theorem extracted_formal_statement_0 (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

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
    set f : ℝ → ℝ := fun τ => rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ))
    set f' : ℝ → ℝ := fun τ => rexp (τ * x) * x - (x + x ^ 2 * (rexp τ - 1))

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))
 ⊢ rexp (τ * x) - (1 + τ * x + x ^ 2 * (rexp τ - 1 - τ)) ≤ 0]

Relevant lemmas:
le_of_sub_nonpos : ∀ {α : Type u} [inst : AddGroup α] [inst_1 : LE α] [inst_2 : AddRightMono α] {a b : α},
  a - b ≤ 0 → a ≤ b