(* Title: HOL/Examples/Knaster_Tarski.thy Author: Makarius Typical textbook proof example. *) section ‹Textbook-style reasoning: the Knaster-Tarski Theorem› theory Knaster_Tarski imports Main begin unbundle lattice_syntax subsection ‹Prose version› text ‹ According to the textbook \<^cite>‹‹pages 93--94› in "davey-priestley"›, the Knaster-Tarski fixpoint theorem is as follows.⁋‹We have dualized the argument, and tuned the notation a little bit.› ❙‹The Knaster-Tarski Fixpoint Theorem.› Let ‹L› be a complete lattice and ‹f: L → L› an order-preserving map. Then ‹⨅{x ∈ L | f(x) ≤ x}› is a fixpoint of ‹f›. ❙‹Proof.› Let ‹H = {x ∈ L | f(x) ≤ x}› and ‹a = ⨅H›. For all ‹x ∈ H› we have ‹a ≤ x›, so ‹f(a) ≤ f(x) ≤ x›. Thus ‹f(a)› is a lower bound of ‹H›, whence ‹f(a) ≤ a›. We now use this inequality to prove the reverse one (!) and thereby complete the proof that ‹a› is a fixpoint. Since ‹f› is order-preserving, ‹f(f(a)) ≤ f(a)›. This says ‹f(a) ∈ H›, so ‹a ≤ f(a)›.› subsection ‹Formal versions› text ‹ The Isar proof below closely follows the original presentation. Virtually all of the prose narration has been rephrased in terms of formal Isar language elements. Just as many textbook-style proofs, there is a strong bias towards forward proof, and several bends in the course of reasoning. › theorem Knaster_Tarski: fixes f :: "'a::complete_lattice ⇒ 'a" assumes "mono f" shows "∃a. f a = a" proof let ?H = "{u. f u ≤ u}" let ?a = "⨅?H" show "f ?a = ?a" proof - { fix x assume "x ∈ ?H" then have "?a ≤ x" by (rule Inf_lower) with ‹mono f› have "f ?a ≤ f x" .. also from ‹x ∈ ?H› have "… ≤ x" .. finally have "f ?a ≤ x" . } then have "f ?a ≤ ?a" by (rule Inf_greatest) { also presume "… ≤ f ?a" finally (order_antisym) show ?thesis . } from ‹mono f› and ‹f ?a ≤ ?a› have "f (f ?a) ≤ f ?a" .. then have "f ?a ∈ ?H" .. then show "?a ≤ f ?a" by (rule Inf_lower) qed qed text ‹ Above we have used several advanced Isar language elements, such as explicit block structure and weak assumptions. Thus we have mimicked the particular way of reasoning of the original text. In the subsequent version the order of reasoning is changed to achieve structured top-down decomposition of the problem at the outer level, while only the inner steps of reasoning are done in a forward manner. We are certainly more at ease here, requiring only the most basic features of the Isar language. › theorem Knaster_Tarski': fixes f :: "'a::complete_lattice ⇒ 'a" assumes "mono f" shows "∃a. f a = a" proof let ?H = "{u. f u ≤ u}" let ?a = "⨅?H" show "f ?a = ?a" proof (rule order_antisym) show "f ?a ≤ ?a" proof (rule Inf_greatest) fix x assume "x ∈ ?H" then have "?a ≤ x" by (rule Inf_lower) with ‹mono f› have "f ?a ≤ f x" .. also from ‹x ∈ ?H› have "… ≤ x" .. finally show "f ?a ≤ x" . qed show "?a ≤ f ?a" proof (rule Inf_lower) from ‹mono f› and ‹f ?a ≤ ?a› have "f (f ?a) ≤ f ?a" .. then show "f ?a ∈ ?H" .. qed qed qed end