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.

中文翻译：

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非传递关系的不动点定理

在本文中，我们开发了有序理论不动点定理的 Isabelle/HOL 库。我们尽可能保持我们的形式化：我们反对几个关于完全阶的众所周知的结果，通常只有反对称或吸引力，反对称或传递性暗示的温和条件。特别地，我们概括了确保在完全关系上存在单调映射的准不动点的各种定理，并证明（准）不动点的集合本身是完备的。该结果概括并加强了 Knaster-Tarski、Bourbaki-Witt、Kleene、Markowsky、Pataraia、Mashburn、Bhatta-George 和 Stouti-Maaden 的定理。