zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_0.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} (t b : ℝ) (ht : t > 0) (hX : AEMeasurable X μ) (hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b)
  (b'_1 : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)) (ω : Ω) (b' : rexp (t * X ω) ≤ rexp (t * b))
  (h_original_goal : ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)) : rexp (t * X ω) ≤ rexp (t * b) := sorry

Tactic state before:
case h
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t b : ℝ
ht : t > 0
hX : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
b'✝ : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
ω : Ω
b' : rexp (t * X ω) ≤ rexp (t * b)
⊢ ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)

Tactic:
rw [(by to_theorem simp : ‖rexp (t * X ω)‖₊ = rexp (t * X ω))]

Tactic state after:
[case h
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t b : ℝ
 ht : t > 0
 hX : AEMeasurable X μ
 hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
 b'✝ : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
 ω : Ω
 b' : rexp (t * X ω) ≤ rexp (t * b)
 ⊢ rexp (t * X ω) ≤ rexp (t * b)]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_1.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} (t b : ℝ) (ht : t > 0) (hX : AEMeasurable X μ) (hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b)
  (b'_1 : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)) (ω : Ω) (b' : rexp (t * X ω) ≤ rexp (t * b)) :
  ↑‖rexp (t * X ω)‖₊ = rexp (t * X ω) := sorry

Tactic state before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t b : ℝ
ht : t > 0
hX : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
b'✝ : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
ω : Ω
b' : rexp (t * X ω) ≤ rexp (t * b)
⊢ ↑‖rexp (t * X ω)‖₊ = rexp (t * X ω)

Tactic:
simp

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_2.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} (t b : ℝ) (ht : t > 0) (hX : AEMeasurable X μ) (hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b)
  (b'_1 : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)) (ω : Ω) (b' : rexp (t * X ω) ≤ rexp (t * b)) :
  rexp (t * X ω) ≤ rexp (t * b) := sorry

Tactic state before:
case h
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t b : ℝ
ht : t > 0
hX : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
b'✝ : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
ω : Ω
b' : rexp (t * X ω) ≤ rexp (t * b)
⊢ rexp (t * X ω) ≤ rexp (t * b)

Tactic:
exact b'

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_3.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} (t b : ℝ) (ht : t > 0) (hX : AEMeasurable X μ) (hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b)
  (b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)) (p : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b))
  (h_original_goal : ∫⁻ (ω : Ω), ↑‖rexp (t * X ω)‖₊ ∂μ ≤ ∫⁻ (x : Ω), ↑‖rexp (t * b)‖₊ ∂μ) :
  ∀ᵐ (a : Ω) ∂μ, ↑‖rexp (t * X a)‖₊ ≤ ↑‖rexp (t * b)‖₊ := sorry

Tactic state before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t b : ℝ
ht : t > 0
hX : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
p : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
⊢ ∫⁻ (ω : Ω), ↑‖rexp (t * X ω)‖₊ ∂μ ≤ ∫⁻ (x : Ω), ↑‖rexp (t * b)‖₊ ∂μ

Tactic:
apply lintegral_mono_ae

Tactic state after:
[case h
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t b : ℝ
 ht : t > 0
 hX : AEMeasurable X μ
 hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
 b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
 p : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
 ⊢ ∀ᵐ (a : Ω) ∂μ, ↑‖rexp (t * X a)‖₊ ≤ ↑‖rexp (t * b)‖₊]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_4.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} (t b : ℝ) (ht : t > 0) (hX : AEMeasurable X μ) (hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b)
  (b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)) (p_1 : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)) (ω : Ω)
  (p : ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)) (h_original_goal : ↑‖rexp (t * X ω)‖₊ ≤ ↑‖rexp (t * b)‖₊) :
  ‖rexp (t * X ω)‖₊ ≤ ‖rexp (t * b)‖₊ := sorry

Tactic state before:
case h
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t b : ℝ
ht : t > 0
hX : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
p✝ : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
ω : Ω
p : ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
⊢ ↑‖rexp (t * X ω)‖₊ ≤ ↑‖rexp (t * b)‖₊

Tactic:
simp only [ENNReal.coe_le_coe]

Tactic state after:
[case h
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t b : ℝ
 ht : t > 0
 hX : AEMeasurable X μ
 hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
 b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
 p✝ : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
 ω : Ω
 p : ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
 ⊢ ‖rexp (t * X ω)‖₊ ≤ ‖rexp (t * b)‖₊]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_5.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} (t b : ℝ) (ht : t > 0) (hX : AEMeasurable X μ) (hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b)
  (b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)) (p_1 : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)) (ω : Ω)
  (p : ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)) : ↑‖rexp (t * b)‖₊ = rexp (t * b) := sorry

