(* Title: CCL/Lfp.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
*)
header {* The Knaster-Tarski Theorem *}
theory Lfp
imports Set
begin
definition
lfp :: "['a set=>'a set] => 'a set" where -- "least fixed point"
"lfp(f) == Inter({u. f(u) <= u})"
(* lfp(f) is the greatest lower bound of {u. f(u) <= u} *)
lemma lfp_lowerbound: "[| f(A) <= A |] ==> lfp(f) <= A"
unfolding lfp_def by blast
lemma lfp_greatest: "[| !!u. f(u) <= u ==> A<=u |] ==> A <= lfp(f)"
unfolding lfp_def by blast
lemma lfp_lemma2: "mono(f) ==> f(lfp(f)) <= lfp(f)"
by (rule lfp_greatest, rule subset_trans, drule monoD, rule lfp_lowerbound, assumption+)
lemma lfp_lemma3: "mono(f) ==> lfp(f) <= f(lfp(f))"
by (rule lfp_lowerbound, frule monoD, drule lfp_lemma2, assumption+)
lemma lfp_Tarski: "mono(f) ==> lfp(f) = f(lfp(f))"
by (rule equalityI lfp_lemma2 lfp_lemma3 | assumption)+
(*** General induction rule for least fixed points ***)
lemma induct:
assumes lfp: "a: lfp(f)"
and mono: "mono(f)"
and indhyp: "!!x. [| x: f(lfp(f) Int {x. P(x)}) |] ==> P(x)"
shows "P(a)"
apply (rule_tac a = a in Int_lower2 [THEN subsetD, THEN CollectD])
apply (rule lfp [THEN [2] lfp_lowerbound [THEN subsetD]])
apply (rule Int_greatest, rule subset_trans, rule Int_lower1 [THEN mono [THEN monoD]],
rule mono [THEN lfp_lemma2], rule CollectI [THEN subsetI], rule indhyp, assumption)
done
(** Definition forms of lfp_Tarski and induct, to control unfolding **)
lemma def_lfp_Tarski: "[| h==lfp(f); mono(f) |] ==> h = f(h)"
apply unfold
apply (drule lfp_Tarski)
apply assumption
done
lemma def_induct:
"[| A == lfp(f); a:A; mono(f);
!!x. [| x: f(A Int {x. P(x)}) |] ==> P(x)
|] ==> P(a)"
apply (rule induct [of concl: P a])
apply simp
apply assumption
apply blast
done
(*Monotonicity of lfp!*)
lemma lfp_mono: "[| mono(g); !!Z. f(Z)<=g(Z) |] ==> lfp(f) <= lfp(g)"
apply (rule lfp_lowerbound)
apply (rule subset_trans)
apply (erule meta_spec)
apply (erule lfp_lemma2)
done
end