theorem extracted_formal_statement_0 (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) : 0 < u := sorry

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
    rw [mul_assoc, ← sq, mul_comm (rexp _)]
    refine mul_le_mul_of_nonneg_left ?_ (sq_nonneg x)
    apply exp_monotone
    refine (mul_le_iff_le_one_right ?_).mpr hx_le_1

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)
 ⊢ 0 < u]

Relevant lemmas:
Real.exp_monotone : Monotone rexp
mul_le_iff_le_one_right : ∀ {α : Type u_3} {a b : α} [inst : MulOneClass α] [inst_1 : Zero α] [inst_2 : Preorder α]
  [inst_3 : PosMulMono α] [inst_4 : PosMulReflectLE α], 0 < a → (a * b ≤ a ↔ b ≤ 1)
sub_nonpos_of_le : ∀ {α : Type u} [inst : AddGroup α] [inst_1 : LE α] [inst_2 : AddRightMono α] {a b : α},
  a ≤ b → a - b ≤ 0
mul_le_mul_of_nonneg_left : ∀ {α : Type u_3} {a b c : α} [inst : Mul α] [inst_1 : Zero α] [inst_2 : Preorder α]
  [inst_3 : PosMulMono α], b ≤ c → 0 ≤ a → a * b ≤ a * c
mul_comm : ∀ {G : Type u_1} [inst : CommMagma G] (a b : G), a * b = b * a
sq : ∀ {M : Type u_2} [inst : Monoid M] (a : M), a ^ 2 = a * a
mul_assoc : ∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * b * c = a * (b * c)
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