Tactic state before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t b : ℝ
ht : t > 0
hX : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
p✝ : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
ω : Ω
p : ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
⊢ ↑‖rexp (t * b)‖₊ = rexp (t * b)

Tactic:
simp

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_6.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} (t b : ℝ) (ht : t > 0) (hX : AEMeasurable X μ) (hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b)
  (b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)) (p_1 : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)) (ω : Ω)
  (p : ↑‖rexp (t * X ω)‖₊ ≤ ↑‖rexp (t * b)‖₊) : ‖rexp (t * X ω)‖₊ ≤ ‖rexp (t * b)‖₊ := sorry

Tactic state before:
case h
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t b : ℝ
ht : t > 0
hX : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
p✝ : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
ω : Ω
p : ↑‖rexp (t * X ω)‖₊ ≤ ↑‖rexp (t * b)‖₊
⊢ ‖rexp (t * X ω)‖₊ ≤ ‖rexp (t * b)‖₊

Tactic:
exact p

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_7.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} (t b : ℝ) (ht : t > 0) (hX : AEMeasurable X μ) (hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b)
  (b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)) (p : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b))
  (p'' : ∫⁻ (ω : Ω), ↑‖rexp (t * X ω)‖₊ ∂μ ≤ ∫⁻ (x : Ω), ↑‖rexp (t * b)‖₊ ∂μ)
  (h_original_goal : ∫⁻ (x : Ω), ↑‖rexp (t * b)‖₊ ∂μ < ⊤) : ↑‖rexp (t * b)‖₊ * μ Set.univ < ⊤ := sorry

Tactic state before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t b : ℝ
ht : t > 0
hX : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
p : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
p'' : ∫⁻ (ω : Ω), ↑‖rexp (t * X ω)‖₊ ∂μ ≤ ∫⁻ (x : Ω), ↑‖rexp (t * b)‖₊ ∂μ
⊢ ∫⁻ (x : Ω), ↑‖rexp (t * b)‖₊ ∂μ < ⊤

Tactic:
simp only [lintegral_const]

Tactic state after:
[Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t b : ℝ
 ht : t > 0
 hX : AEMeasurable X μ
 hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
 b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
 p : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
 p'' : ∫⁻ (ω : Ω), ↑‖rexp (t * X ω)‖₊ ∂μ ≤ ∫⁻ (x : Ω), ↑‖rexp (t * b)‖₊ ∂μ
 ⊢ ↑‖rexp (t * b)‖₊ * μ Set.univ < ⊤]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_8.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} (t b : ℝ) (ht : t > 0) (hX : AEMeasurable X μ) (hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b)
  (b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)) (p : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b))
  (p'' : ∫⁻ (ω : Ω), ↑‖rexp (t * X ω)‖₊ ∂μ ≤ ∫⁻ (x : Ω), ↑‖rexp (t * b)‖₊ ∂μ) :
  ↑‖rexp (t * b)‖₊ * μ Set.univ < ⊤ := sorry

Tactic state before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t b : ℝ
ht : t > 0
hX : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
b' : ∀ᵐ (ω : Ω) ∂μ, rexp (t * X ω) ≤ rexp (t * b)
p : ∀ᵐ (ω : Ω) ∂μ, ↑‖rexp (t * X ω)‖₊ ≤ rexp (t * b)
p'' : ∫⁻ (ω : Ω), ↑‖rexp (t * X ω)‖₊ ∂μ ≤ ∫⁻ (x : Ω), ↑‖rexp (t * b)‖₊ ∂μ
⊢ ↑‖rexp (t * b)‖₊ * μ Set.univ < ⊤

Tactic:
apply ENNReal.mul_lt_top ENNReal.coe_lt_top (IsFiniteMeasure.measure_univ_lt_top)

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_9.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} (t b : ℝ) (ht : t > 0) (hX : AEMeasurable X μ) (hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b)
  (hm1 : HasFiniteIntegral (fun ω => rexp (t * X ω)) μ) : Integrable (fun ω => rexp (t * X ω)) μ := sorry

Tactic state before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t b : ℝ
ht : t > 0
hX : AEMeasurable X μ
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
hm1 : HasFiniteIntegral (fun ω => rexp (t * X ω)) μ
⊢ Integrable (fun ω => rexp (t * X ω)) μ

