zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_73.{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) :
  AEMeasurable (fun a => X a ^ 2) μ := sorry

Tactic state before:
case hF'_meas.hg
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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
⊢ AEMeasurable (fun a => X a ^ 2) μ

Tactic:
exact hX.pow_const 2

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_74.{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_1 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (ω : Ω) (x : ℝ) (h : x ∈ Metric.ball t (|t| + 1))
  (h_original_goal : |rexp (x * X ω) * X ω ^ 2| ≤ rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2) :
  |rexp (x * X ω) * X ω ^ 2| ≤ rexp ((2 * |t| + 1) * |X ω|) * X ω ^ 2 := sorry

Tactic state before:
case h_bound
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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
bound : Ω → ℝ := fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2
ω : Ω
x : ℝ
h : x ∈ Metric.ball t (|t| + 1)
⊢ |rexp (x * X ω) * X ω ^ 2| ≤ rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2

Tactic:
simp only [sq_abs]

Tactic state after:
[case h_bound
 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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 bound : Ω → ℝ := fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2
 ω : Ω
 x : ℝ
 h : x ∈ Metric.ball t (|t| + 1)
 ⊢ |rexp (x * X ω) * X ω ^ 2| ≤ rexp ((2 * |t| + 1) * |X ω|) * X ω ^ 2]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_75.{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_1 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (ω : Ω) (x : ℝ) (h : x ∈ Metric.ball t (|t| + 1)) : |rexp (x * X ω)| * |X ω ^ 2| = rexp (x * X ω) * X ω ^ 2 := 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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
bound : Ω → ℝ := fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2
ω : Ω
x : ℝ
h : x ∈ Metric.ball t (|t| + 1)
⊢ |rexp (x * X ω)| * |X ω ^ 2| = rexp (x * X ω) * X ω ^ 2

Tactic:
simp

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_76.{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_1 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b)
  (ω : Ω) (x : ℝ) (h : x ∈ Metric.ball t (|t| + 1)) (h_original_goal : rexp (x * X ω) ≤ rexp ((2 * |t| + 1) * |X ω|)) :
  x * X ω ≤ (2 * |t| + 1) * |X ω| := 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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
bound : Ω → ℝ := fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2
ω : Ω
x : ℝ
h : x ∈ Metric.ball t (|t| + 1)
⊢ rexp (x * X ω) ≤ rexp ((2 * |t| + 1) * |X ω|)

Tactic:
simp only [exp_le_exp]

Tactic state after:
[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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 bound : Ω → ℝ := fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2
 ω : Ω
 x : ℝ
 h : x ∈ Metric.ball t (|t| + 1)
 ⊢ x * X ω ≤ (2 * |t| + 1) * |X ω|]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_77.{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)
  (h_original_goal : Integrable (fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2) μ) :
  Integrable (fun x => |X x| ^ 2) μ := sorry

Tactic state before:
case bound_integrable
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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
⊢ Integrable (fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2) μ

Tactic:
apply MeasureTheory.Integrable.bdd_mul'

Tactic state after:
[case bound_integrable.hg
 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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ⊢ Integrable (fun x => |X x| ^ 2) μ,
 case bound_integrable.hf
 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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ⊢ AEStronglyMeasurable (fun x => rexp ((2 * |t| + 1) * |X x|)) μ,
 case bound_integrable.hf_bound
 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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ⊢ ∀ᵐ (x : Ω) ∂μ, ‖rexp ((2 * |t| + 1) * |X x|)‖ ≤ ?bound_integrable.c,
 case bound_integrable.c
 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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ⊢ ℝ]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_78.{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)
  (h_original_goal : Integrable (fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2) μ) :
  AEStronglyMeasurable (fun x => rexp ((2 * |t| + 1) * |X x|)) μ := sorry

Tactic state before:
case bound_integrable
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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
⊢ Integrable (fun ω => rexp ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2) μ

Tactic:
apply MeasureTheory.Integrable.bdd_mul'

Tactic state after:
[case bound_integrable.hg
 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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ⊢ Integrable (fun x => |X x| ^ 2) μ,
 case bound_integrable.hf
 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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ⊢ AEStronglyMeasurable (fun x => rexp ((2 * |t| + 1) * |X x|)) μ,
 case bound_integrable.hf_bound
 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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ⊢ ∀ᵐ (x : Ω) ∂μ, ‖rexp ((2 * |t| + 1) * |X x|)‖ ≤ ?bound_integrable.c,
 case bound_integrable.c
 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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ⊢ ℝ]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_79.{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 : Integrable (fun x => |X x| ^ 2) μ) : Integrable (fun x => X x ^ 2) μ := sorry

