# Concept Lattice
#
# Formal concepts are partially ordered with regard to inclusion of their extents
# or (which is equivalent) inverse inclusion of their intent.
class ConceptLattice[O: Object, A: Object]
	super POSet[FormalConcept[O, A]]

	# Build `self` from a set of formal `concepts`.
	init from_concepts(concepts: Set[FormalConcept[O, A]]) do
		for c in concepts do
			add_node c
		end
		for c1 in concepts do
			for c2 in concepts do
				if c1 == c2 then continue
				if not is_lower_neighbour(c1, c2, concepts) then continue
				add_edge(c2, c1)
			end
		end
	end

	# Is `sub` the greatest lower bound of `sup` considering all `concepts`?
	fun is_lower_neighbour(sub, sup: FormalConcept[O, A], concepts: Set[FormalConcept[O, A]]): Bool
	do
		if sub == sup then return false
		if not sub.is_subconcept(sup) then return false
		for concept in concepts do
			if sub == concept then continue
			if sup == concept then continue
			if not sub.is_subconcept(concept) then continue
			if not concept.is_subconcept(sup) then continue
			return false
		end
		return true
	end
end