English
A simple bounded order carries a natural preorder given by ≤; reflexivity and transitivity hold.
Русский
Простой ограниченный порядок содержит естественный предраспорядок, задаваемый ≤; рефлексивность и транзитивность выполняются.
LaTeX
$$$\le$ is a preorder on $\alpha$$$
Lean4
/-- A simple `BoundedOrder` induces a preorder. This is not an instance to prevent loops. -/
protected def preorder {α} [LE α] [BoundedOrder α] [IsSimpleOrder α] : Preorder α
where
le := (· ≤ ·)
le_refl a := by rcases eq_bot_or_eq_top a with (rfl | rfl) <;> simp
le_trans a b
c := by
rcases eq_bot_or_eq_top a with (rfl | rfl)
· simp
· rcases eq_bot_or_eq_top b with (rfl | rfl)
· rcases eq_bot_or_eq_top c with (rfl | rfl) <;> simp
· simp
