theorem extracted_formal_statement_0 : 0 ≤ 2⁻¹ := sorry

State before:
Ω : Type u
inst✝¹ : MeasurableSpace Ω
μ : autoParam (Measure Ω) _auto✝
inst✝ : IsProbabilityMeasure μ
t a b : ℝ
X : Ω → ℝ
ht : 0 ≤ t
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
h0 : ∫ (x : Ω), X x ∂μ = 0
w : ¬t = 0
f : ℝ → ℝ := fun t => cgf X μ t
hf : f 0 = 0
f' : ℝ → ℝ := fun t => ∫ (x : Ω), X x ∂Measure.tilted μ fun ω => t * X ω
hf' : f' 0 = 0
f'' : ℝ → ℝ := fun t => Var[X; Measure.tilted μ fun ω => t * X ω]
c : ℝ
left✝ : 0 < c ∧ c < t
cq' : f t = f'' c * t ^ 2 / 2
⊢ f'' c * t ^ 2 / 2 ≤ ((b - a) / 2) ^ 2 * t ^ 2 / 2

Tactic: apply mul_le_mul_of_nonneg_right
    apply mul_le_mul_of_nonneg_right
    dsimp [f'']

State after:
[case h.h
 Ω : Type u
 inst✝¹ : MeasurableSpace Ω
 μ : autoParam (Measure Ω) _auto✝
 inst✝ : IsProbabilityMeasure μ
 t a b : ℝ
 X : Ω → ℝ
 ht : 0 ≤ t
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 h0 : ∫ (x : Ω), X x ∂μ = 0
 w : ¬t = 0
 f : ℝ → ℝ := fun t => cgf X μ t
 hf : f 0 = 0
 f' : ℝ → ℝ := fun t => ∫ (x : Ω), X x ∂Measure.tilted μ fun ω => t * X ω
 hf' : f' 0 = 0
 f'' : ℝ → ℝ := fun t => Var[X; Measure.tilted μ fun ω => t * X ω]
 c : ℝ
 left✝ : 0 < c ∧ c < t
 cq' : f t = f'' c * t ^ 2 / 2
 ⊢ Var[X; Measure.tilted μ fun ω => c * X ω] ≤ ((b - a) / 2) ^ 2,
 case h.a0
 Ω : Type u
 inst✝¹ : MeasurableSpace Ω
 μ : autoParam (Measure Ω) _auto✝
 inst✝ : IsProbabilityMeasure μ
 t a b : ℝ
 X : Ω → ℝ
 ht : 0 ≤ t
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 h0 : ∫ (x : Ω), X x ∂μ = 0
 w : ¬t = 0
 f : ℝ → ℝ := fun t => cgf X μ t
 hf : f 0 = 0
 f' : ℝ → ℝ := fun t => ∫ (x : Ω), X x ∂Measure.tilted μ fun ω => t * X ω
 hf' : f' 0 = 0
 f'' : ℝ → ℝ := fun t => Var[X; Measure.tilted μ fun ω => t * X ω]
 c : ℝ
 left✝ : 0 < c ∧ c < t
 cq' : f t = f'' c * t ^ 2 / 2
 ⊢ 0 ≤ t ^ 2,
 case a0
 Ω : Type u
 inst✝¹ : MeasurableSpace Ω
 μ : autoParam (Measure Ω) _auto✝
 inst✝ : IsProbabilityMeasure μ
 t a b : ℝ
 X : Ω → ℝ
 ht : 0 ≤ t
 hX : AEMeasurable X μ
 h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
 h0 : ∫ (x : Ω), X x ∂μ = 0
 w : ¬t = 0
 f : ℝ → ℝ := fun t => cgf X μ t
 hf : f 0 = 0
 f' : ℝ → ℝ := fun t => ∫ (x : Ω), X x ∂Measure.tilted μ fun ω => t * X ω
 hf' : f' 0 = 0
 f'' : ℝ → ℝ := fun t => Var[X; Measure.tilted μ fun ω => t * X ω]
 c : ℝ
 left✝ : 0 < c ∧ c < t
 cq' : f t = f'' c * t ^ 2 / 2
 ⊢ 0 ≤ 2⁻¹]

Relevant lemmas:
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