Tactic state before:
case bound_integrable.hg
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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
⊢ Integrable (fun x => |X x| ^ 2) μ

Tactic:
simp only [sq_abs]

Tactic state after:
[case bound_integrable.hg
 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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ⊢ Integrable (fun x => X x ^ 2) μ]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_80.{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 : Integrable (fun x => X x ^ 2) μ) : Integrable (fun ω => X ω * X ω) μ := sorry

Tactic state before:
case bound_integrable.hg
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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
⊢ Integrable (fun x => X x ^ 2) μ

Tactic:
rw [(by ext ω; ring : (fun ω ↦ X ω ^ 2) = (fun ω ↦ X ω * X ω))]

Tactic state after:
[case bound_integrable.hg
 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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ⊢ Integrable (fun ω => X ω * X ω) μ]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_81.{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_1 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) :
  let c := ‖a‖ ⊔ ‖b‖;
  ∀ (x : Ω), X x ∈ Set.Icc a b → ‖X x‖ ≤ |c| := sorry

Tactic state before:
case h
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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
x : Ω
h : X x ∈ Set.Icc a b
⊢ ‖X x‖ ≤ ?m.70996

Tactic:
exact
  (by to_theorem simp; to_theorem exact le_trans' (le_abs_self (max ‖a‖ ‖b‖)) (abs_le_max_abs_abs h.1 h.2) :
    ‖X x‖ ≤ |c|)

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_82.{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_1 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) :
  let c := ‖a‖ ⊔ ‖b‖;
  ∀ (x : Ω), X x ∈ Set.Icc a b → ‖X x‖ ≤ |c| → |X x| ≤ |c| := 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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
x : Ω
h : X x ∈ Set.Icc a b
⊢ ‖X x‖ ≤ |c|

Tactic:
simp

Tactic state after:
[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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 x : Ω
 h : X x ∈ Set.Icc a b
 ⊢ |X x| ≤ |c|]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_83.{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_1 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) :
  let c := ‖a‖ ⊔ ‖b‖;
  ∀ (x : Ω), X x ∈ Set.Icc a b → |X x| ≤ |c| := 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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
x : Ω
h : X x ∈ Set.Icc a b
⊢ |X x| ≤ |c|

Tactic:
exact le_trans' (le_abs_self (max ‖a‖ ‖b‖)) (abs_le_max_abs_abs h.1 h.2)

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_84.{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)
  (h_original_goal : AEStronglyMeasurable (fun x => rexp ((2 * |t| + 1) * |X x|)) μ) :
  AEMeasurable (fun ω => |X ω|) μ := sorry

Tactic state before:
case bound_integrable.hf
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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
⊢ AEStronglyMeasurable (fun x => rexp ((2 * |t| + 1) * |X x|)) μ

Tactic:
apply aemeasurable_expt (2 * |t| + 1) _

Tactic state after:
[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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ⊢ AEMeasurable (fun ω => |X ω|) μ]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_85 : Measurable abs := 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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
⊢ AEMeasurable (fun ω => |X ω|) μ

Tactic:
apply Measurable.comp_aemeasurable' _ hX

Tactic state after:
[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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ⊢ Measurable abs]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_86 : Measurable abs := 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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
⊢ Measurable abs

Tactic:
apply measurable_norm

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_87.{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_1 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) :
  let c := ‖a‖ ⊔ ‖b‖;
  ∀ (ω : Ω),
    X ω ∈ Set.Icc a b →
      ‖rexp ((2 * |t| + 1) * |X ω|)‖ ≤ ‖rexp ((2 * |t| + 1) * c)‖ → (2 * |t| + 1) * |X ω| ≤ (2 * |t| + 1) * c := sorry

Tactic state before:
case h
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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
ω : Ω
h : X ω ∈ Set.Icc a b
⊢ ‖rexp ((2 * |t| + 1) * |X ω|)‖ ≤ ‖rexp ((2 * |t| + 1) * c)‖

Tactic:
simp only [norm_eq_abs, abs_exp, exp_le_exp]

Tactic state after:
[case h
 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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ω : Ω
 h : X ω ∈ Set.Icc a b
 ⊢ (2 * |t| + 1) * |X ω| ≤ (2 * |t| + 1) * c]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_88.{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_1 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) :
  let c := ‖a‖ ⊔ ‖b‖;
  ∀ (ω : Ω),
    X ω ∈ Set.Icc a b → (2 * |t| + 1) * |X ω| ≤ (2 * |t| + 1) * c → (2 * |t| + 1) * |X ω| ≤ (2 * |t| + 1) * |c| := sorry

Tactic state before:
case h
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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
ω : Ω
h : X ω ∈ Set.Icc a b
⊢ (2 * |t| + 1) * |X ω| ≤ (2 * |t| + 1) * c

Tactic:
rw [← (abs_eq_self.mpr (le_trans' (le_max_left |a| ‖b‖) (abs_nonneg a)) : |c| = c)]

Tactic state after:
[case h
 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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ω : Ω
 h : X ω ∈ Set.Icc a b
 ⊢ (2 * |t| + 1) * |X ω| ≤ (2 * |t| + 1) * |c|]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_89.{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_1 : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) :
  let c := ‖a‖ ⊔ ‖b‖;
  ∀ (ω : Ω), X ω ∈ Set.Icc a b → (2 * |t| + 1) * |X ω| ≤ (2 * |t| + 1) * |c| := sorry

Tactic state before:
case h
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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
ω : Ω
h : X ω ∈ Set.Icc a b
⊢ (2 * |t| + 1) * |X ω| ≤ (2 * |t| + 1) * |c|

Tactic:
exact
  mul_le_mul_of_nonneg_left (le_trans' (le_abs_self (max ‖a‖ ‖b‖)) (abs_le_max_abs_abs h.1 h.2))
    (by linarith [abs_nonneg t] : 0 ≤ 2 * |t| + 1)

Tactic state after:
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_90.{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)
  (ω : Ω) (x : ℝ) (a_1 : x ∈ Metric.ball t (|t| + 1))
  (this : HasDerivAt (fun x => rexp (x * X ω)) (rexp (x * X ω) * X ω) x)
  (h_original_goal : HasDerivAt (fun x => rexp (x * X ω) * X ω) (rexp (x * X ω) * X ω ^ 2) x) :
  HasDerivAt (fun x => rexp (x * X ω) * X ω) (rexp (x * X ω) * X ω * X ω) x := 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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
ω : Ω
x : ℝ
a✝ : x ∈ Metric.ball t (|t| + 1)
this : HasDerivAt (fun x => rexp (x * X ω)) (rexp (x * X ω) * X ω) x
⊢ HasDerivAt (fun x => rexp (x * X ω) * X ω) (rexp (x * X ω) * X ω ^ 2) x

Tactic:
rw [(by ring : rexp (x * X ω) * X ω ^ 2 = (rexp (x * X ω) * X ω) * X ω)]

Tactic state after:
[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) * X ω
 e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
 ω : Ω
 x : ℝ
 a✝ : x ∈ Metric.ball t (|t| + 1)
 this : HasDerivAt (fun x => rexp (x * X ω)) (rexp (x * X ω) * X ω) x
 ⊢ HasDerivAt (fun x => rexp (x * X ω) * X ω) (rexp (x * X ω) * X ω * X ω) x]
import Mathlib
import FoML


open MeasureTheory Filter Finset Real

open scoped MeasureTheory ProbabilityTheory ENNReal NNReal

theorem extracted_formal_statement_91.{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)
  (ω : Ω) (x : ℝ) (a_1 : x ∈ Metric.ball t (|t| + 1))
  (this : HasDerivAt (fun x => rexp (x * X ω)) (rexp (x * X ω) * X ω) x) :
  HasDerivAt (fun x => rexp (x * X ω) * X ω) (rexp (x * X ω) * X ω * X ω) x := 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) * X ω
e' : ℝ → Ω → ℝ := fun t ω => rexp (t * X ω) * X ω ^ 2
ω : Ω
x : ℝ
a✝ : x ∈ Metric.ball t (|t| + 1)
this : HasDerivAt (fun x => rexp (x * X ω)) (rexp (x * X ω) * X ω) x
⊢ HasDerivAt (fun x => rexp (x * X ω) * X ω) (rexp (x * X ω) * X ω * X ω) x

Tactic:
exact HasDerivAt.mul_const this (X ω)

Tactic state after: