# Theory Lfp

```(*  Title:      CCL/Lfp.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section ‹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  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
```