zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_55.{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)
  (h2 : Integrable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ)
  (h3 : HasFiniteIntegral ((fun ω => 1 + 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 + 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 : Integrable (fun ω => 1 + τ * X i ω) μ
h2 : Integrable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ
h3 : HasFiniteIntegral ((fun ω => 1 + 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 + 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_56.{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)
  (h2 : Integrable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ)
  (h3 : HasFiniteIntegral ((fun ω => 1 + 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 + τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ))
  (h_original_goal : HasFiniteIntegral (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) μ) :
  HasFiniteIntegral ((fun ω => 1 + τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) 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
h2 : Integrable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ
h3 : HasFiniteIntegral ((fun ω => 1 + 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 + τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)
⊢ HasFiniteIntegral (fun ω => 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) μ

Tactic:
rw [this]

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
 h1 : Integrable (fun ω => 1 + τ * X i ω) μ
 h2 : Integrable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ
 h3 : HasFiniteIntegral ((fun ω => 1 + 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 + τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)
 ⊢ HasFiniteIntegral ((fun ω => 1 + 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_57.{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)
  (h2 : Integrable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ)
  (h3 : HasFiniteIntegral ((fun ω => 1 + 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 + τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) :
  HasFiniteIntegral ((fun ω => 1 + τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) 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
h2 : Integrable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ
h3 : HasFiniteIntegral ((fun ω => 1 + 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 + τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)
⊢ HasFiniteIntegral ((fun ω => 1 + τ * X i ω) + fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ

Tactic:
exact h3

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_58.{u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (ε τ : ℝ) (hτ : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  (Integrable (fun ω => rexp (τ * S ω)) μ → (μ {ω | ε ≤ S ω}).toReal ≤ rexp (-τ * ε) * mgf S μ τ) →
    Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
      (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ := sorry

Tactic state before:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
σ ε τ : ℝ
hτ : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
ht_sq : 0 ≤ σ ^ 2
chernoff_bound : Integrable (fun ω => rexp (τ * S ω)) μ → (μ {ω | ε ≤ S ω}).toReal ≤ rexp (-τ * ε) * mgf S μ τ
⊢ Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
    (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ

Tactic:
exact chernoff_bound

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_59.{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))
  (hXi : iIndepFun X μ) (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1)
  (hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0) (σ : ℝ) (hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ)
  (ε τ : ℝ) (hτ : τ ≥ 0)
  (chernoff_bound :
    τ ≥ 0 →
      Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
        (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ)
  (h_chernoff :
    Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
      (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ) :
  Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ := sorry

Tactic state before:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hXi : iIndepFun X μ
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0
σ : ℝ
hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ
ε τ : ℝ
hτ : τ ≥ 0
chernoff_bound :
  τ ≥ 0 →
    Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
      (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
h_chernoff :
  Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
    (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
⊢ Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ

Tactic:
exact sum_integrable μ hXm hX_bdd τ hτ

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_60.{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))
  (hXi : iIndepFun X μ) (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1)
  (hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0) (σ : ℝ) (hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ)
  (ε τ : ℝ) (hτ : τ ≥ 0)
  (chernoff_bound :
    τ ≥ 0 →
      Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
        (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ)
  (h_chernoff :
    Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
      (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ)
  (h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ) (i : Fin n) : Measurable (X i) := sorry

Tactic state before:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hXi : iIndepFun X μ
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0
σ : ℝ
hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ
ε τ : ℝ
hτ : τ ≥ 0
chernoff_bound :
  τ ≥ 0 →
    Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
      (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
h_chernoff :
  Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
    (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ
t : ℝ
i : Fin n
⊢ Measurable (X i)

Tactic:
exact hXm i

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_61.{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))
  (hXi : iIndepFun X μ) (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1)
  (hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0) (σ : ℝ) (hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ)
  (ε τ : ℝ) (hτ : τ ≥ 0)
  (chernoff_bound :
    τ ≥ 0 →
      Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
        (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ)
  (h_chernoff :
    Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
      (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ)
  (h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ) (t : ℝ) (hXmeas : ∀ (i : Fin n), Measurable (X i))
  (h_original_goal : mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t) :
  mgf (fun ω => ∑ i, X i ω) μ t = mgf (∑ i, X i) μ t := sorry

Tactic state before:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hXi : iIndepFun X μ
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0
σ : ℝ
hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ
ε τ : ℝ
hτ : τ ≥ 0
chernoff_bound :
  τ ≥ 0 →
    Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
      (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
h_chernoff :
  Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
    (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ
t : ℝ
hXmeas : ∀ (i : Fin n), Measurable (X i)
⊢ mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t

Tactic:
rw [← iIndepFun.mgf_sum hXi hXmeas Finset.univ]

Tactic state after:
[Ω : Type u_1
 inst✝¹ : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hXi : iIndepFun X μ
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0
 σ : ℝ
 hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ
 ε τ : ℝ
 hτ : τ ≥ 0
 chernoff_bound :
   τ ≥ 0 →
     Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
       (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
 h_chernoff :
   Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
     (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
 h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ
 t : ℝ
 hXmeas : ∀ (i : Fin n), Measurable (X i)
 ⊢ mgf (fun ω => ∑ i, X i ω) μ t = mgf (∑ i, X i) μ t]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_62.{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))
  (hXi : iIndepFun X μ) (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1)
  (hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0) (σ : ℝ) (hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ)
  (ε τ : ℝ) (hτ : τ ≥ 0)
  (chernoff_bound :
    τ ≥ 0 →
      Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
        (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ)
  (h_chernoff :
    Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
      (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ)
  (h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ) (hXmeas : ∀ (i : Fin n), Measurable (X i)) (x : Ω) :
  ∑ i, X i x = (∑ i, X i) x := sorry

Tactic state before:
case e_X.h
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hXi : iIndepFun X μ
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0
σ : ℝ
hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ
ε τ : ℝ
hτ : τ ≥ 0
chernoff_bound :
  τ ≥ 0 →
    Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
      (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
h_chernoff :
  Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
    (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ
t : ℝ
hXmeas : ∀ (i : Fin n), Measurable (X i)
x : Ω
⊢ ∑ i, X i x = (∑ i, X i) x

Tactic:
rw [Finset.sum_apply]

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_63.{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))
  (hXi : iIndepFun X μ) (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1)
  (hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0) (σ : ℝ) (hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ)
  (ε τ : ℝ) (hτ : τ ≥ 0)
  (h_chernoff :
    Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
      (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ)
  (h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ)
  (h_ysum_mgf : ∀ (t : ℝ), mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t)
  (chernoff_bound : (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-(τ * ε)) * ∏ i, mgf (X i) μ τ)
  (h_original_goal :
    rexp (-(τ * ε)) * ∏ i, ∫ (x : Ω), (fun ω => rexp (τ * X i ω)) x ∂μ ≤
      rexp (-(τ * ε) + ↑n * σ ^ 2 * (rexp τ - 1 - τ))) :
  rexp (-(τ * ε)) * ∏ x, ∫ (x_1 : Ω), rexp (τ * X x x_1) ∂μ ≤ rexp (-(τ * ε) + ↑n * σ ^ 2 * (rexp τ - 1 - τ)) := sorry

Tactic state before:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hXi : iIndepFun X μ
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0
σ : ℝ
hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ
ε τ : ℝ
hτ : τ ≥ 0
h_chernoff :
  Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
    (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ
h_ysum_mgf : ∀ (t : ℝ), mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t
chernoff_bound : (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-(τ * ε)) * ∏ i, mgf (X i) μ τ
⊢ rexp (-(τ * ε)) * ∏ i, ∫ (x : Ω), (fun ω => rexp (τ * X i ω)) x ∂μ ≤ rexp (-(τ * ε) + ↑n * σ ^ 2 * (rexp τ - 1 - τ))

Tactic:
simp only

Tactic state after:
[Ω : Type u_1
 inst✝¹ : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hXi : iIndepFun X μ
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0
 σ : ℝ
 hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ
 ε τ : ℝ
 hτ : τ ≥ 0
 h_chernoff :
   Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
     (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
 h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ
 h_ysum_mgf : ∀ (t : ℝ), mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t
 chernoff_bound : (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-(τ * ε)) * ∏ i, mgf (X i) μ τ
 ⊢ rexp (-(τ * ε)) * ∏ x, ∫ (x_1 : Ω), rexp (τ * X x x_1) ∂μ ≤ rexp (-(τ * ε) + ↑n * σ ^ 2 * (rexp τ - 1 - τ))]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_64.{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))
  (hXi : iIndepFun X μ) (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1)
  (hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0) (σ : ℝ) (hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ)
  (ε τ : ℝ) (hτ : τ ≥ 0)
  (h_chernoff :
    Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
      (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ)
  (h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ)
  (h_ysum_mgf : ∀ (t : ℝ), mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t)
  (chernoff_bound : (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-(τ * ε)) * ∏ i, mgf (X i) μ τ) (i : Fin n) (ω : Ω)
  (hω : |X i ω| ≤ 1) : |X i ω| ≤ 1 := sorry

Tactic state before:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hXi : iIndepFun X μ
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0
σ : ℝ
hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ
ε τ : ℝ
hτ : τ ≥ 0
h_chernoff :
  Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
    (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ
h_ysum_mgf : ∀ (t : ℝ), mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t
chernoff_bound : (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-(τ * ε)) * ∏ i, mgf (X i) μ τ
i : Fin n
ω : Ω
hω : |X i ω| ≤ 1
x : ℝ := X i ω
⊢ |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_65.{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))
  (hXi : iIndepFun X μ) (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1)
  (hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0) (σ : ℝ) (hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ)
  (ε τ : ℝ) (hτ : τ ≥ 0)
  (h_chernoff :
    Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
      (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ)
  (h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ)
  (h_ysum_mgf : ∀ (t : ℝ), mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t)
  (chernoff_bound : (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-(τ * ε)) * ∏ i, mgf (X i) μ τ) (i : Fin n) (ω : Ω)
  (hω : |X i ω| ≤ 1) :
  let x := X i ω;
  |x| ≤ 1 → rexp (τ * x) ≤ 1 + τ * x + x ^ 2 * (rexp τ - 1 - τ) := sorry

Tactic state before:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hXi : iIndepFun X μ
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0
σ : ℝ
hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ
ε τ : ℝ
hτ : τ ≥ 0
h_chernoff :
  Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
    (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ
h_ysum_mgf : ∀ (t : ℝ), mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t
chernoff_bound : (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-(τ * ε)) * ∏ i, mgf (X i) μ τ
i : Fin n
ω : Ω
hω : |X i ω| ≤ 1
x : ℝ := X i ω
h_x_abs_le_1 : |x| ≤ 1
⊢ rexp (τ * x) ≤ 1 + τ * x + x ^ 2 * (rexp τ - 1 - τ)

Tactic:
exact exp_bound x τ h_x_abs_le_1 hτ

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_66.{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))
  (hXi : iIndepFun X μ) (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1)
  (hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0) (σ : ℝ) (hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ)
  (ε τ : ℝ) (hτ : τ ≥ 0)
  (h_chernoff :
    Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
      (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ)
  (h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ)
  (h_ysum_mgf : ∀ (t : ℝ), mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t)
  (chernoff_bound : (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-(τ * ε)) * ∏ i, mgf (X i) μ τ) (i : Fin n) (ω : Ω)
  (hω : |X i ω| ≤ 1) (h_exp_bound : rexp (τ * X i ω) ≤ 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)) :
  rexp (τ * X i ω) ≤ 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ) := sorry

Tactic state before:
case h
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hXi : iIndepFun X μ
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
hμX : ∀ (i : Fin n), ∫ (x : Ω), X i x ∂μ = 0
σ : ℝ
hσ : ∀ (i : Fin n), σ ^ 2 = ∫ (x : Ω), (X i ^ 2) x ∂μ
ε τ : ℝ
hτ : τ ≥ 0
h_chernoff :
  Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ →
    (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-τ * ε) * mgf (fun ω => ∑ i, X i ω) μ τ
h_integ : Integrable (fun ω => rexp (τ * ∑ i, X i ω)) μ
h_ysum_mgf : ∀ (t : ℝ), mgf (fun ω => ∑ i, X i ω) μ t = ∏ i, mgf (X i) μ t
chernoff_bound : (μ {ω | ε ≤ ∑ i, X i ω}).toReal ≤ rexp (-(τ * ε)) * ∏ i, mgf (X i) μ τ
i : Fin n
ω : Ω
hω : |X i ω| ≤ 1
x : ℝ := X i ω
h_x_abs_le_1 : |x| ≤ 1
h_exp_bound : rexp (τ * X i ω) ≤ 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)
⊢ rexp (τ * X i ω) ≤ 1 + τ * X i ω + X i ω ^ 2 * (rexp τ - 1 - τ)

Tactic:
exact h_exp_bound

Tactic state after: