(* Title: HOL/Isar_examples/KnasterTarski.thy
ID: $Id$
Author: Markus Wenzel, TU Muenchen
Typical textbook proof example.
*)
theory KnasterTarski = Main:;
theorems [dest] = monoD; (* FIXME [dest!!] *)
text {*
The proof of Knaster-Tarski below closely follows the presentation in
'Introduction to Lattices' and Order by Davey/Priestley, pages
93--94. All of their narration has been rephrased in terms of formal
Isar language elements. Just as many textbook-style proofs, there is
a strong bias towards forward reasoning, and little hierarchical
structure.
*};
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;
assume mem: "x : ??H";
hence "??a <= x"; by (rule Inter_lower);
with mono; have "f ??a <= f x"; ..;
also; from mem; have "... <= x"; ..;
finally; have "f ??a <= x"; .;
}};
hence ge: "f ??a <= ??a"; by (rule Inter_greatest);
{{;
also; presume "... <= f ??a";
finally (order_antisym); show ??thesis; .;
}};
from mono ge; have "f (f ??a) <= f ??a"; ..;
hence "f ??a : ??H"; ..;
thus "??a <= f ??a"; by (rule Inter_lower);
qed;
qed;
end;