English
If s is a lower set in the index order, then any two partial equivalences on s must coincide; the data below s is uniquely determined by the rest of the system.
Русский
Если s является нижним множеством по порядку индексов, то любые два частичных эквиваленций на s совпадают; данные ниже s определены однозначно остальной частью системы.
LaTeX
$$$\\text{IsLowerSet}(s) \\Rightarrow \\forall e_1 e_2 : \\mathrm{PEquivOn}(f, equivSucc, s), e_1 = e_2$$$
Lean4
/-- Two elements of a lattice are disjoint if their inf is the bottom element.
(This generalizes disjoint sets, viewed as members of the subset lattice.)
Note that we define this without reference to `⊓`, as this allows us to talk about orders where
the infimum is not unique, or where implementing `Inf` would require additional `Decidable`
arguments. -/
def Disjoint (a b : α) : Prop :=
∀ ⦃x⦄, x ≤ a → x ≤ b → x ≤ ⊥
