zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_0 (x : ℝ) (hx : -1 ≤ x ∧ x ≤ 1) : x ≤ 1 := sorry

Tactic state before:
x τ : ℝ
hx : -1 ≤ x ∧ x ≤ 1
hτ : 0 ≤ τ
⊢ x ≤ 1

Tactic:
exact hx.2

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_1 (x τ : ℝ) (hx : |x| ≤ 1) (hτ : 0 ≤ τ) (hx_le_1 : x ≤ 1)
  (h_original_goal : rexp (τ * x) ≤ 1 + τ * x + x ^ 2 * (rexp τ - 1 - τ)) :
  rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ)) ≤ 0 := sorry

Tactic state before:
x τ : ℝ
hx : |x| ≤ 1
hτ : 0 ≤ τ
hx_le_1 : x ≤ 1
⊢ rexp (τ * x) ≤ 1 + τ * x + x ^ 2 * (rexp τ - 1 - τ)

Tactic:
apply le_of_sub_nonpos

Tactic state after:
[case a
 x τ : ℝ
 hx : |x| ≤ 1
 hτ : 0 ≤ τ
 hx_le_1 : x ≤ 1
 ⊢ rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ)) ≤ 0]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_2 (x : ℝ) (hx : |x| ≤ 1) (hx_le_1 : x ≤ 1) :
  let f := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ));
  f 0 = 0 := sorry

Tactic state before:
x τ : ℝ
hx : |x| ≤ 1
hτ : 0 ≤ τ
hx_le_1 : x ≤ 1
f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
⊢ f 0 = 0

Tactic:
simp [f]

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_3 (x : ℝ) (hx : |x| ≤ 1) (hx_le_1 : x ≤ 1) :
  let f' := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1));
  f' 0 = 0 := sorry

Tactic state before:
x τ : ℝ
hx : |x| ≤ 1
hτ : 0 ≤ τ
hx_le_1 : x ≤ 1
f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
hf0 : f 0 = 0
⊢ f' 0 = 0

Tactic:
simp [f']

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_4 (x τ : ℝ) (hx : |x| ≤ 1) (hτ : 0 ≤ τ) (hx_le_1 : x ≤ 1) :
  let f := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ));
  f 0 = 0 → f τ ≤ 0 → f τ ≤ f 0 := sorry

Tactic state before:
case a
x τ : ℝ
hx : |x| ≤ 1
hτ : 0 ≤ τ
hx_le_1 : x ≤ 1
f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
hf0 : f 0 = 0
hf'0 : f' 0 = 0
⊢ f τ ≤ 0

Tactic:
rw [← hf0]

Tactic state after:
[case a
 x τ : ℝ
 hx : |x| ≤ 1
 hτ : 0 ≤ τ
 hx_le_1 : x ≤ 1
 f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
 f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
 f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
 hf0 : f 0 = 0
 hf'0 : f' 0 = 0
 ⊢ f τ ≤ f 0]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_5 (x : ℝ) (hx : |x| ≤ 1) (hx_le_1 : x ≤ 1) :
  let f' := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1));
  f' 0 = 0 → ∀ t ∈ interior (Ici 0), f' t ≤ 0 → f' t ≤ f' 0 := sorry

Tactic state before:
case a
x τ : ℝ
hx : |x| ≤ 1
hτ : 0 ≤ τ
hx_le_1 : x ≤ 1
f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
hf0 : f 0 = 0
hf'0 : f' 0 = 0
t : ℝ
ht : t ∈ interior (Ici 0)
⊢ f' t ≤ 0

Tactic:
rw [← hf'0]

Tactic state after:
[case a
 x τ : ℝ
 hx : |x| ≤ 1
 hτ : 0 ≤ τ
 hx_le_1 : x ≤ 1
 f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
 f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
 f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
 hf0 : f 0 = 0
 hf'0 : f' 0 = 0
 t : ℝ
 ht : t ∈ interior (Ici 0)
 ⊢ f' t ≤ f' 0]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_6 (x : ℝ) (hx : |x| ≤ 1) (hx_le_1 : x ≤ 1) (u : ℝ) (hu : u ∈ interior (Ici 0))
  (h_original_goal : rexp (u * x) * x * x - x ^ 2 * rexp u ≤ 0) : rexp (u * x) * x * x ≤ x ^ 2 * rexp u := sorry

Tactic state before:
case a
x τ : ℝ
hx : |x| ≤ 1
hτ : 0 ≤ τ
hx_le_1 : x ≤ 1
f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
hf0 : f 0 = 0
hf'0 : f' 0 = 0
t : ℝ
ht : t ∈ interior (Ici 0)
u : ℝ
hu : u ∈ interior (Ici 0)
⊢ rexp (u * x) * x * x - x ^ 2 * rexp u ≤ 0

Tactic:
apply sub_nonpos_of_le

Tactic state after:
[case a.a
 x τ : ℝ
 hx : |x| ≤ 1
 hτ : 0 ≤ τ
 hx_le_1 : x ≤ 1
 f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
 f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
 f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
 hf0 : f 0 = 0
 hf'0 : f' 0 = 0
 t : ℝ
 ht : t ∈ interior (Ici 0)
 u : ℝ
 hu : u ∈ interior (Ici 0)
 ⊢ rexp (u * x) * x * x ≤ x ^ 2 * rexp u]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_7 (x : ℝ) (hx : |x| ≤ 1) (hx_le_1 : x ≤ 1) (u : ℝ) (hu : u ∈ interior (Ici 0))
  (h_original_goal : rexp (u * x) * x * x ≤ x ^ 2 * rexp u) : x ^ 2 * rexp (u * x) ≤ x ^ 2 * rexp u := sorry

Tactic state before:
case a.a
x τ : ℝ
hx : |x| ≤ 1
hτ : 0 ≤ τ
hx_le_1 : x ≤ 1
f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
hf0 : f 0 = 0
hf'0 : f' 0 = 0
t : ℝ
ht : t ∈ interior (Ici 0)
u : ℝ
hu : u ∈ interior (Ici 0)
⊢ rexp (u * x) * x * x ≤ x ^ 2 * rexp u

Tactic:
rw [mul_assoc, ← sq, mul_comm (rexp _)]

Tactic state after:
[case a.a
 x τ : ℝ
 hx : |x| ≤ 1
 hτ : 0 ≤ τ
 hx_le_1 : x ≤ 1
 f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
 f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
 f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
 hf0 : f 0 = 0
 hf'0 : f' 0 = 0
 t : ℝ
 ht : t ∈ interior (Ici 0)
 u : ℝ
 hu : u ∈ interior (Ici 0)
 ⊢ x ^ 2 * rexp (u * x) ≤ x ^ 2 * rexp u]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_8 (x : ℝ) (hx : |x| ≤ 1) (hx_le_1 : x ≤ 1) (u : ℝ) (hu : u ∈ interior (Ici 0))
  (h_original_goal : rexp (u * x) ≤ rexp u) : u * x ≤ u := sorry

Tactic state before:
case a.a
x τ : ℝ
hx : |x| ≤ 1
hτ : 0 ≤ τ
hx_le_1 : x ≤ 1
f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
hf0 : f 0 = 0
hf'0 : f' 0 = 0
t : ℝ
ht : t ∈ interior (Ici 0)
u : ℝ
hu : u ∈ interior (Ici 0)
⊢ rexp (u * x) ≤ rexp u

Tactic:
apply exp_monotone

Tactic state after:
[case a.a.a
 x τ : ℝ
 hx : |x| ≤ 1
 hτ : 0 ≤ τ
 hx_le_1 : x ≤ 1
 f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
 f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))
 f'' : ℝ → ℝ := fun τ => rexp (τ * x) * x * x - x ^ 2 * rexp τ
 hf0 : f 0 = 0
 hf'0 : f' 0 = 0
 t : ℝ
 ht : t ∈ interior (Ici 0)
 u : ℝ
 hu : u ∈ interior (Ici 0)
 ⊢ u * x ≤ u]
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_9.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) :
  let I := Finset.univ;
  ∀ i ∈ I, Measurable (X i) := sorry

Tactic state before:
Ω : Type u
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
I : Finset (Fin n) := Finset.univ
i : Fin n
hi : i ∈ I
⊢ 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_10.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) :
  let I := Finset.univ;
  (∀ i ∈ I, Measurable (X i)) → (Measurable fun a => ∑ i ∈ I, X i a) → Measurable fun ω => ∑ i, X i ω := sorry

Tactic state before:
Ω : Type u
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
hlam : τ ≥ 0
S : Ω → ℝ := fun ω => ∑ i, X i ω
I : Finset (Fin n) := Finset.univ
hXm' : ∀ i ∈ I, Measurable (X i)
h : Measurable fun a => ∑ i ∈ I, X i a
⊢ Measurable fun ω => ∑ i, X i ω

Tactic:
exact h

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_11.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S → (Measurable fun ω => rexp (τ * S ω)) → AEStronglyMeasurable (fun ω => rexp (τ * S ω)) μ := sorry

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

Tactic:
exact f_measω.aestronglyMeasurable

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_12.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      ∀ (i : Fin n) (ω : Ω), |X i ω| ≤ 1 → ‖rexp (τ * X i ω)‖ ≤ rexp τ → rexp (τ * X i ω) ≤ rexp τ := sorry

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

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

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


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_13.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      ∀ (i : Fin n) (ω : Ω), |X i ω| ≤ 1 → rexp (τ * X i ω) ≤ rexp τ → τ * X i ω ≤ τ := sorry

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

Tactic:
rw [exp_le_exp]

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


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_14.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) →
        (∀ᵐ (ω : Ω) ∂μ, ∀ (i : Fin n), |X i ω| ≤ 1) → ∀ (i : Fin n), ∀ᵐ (a : Ω) ∂μ, |X i a| ≤ 1 := sorry

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

Tactic:
rw [ae_all_iff]

Tactic state after:
[Ω : Type u
 inst : MeasurableSpace Ω
 μ : Measure Ω
 inst_1 : IsProbabilityMeasure μ
 n : ℕ
 X : Fin n → Ω → ℝ
 hXm : ∀ (i : Fin n), Measurable (X i)
 hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
 τ : ℝ
 hlam : τ ≥ 0
 S : Ω → ℝ := fun ω => ∑ i, X i ω
 hS_meas : Measurable S
 f : ℝ → ℝ := fun x => rexp (τ * x)
 f_meas : Measurable f
 f_measω : Measurable fun ω => rexp (τ * S ω)
 h_bound : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ
 ⊢ ∀ (i : Fin n), ∀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_15.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) → ∀ (i : Fin n), ∀ᵐ (a : Ω) ∂μ, |X i a| ≤ 1 := sorry

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

Tactic:
exact hX_bdd

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_16.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) →
        (∀ᵐ (ω : Ω) ∂μ, ∀ (i : Fin n), |X i ω| ≤ 1) →
          ∀ (ω : Ω),
            (∀ (i : Fin n), |X i ω| ≤ 1) → ‖rexp (τ * S ω)‖ ≤ rexp (↑n * τ) → rexp (τ * S ω) ≤ rexp (↑n * τ) := sorry

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

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

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


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_17.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) →
        (∀ᵐ (ω : Ω) ∂μ, ∀ (i : Fin n), |X i ω| ≤ 1) →
          ∀ (ω : Ω), (∀ (i : Fin n), |X i ω| ≤ 1) → rexp (τ * S ω) ≤ rexp (↑n * τ) → τ * S ω ≤ ↑n * τ := sorry

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

Tactic:
rw [exp_le_exp]

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


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_18.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) →
        (∀ᵐ (ω : Ω) ∂μ, ∀ (i : Fin n), |X i ω| ≤ 1) →
          ∀ (ω : Ω), (∀ (i : Fin n), |X i ω| ≤ 1) → ∀ i ∈ Finset.univ, X i ω ≤ |X i ω| := sorry

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

Tactic:
exact le_abs_self (X _ ω)

Tactic state after:
import Mathlib
import FoML


open scoped Topology

open MeasureTheory ProbabilityTheory BigOperators ENNReal Real Set Filter Asymptotics

theorem extracted_formal_statement_19.{u} {Ω : Type u} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (hlam : τ ≥ 0) :
  let S := fun ω => ∑ i, X i ω;
  Measurable S →
    (Measurable fun ω => rexp (τ * S ω)) →
      (∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, ‖rexp (τ * X i ω)‖ ≤ rexp τ) →
        (∀ᵐ (ω : Ω) ∂μ, ∀ (i : Fin n), |X i ω| ≤ 1) →
          ∀ (ω : Ω), (∀ (i : Fin n), |X i ω| ≤ 1) → ∀ i ∈ Finset.univ, |X i ω| ≤ 1 := sorry

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

Tactic:
exact hω _

Tactic state after: