FIXED-POINT THEOREMS FOR NON-TRANSITIVE RELATIONS

In this paper, we develop an Isabelle/HOL library of order-theoretic fixed-point theorems. We keep our formalization as general as possible: we reprove several well-known results about complete orders, often with only antisymmetry or attractivity, a mild condition implied by either antisymmetry or transitivity. In particular, we generalize various theorems ensuring the existence of a quasi-fixed point of monotone maps over complete relations, and show that the set of (quasi-)fixed points is itself complete. This result generalizes and strengthens theorems of Knaster–Tarski, Bourbaki–Witt, Kleene, Markowsky, Pataraia, Mashburn, Bhatta–George, and Stouti–Maaden.