src/HOL/Isar_examples/KnasterTarski.thy
 author wenzelm Thu, 01 Jul 1999 21:30:18 +0200 changeset 6882 fe4e3d26fa8f child 6883 f898679685b7 permissions -rw-r--r--
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(*  Title:      HOL/Isar_examples/KnasterTarski.thy
ID:         \$Id\$
Author:     Markus Wenzel, TU Muenchen

Typical textbook proof example.
*)

theory KnasterTarski = Main:;

(*

text {*
The proof of Knaster-Tarski below closely follows the presentation in
'Introduction to Lattices and Order' by Davey/Priestley, pages
93--94.  Only one statement of their narration has not been rephrased
in formal Isar language elements, but left as a comment.  Also note
that Davey/Priestley do not point out non-emptyness of the set ??H,
(which is obvious, but not vacous).
*};
*)

theorems [dest] = monoD;  (* FIXME [dest!!] *)

theorem KnasterTarski: "mono f ==> EX a::'a set. f a = a";
proof;
let ??H = "{u. f u <= u}";
let ??a = "Inter ??H";

assume mono: "mono f";
show "f ??a = ??a";
proof same;
fix x;
presume mem: "x : ??H";
hence "??a <= x"; by (rule Inter_lower);
with mono; have "f ??a <= f x"; ..;
also; from mem; have "f x <= x"; ..;
finally; have "f ??a <= x"; .;
hence ge: "f ??a <= ??a"; by (rule Inter_greatest);
(* text {* We now use this inequality to prove the reverse one (!)
and thereby complete the proof that @term{??a} is a fixpoint. *};  *)
with mono; have "f (f ??a) <= f ??a"; ..;
hence "f ??a : ??H"; ..;
hence "??a <= f ??a"; by (rule Inter_lower);
also (order_antisym); note ge;
finally; show "f ??a = ??a"; proof same;
next;
have "f UNIV <= UNIV"; by (rule subset_UNIV);
thus "UNIV : ??H"; ..;
qed;
qed;

end;
```