English
For any real r, normalization scales by the sign of r: normalize(r x) = sign(r) · normalize(x).
Русский
Для любого вещественного r нормирование масштабирует на знак r: normalize(r x) = sign(r) · normalize(x).
LaTeX
$$$\\operatorname{normalize}(r x) = \\operatorname{sign}(r)\\,\\operatorname{normalize}(x)$$$
Lean4
theorem normalize_smul (r : ℝ) (x : V) : normalize (r • x) = (SignType.sign r : ℝ) • normalize x :=
by
rcases lt_trichotomy 0 r with (h_pos | rfl | h_neg)
· simp [normalize_smul_of_pos, h_pos]
· simp
· simp [normalize_smul_of_neg, h_neg]