Tactic:
exact
  ⟨aestronglyMeasurable_iff_aemeasurable.mpr <| measurable_exp.comp_aemeasurable' (hX.const_mul t), hm1⟩

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_10.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} {t a b : ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (ht : 0 < t) : Integrable (fun ω => rexp (t * X ω)) μ := sorry

Tactic state before:
case inr.inr
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t a b : ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
ht : 0 < t
⊢ Integrable (fun ω => rexp (t * X ω)) μ

Tactic:
exact integrable_expt t b ht hX (h.mono fun ω h ↦ h.2)

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_11.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} {t a b : ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (ht : t = 0) (h_original_goal : Integrable (fun ω => rexp (t * X ω)) μ) :
  Integrable (fun ω => rexp (0 * X ω)) μ := sorry

Tactic state before:
case inr.inl
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t a b : ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
ht : t = 0
⊢ Integrable (fun ω => rexp (t * X ω)) μ

Tactic:
rw [ht]

Tactic state after:
[case inr.inl
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t a b : ℝ
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 ht : t = 0
 ⊢ Integrable (fun ω => rexp (0 * X ω)) μ]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_12.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} {a b : ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) :
  Integrable (fun ω => rexp (0 * X ω)) μ := sorry

Tactic state before:
case inr.inl
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t a b : ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
ht : t = 0
⊢ Integrable (fun ω => rexp (0 * X ω)) μ

Tactic:
simp only [zero_mul, exp_zero, integrable_const]

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_13.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} {t a b : ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (ht : t < 0) (h_original_goal : Integrable (fun ω => rexp (t * X ω)) μ) :
  Integrable (fun ω => rexp (-t * -X ω)) μ := sorry

Tactic state before:
case inl
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t a b : ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
ht : t < 0
⊢ Integrable (fun ω => rexp (t * X ω)) μ

Tactic:
rw [(by ext ω; to_theorem rw [(by ring : -t * (-X ω) = t * X ω)] :
    (fun ω ↦ rexp (t * X ω)) = (fun ω ↦ rexp (-t * (-X ω))))]

Tactic state after:
[case inl
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t a b : ℝ
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 ht : t < 0
 ⊢ Integrable (fun ω => rexp (-t * -X ω)) μ]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_14.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} {t a b : ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (ht : t < 0) (ω : Ω) : rexp (t * X ω) = rexp (-t * -X ω) := sorry

Tactic state before:
case h
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t a b : ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
ht : t < 0
ω : Ω
⊢ rexp (t * X ω) = rexp (-t * -X ω)

Tactic:
rw [(by ring : -t * (-X ω) = t * X ω)]

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_15.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} {t a b : ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (ht : t < 0) (h_original_goal : Integrable (fun ω => rexp (-t * -X ω)) μ) : -t > 0 := sorry

Tactic state before:
case inl
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t a b : ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
ht : t < 0
⊢ Integrable (fun ω => rexp (-t * -X ω)) μ

Tactic:
apply integrable_expt (-t) _ _ hX.neg

Tactic state after:
[case inl
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t a b : ℝ
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 ht : t < 0
 ⊢ ∀ᵐ (ω : Ω) ∂μ, -X ω ≤ ?m.21027,
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t a b : ℝ
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 ht : t < 0
 ⊢ ℝ,
 Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 X : Ω → ℝ
 t a b : ℝ
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 ht : t < 0
 ⊢ -t > 0]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_16.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] {X : Ω → ℝ} {a b : ℝ} (hX : AEMeasurable X μ) (h_1 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (ω : Ω) (h : X ω ∈ Set.Icc a b) : -X ω ≤ -a := sorry

Tactic state before:
case h
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t a b : ℝ
hX : AEMeasurable X μ
h✝ : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
ht : t < 0
ω : Ω
h : X ω ∈ Set.Icc a b
⊢ -X ω ≤ ?m.21027

Tactic:
apply neg_le_neg h.1

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_17 {t : ℝ} (ht : t < 0) : -t > 0 := sorry

Tactic state before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
t a b : ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
ht : t < 0
⊢ -t > 0

Tactic:
exact Left.neg_pos_iff.mpr ht

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_18.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsProbabilityMeasure μ] (a b t : ℝ) {X : Ω → ℝ} (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) (hX : AEMeasurable X μ)
  (x : IsProbabilityMeasure (μ.tilted fun ω => t * X ω)) :
  Var[X; μ.tilted fun ω => t * X ω] ≤ ((b - a) / 2) ^ 2 := sorry

Tactic state before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
a b t : ℝ
X : Ω → ℝ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hX : AEMeasurable X μ
x✝ : IsProbabilityMeasure (μ.tilted fun ω => t * X ω)
⊢ Var[X; μ.tilted fun ω => t * X ω] ≤ ((b - a) / 2) ^ 2

Tactic:
exact
  variance_le_sq_of_bounded ((tilted_absolutelyContinuous μ fun ω ↦ t * X ω) h)
    (hX.mono_ac (tilted_absolutelyContinuous μ fun ω ↦ t * X ω))

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_19.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (h_original_goal : HasFiniteIntegral X μ) : ∀ᵐ (a_1 : Ω) ∂μ, ‖X a_1‖ ≤ ‖a‖ ⊔ ‖b‖ := sorry

Tactic state before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
⊢ HasFiniteIntegral X μ

Tactic:
apply (hasFiniteIntegral_const (max ‖a‖ ‖b‖)).mono'

Tactic state after:
[Ω : Type u_1
 m : MeasurableSpace Ω
 μ : Measure Ω
 inst✝ : IsFiniteMeasure μ
 a b : ℝ
 X : Ω → ℝ
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 ⊢ ∀ᵐ (a_1 : Ω) ∂μ, ‖X a_1‖ ≤ ‖a‖ ⊔ ‖b‖]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_20.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (m1 : HasFiniteIntegral X μ) : Integrable X μ := sorry

Tactic state before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
m1 : HasFiniteIntegral X μ
⊢ Integrable X μ

Tactic:
exact ⟨aestronglyMeasurable_iff_aemeasurable.mpr hX, m1⟩

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_21.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (t a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) :
  let e := fun t ω => rexp (t * X ω);
  let e' := fun t ω => rexp (t * X ω) * X ω;
  Integrable (e' t) μ ∧ HasDerivAt (fun t => ∫ (x : Ω), e t x ∂μ) (∫ (x : Ω), e' t x ∂μ) t →
    Integrable (fun ω => rexp (t * X ω) * X ω) μ →
      HasDerivAt (fun t => ∫ (x : Ω), rexp (t * X x) ∂μ) (∫ (x : Ω), rexp (t * X x) * X x ∂μ) t := sorry

Tactic state before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
t a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
c : ℝ := ‖a‖ ⊔ ‖b‖
e : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω)
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω
this : Integrable (e' t) μ ∧ HasDerivAt (fun t => ∫ (x : Ω), e t x ∂μ) (∫ (x : Ω), e' t x ∂μ) t
hint : Integrable (fun ω => rexp (t * X ω) * X ω) μ
⊢ HasDerivAt (fun t => ∫ (x : Ω), rexp (t * X x) ∂μ) (∫ (x : Ω), rexp (t * X x) * X x ∂μ) t

Tactic:
apply this.2

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_25 (t : ℝ) : 0 < |t| + 1 := sorry

Tactic state before:
case ε_pos
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
t a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
c : ℝ := ‖a‖ ⊔ ‖b‖
e : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω)
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω
⊢ 0 < ?ε

Tactic:
exact LE.le.trans_lt (abs_nonneg t) (lt_add_one |t|)

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_26.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (t a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) :
  let e := fun t ω => rexp (t * X ω);
  ∀ᶠ (x : ℝ) in nhds t, AEStronglyMeasurable (e x) μ := sorry

Tactic state before:
case hF_meas
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
t a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
c : ℝ := ‖a‖ ⊔ ‖b‖
e : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω)
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω
⊢ ∀ᶠ (x : ℝ) in nhds t, AEStronglyMeasurable (e x) μ

Tactic:
exact
  Filter.Eventually.of_forall
    (by intro t'; to_theorem apply aemeasurable_expt _ hX : ∀ (x : ℝ), AEStronglyMeasurable (e x) μ)

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_27.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] (a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) :
  let e := fun t ω => rexp (t * X ω);
  ∀ (t' : ℝ), AEStronglyMeasurable (e t') μ := sorry

Tactic state before:
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
t a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
c : ℝ := ‖a‖ ⊔ ‖b‖
e : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω)
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω
t' : ℝ
⊢ AEStronglyMeasurable (e t') μ

Tactic:
apply aemeasurable_expt _ hX

Tactic state after: