theorem extracted_formal_statement_11.{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 ((2 * |t| + 1) * |X ω|) * |X ω| ^ 2 := sorry

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]
    show |rexp (x * X ω) * X ω ^ 2| ≤ rexp ((2 * |t| + 1) * |X ω|) * X ω ^ 2
    calc
  _ = |rexp (x * X ω)| * |(X ω ^ 2)| := abs_mul (rexp (x * X ω)) (X ω ^ 2)
  _ = rexp (x * X ω) * (X ω ^ 2) := by simp
  _ ≤ rexp ((2 * |t| + 1) * |X ω|) * X ω ^ 2 :=
    by
    suffices rexp (x * X ω) ≤ rexp ((2 * |t| + 1) * |X ω|) from mul_le_mul_of_nonneg_right this (sq_nonneg (X ω))
    simp only [exp_le_exp]
    calc
      _ ≤ |x * X ω| := le_abs_self (x * X ω)
      _ = |x| * |X ω| := (abs_mul x (X ω))
      _ ≤ (2 * |t| + 1) * |X ω| :=
        by
        suffices |x| ≤ 2 * |t| + 1 from mul_le_mul_of_nonneg_right this (abs_nonneg (X ω))
        simp only [Metric.mem_ball] at h
        linarith [le_trans' (le_of_lt h) (abs_sub_abs_le_abs_sub x t)]

State after:

Relevant lemmas:
abs_nonneg : ∀ {α : Type u_1} [inst : Lattice α] [inst_1 : AddGroup α] [inst_2 : AddLeftMono α]
  [inst_3 : AddRightMono α] (a : α), 0 ≤ |a|
le_trans' : ∀ {α : Type u_2} [inst : Preorder α] {a b c : α}, b ≤ c → a ≤ b → a ≤ c
mul_le_mul_of_nonneg_right : ∀ {α : Type u_3} {a b c : α} [inst : Mul α] [inst_1 : Zero α] [inst_2 : Preorder α]
  [inst_3 : MulPosMono α], b ≤ c → 0 ≤ a → b * a ≤ c * a
le_of_lt : ∀ {α : Type u_1} [inst : Preorder α] {a b : α}, a < b → a ≤ b
Metric.mem_ball : ∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α} {ε : ℝ}, y ∈ Metric.ball x ε ↔ dist y x < ε
Real.exp_le_exp : ∀ {x y : ℝ}, rexp x ≤ rexp y ↔ x ≤ y
abs_mul : ∀ {α : Type u_1} [inst : LinearOrderedRing α] (a b : α), |a * b| = |a| * |b|
sq_abs : ∀ {α : Type u_1} [inst : LinearOrderedRing α] (a : α), |a| ^ 2 = a ^ 2
le_abs_self : ∀ {α : Type u_1} [inst : Lattice α] [inst_1 : AddGroup α] (a : α), a ≤ |a|
sq_nonneg : ∀ {α : Type u} [inst : Semiring α] [inst_1 : LinearOrder α] [inst_2 : IsRightCancelAdd α]
  [inst_3 : ZeroLEOneClass α] [inst_4 : ExistsAddOfLE α] [inst_5 : PosMulMono α] [inst_6 : AddLeftStrictMono α] (a : α),
  0 ≤ a ^ 2
abs_sub_abs_le_abs_sub : ∀ {G : Type u_1} [inst : LinearOrderedAddCommGroup G] (a b : G), |a| - |b| ≤ |a - b|