEStronglyMeasurable (fun ω => X i ω ^ 2) μ)
  (h4 : AEStronglyMeasurable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ)
  (h5 : AEStronglyMeasurable ((fun x => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ)
  (this :
    (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) =
      (fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) :
  AEStronglyMeasurable ((fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ := sorry

Tactic state before:
case 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
i : Fin n
h1 : AEStronglyMeasurable (fun x => 1) μ
h2 : AEStronglyMeasurable (fun ω => τ * X i ω) μ
h3 : AEStronglyMeasurable (fun ω => X i ω ^ 2) μ
h4 : AEStronglyMeasurable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ
h5 : AEStronglyMeasurable ((fun x => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ
this :
  (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) =
    (fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))
⊢ AEStronglyMeasurable ((fun ω => 1) + ((fun ω => τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))) μ

Tactic:
exact h5

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_40.{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) (h_const : Integrable (fun x => 1) μ) :
  AEStronglyMeasurable (fun ω => X i ω) μ := sorry

Tactic state before:
case 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
i : Fin n
h_const : Integrable (fun x => 1) μ
⊢ AEStronglyMeasurable (fun ω => X i ω) μ

Tactic:
exact (hXm i).aestronglyMeasurable

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_41.{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) (h_const : Integrable (fun x => 1) μ) (ω : Ω)
  (hω : |X i ω| ≤ 1) : ‖X i ω‖ ≤ 1 := 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
i : Fin n
h_const : Integrable (fun x => 1) μ
ω : Ω
hω : |X i ω| ≤ 1
⊢ ‖X i ω‖ ≤ 1

Tactic:
exact hω

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_42.{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) (h_const : Integrable (fun x => 1) μ)
  (h_bound : ∀ᵐ (ω : Ω) ∂μ, ‖X i ω‖ ≤ 1) (h_original_goal : HasFiniteIntegral (fun ω => X i ω) μ) :
  ∀ᵐ (a : Ω) ∂μ, ‖X i a‖ ≤ 1 := sorry

Tactic state before:
case 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
i : Fin n
h_const : Integrable (fun x => 1) μ
h_bound : ∀ᵐ (ω : Ω) ∂μ, ‖X i ω‖ ≤ 1
⊢ HasFiniteIntegral (fun ω => X i ω) μ

Tactic:
apply hasFiniteIntegral_of_bounded (C := 1)

Tactic state after:
[case 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
 i : Fin n
 h_const : Integrable (fun x => 1) μ
 h_bound : ∀ᵐ (ω : Ω) ∂μ, ‖X i ω‖ ≤ 1
 ⊢ ∀ᵐ (a : Ω) ∂μ, ‖X i a‖ ≤ 1]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_43.{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) (h_const : Integrable (fun x => 1) μ)
  (h_bound : ∀ᵐ (ω : Ω) ∂μ, ‖X i ω‖ ≤ 1) : ∀ᵐ (a : Ω) ∂μ, ‖X i a‖ ≤ 1 := sorry

Tactic state before:
case 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
i : Fin n
h_const : Integrable (fun x => 1) μ
h_bound : ∀ᵐ (ω : Ω) ∂μ, ‖X i ω‖ ≤ 1
⊢ ∀ᵐ (a : Ω) ∂μ, ‖X i a‖ ≤ 1

Tactic:
exact h_bound

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_44.{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) (i : Fin n) (h_const : Integrable (fun x => 1) μ)
  (hX_int : Integrable (fun ω => X i ω) μ) (h_mul' : Integrable (fun x => τ * X i x) μ) :
  Integrable (fun ω => τ * X i ω) μ := 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
i : Fin n
h_const : Integrable (fun x => 1) μ
hX_int : Integrable (fun ω => X i ω) μ
h_mul' : Integrable (fun x => τ * X i x) μ
⊢ Integrable (fun ω => τ * X i ω) μ

Tactic:
exact h_mul'

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_45.{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) (i : Fin n) (h_const : Integrable (fun x => 1) μ)
  (h_mul : Integrable (fun ω => τ * X i ω) μ) (this : HasFiniteIntegral ((fun x => 1) + fun ω => τ * X i ω) μ)
  (h_original_goal : Integrable (fun ω => 1 + τ * X i ω) μ) : Integrable (fun x => τ * X i x) μ := 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
i : Fin n
h_const : Integrable (fun x => 1) μ
h_mul : Integrable (fun ω => τ * X i ω) μ
this : HasFiniteIntegral ((fun x => 1) + fun ω => τ * X i ω) μ
⊢ Integrable (fun ω => 1 + τ * X i ω) μ

Tactic:
simp [<- Pi.add_apply]

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
 i : Fin n
 h_const : Integrable (fun x => 1) μ
 h_mul : Integrable (fun ω => τ * X i ω) μ
 this : HasFiniteIntegral ((fun x => 1) + fun ω => τ * X i ω) μ
 ⊢ Integrable (fun x => τ * X i x) μ]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_46.{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) (i : Fin n) (h_const : Integrable (fun x => 1) μ)
  (h_mul : Integrable (fun ω => τ * X i ω) μ) (this : HasFiniteIntegral ((fun x => 1) + fun ω => τ * X i ω) μ) :
  Integrable (fun x => τ * X i x) μ := 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
i : Fin n
h_const : Integrable (fun x => 1) μ
h_mul : Integrable (fun ω => τ * X i ω) μ
this : HasFiniteIntegral ((fun x => 1) + fun ω => τ * X i ω) μ
⊢ Integrable (fun x => τ * X i x) μ

Tactic:
exact h_mul

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_47.{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) (i : Fin n)
  (h1 : Integrable (fun ω => 1 + τ * X i ω) μ) : AEStronglyMeasurable (fun ω => X i ω ^ 2) μ := sorry

Tactic state before:
case 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
i : Fin n
h1 : Integrable (fun ω => 1 + τ * X i ω) μ
⊢ AEStronglyMeasurable (fun ω => X i ω ^ 2) μ

Tactic:
exact (Measurable.pow (hXm i) (by measurability)).aestronglyMeasurable

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_48.{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) (i : Fin n) (h1 : Integrable (fun ω => 1 + τ * X i ω) μ)
  (a : Ω) (ha : |X i a| ≤ 1) (h_original_goal : X i a ^ 2 ≤ 1) : |X i a| ≤ 1 := 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
i : Fin n
h1 : Integrable (fun ω => 1 + τ * X i ω) μ
a : Ω
ha : |X i a| ≤ 1
⊢ X i a ^ 2 ≤ 1

Tactic:
rw [sq_le_one_iff_abs_le_one]

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
 i : Fin n
 h1 : Integrable (fun ω => 1 + τ * X i ω) μ
 a : Ω
 ha : |X i a| ≤ 1
 ⊢ |X i a| ≤ 1]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_49.{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) (i : Fin n) (h1 : Integrable (fun ω => 1 + τ * X i ω) μ)
  (a : Ω) (ha : |X i a| ≤ 1) : |X i a| ≤ 1 := 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
i : Fin n
h1 : Integrable (fun ω => 1 + τ * X i ω) μ
a : Ω
ha : |X i a| ≤ 1
⊢ |X i a| ≤ 1

Tactic:
exact ha

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_50.{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) (i : Fin n) (h1 : Integrable (fun ω => 1 + τ * X i ω) μ)
  (h_bound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≤ 1) (h_lbound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≥ 0) (a : Ω) (ha : X i a ^ 2 ≤ 1)
  (hlb : X i a ^ 2 ≥ 0) (h_original_goal : X i a ^ 2 ∈ Icc 0 1) : 0 ≤ X i a ^ 2 ∧ X i a ^ 2 ≤ 1 := 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
i : Fin n
h1 : Integrable (fun ω => 1 + τ * X i ω) μ
h_bound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≤ 1
h_lbound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≥ 0
a : Ω
ha : X i a ^ 2 ≤ 1
hlb : X i a ^ 2 ≥ 0
⊢ X i a ^ 2 ∈ Icc 0 1

Tactic:
simp only [Set.mem_Icc]

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
 i : Fin n
 h1 : Integrable (fun ω => 1 + τ * X i ω) μ
 h_bound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≤ 1
 h_lbound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≥ 0
 a : Ω
 ha : X i a ^ 2 ≤ 1
 hlb : X i a ^ 2 ≥ 0
 ⊢ 0 ≤ X i a ^ 2 ∧ X i a ^ 2 ≤ 1]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_51.{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) (i : Fin n) (h1 : Integrable (fun ω => 1 + τ * X i ω) μ)
  (h_bound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≤ 1) (h_lbound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≥ 0) (a : Ω) (ha : X i a ^ 2 ≤ 1)
  (hlb : X i a ^ 2 ≥ 0) : 0 ≤ X i a ^ 2 := 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
i : Fin n
h1 : Integrable (fun ω => 1 + τ * X i ω) μ
h_bound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≤ 1
h_lbound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≥ 0
a : Ω
ha : X i a ^ 2 ≤ 1
hlb : X i a ^ 2 ≥ 0
⊢ 0 ≤ X i a ^ 2

Tactic:
exact hlb

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_52.{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) (i : Fin n) (h1 : Integrable (fun ω => 1 + τ * X i ω) μ)
  (h_bound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≤ 1) (h_lbound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≥ 0) (a : Ω) (ha : X i a ^ 2 ≤ 1)
  (hlb : X i a ^ 2 ≥ 0) : X i a ^ 2 ≤ 1 := 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
i : Fin n
h1 : Integrable (fun ω => 1 + τ * X i ω) μ
h_bound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≤ 1
h_lbound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≥ 0
a : Ω
ha : X i a ^ 2 ≤ 1
hlb : X i a ^ 2 ≥ 0
⊢ X i a ^ 2 ≤ 1

Tactic:
exact ha

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_53.{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) (i : Fin n) (h1 : Integrable (fun ω => 1 + τ * X i ω) μ)
  (h_bound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≤ 1) (h_lbound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≥ 0)
  (h_box : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ∈ Icc 0 1) : HasFiniteIntegral (fun ω => X i ω ^ 2) μ := sorry

Tactic state before:
case 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
i : Fin n
h1 : Integrable (fun ω => 1 + τ * X i ω) μ
h_bound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≤ 1
h_lbound : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ≥ 0
h_box : ∀ᵐ (a : Ω) ∂μ, X i a ^ 2 ∈ Icc 0 1
⊢ HasFiniteIntegral (fun ω => X i ω ^ 2) μ

Tactic:
apply HasFiniteIntegral.of_mem_Icc (a := 0) (b := 1) h_box

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_54.{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) (i : Fin n) (h1 : Integrable (fun ω => 1 + τ * X i ω) μ)
  (h_int : Integrable (fun ω => X i ω ^ 2) μ) : Integrable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ := 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
i : Fin n
h1 : Integrable (fun ω => 1 + τ * X i ω) μ
h_int : Integrable (fun ω => X i ω ^ 2) μ
⊢ Integrable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ

Tactic:
apply Integrable.mul_const h_int (rexp τ - 1 - τ)

Tactic state after: