theorem extracted_formal_statement_18 (x : ℝ) (xpos : x ≤ 0) (xzero : x ≠ 0) (xneg : x < 0)
  (h_original_goal : x.sign * |x| = x) : x ≤ 0 := sorry

State before:
case neg
x : ℝ
xpos : x ≤ 0
xzero : x ≠ 0
xneg : x < 0
⊢ x.sign * |x| = x

Tactic: rw [Real.sign_of_neg xneg]
    simp
    rw [neg_eq_iff_eq_neg, abs_eq_neg_self]

State after:
[case neg
 x : ℝ
 xpos : x ≤ 0
 xzero : x ≠ 0
 xneg : x < 0
 ⊢ x ≤ 0]

Relevant lemmas:
Real.sign_of_neg : ∀ {r : ℝ}, r < 0 → r.sign = -1
neg_eq_iff_eq_neg : ∀ {G : Type u_3} [inst : InvolutiveNeg G] {a b : G}, -a = b ↔ a = -b
abs_eq_neg_self : ∀ {G : Type u_1} [inst : LinearOrderedAddCommGroup G] {a : G}, |a| = -a ↔ a ≤ 0