Fixpoints in Complete Lattices 1

Theorem (5) states that if an iterate of a function has a unique xpoint then it is also the xpoint of the function. It has been included here in response to P. Andrews claim that such a proof in set theory takes thousands of lines when one starts with the axioms. While probably true, such a claim is misleading about the usefulness of proof-checking systems based on set theory. Next, we prove the existence of the least and the greatest xpoints for-monotone functions from a powerset to a powerset of a set. Scheme Knaster is the Knaster theorem about the existence of xpoints, cf. 14]. Theorem (11) is the Banach decomposition theorem which is then used to prove the Schrr oder-Bernstein theorem (12) (we followed Paulson's development of these theorems in Isabelle 16]). It is interesting to note that the last theorem when stated in Mizar in terms of cardinals becomes trivial to prove as in the Mizar development of cardinals the relation is synonymous with. Section 3 introduces the notion of the lattice of a lattice subset provided the subset has lubs and glbs. The main theorem of Section 4 is the Tarski theorem (43) that every monotone function f over a complete lattice L has a complete lattice of xpoints. As the consequence of this theorem we get the existence of the least xpoint equal to f (?L) for some ordinal with cardinality not bigger than the cardinality of the carrier of L, cf. 14], and analogously the existence of the greatest xpoint equal to f (>L). Section 5 connects the xpoint properties of monotone functions over complete lattices with the xpoints of-monotone functions over the lattice of subsets of a set (Boolean lattice). In this paper f, g, h are functions. The following propositions are true: (1) If f is one-to-one and g is one-to-one and rng f misses rng g; then f+g is one-to-one. (2) If dom f misses dom g; then f g is a function.