# Theory Gfp

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

section ‹Greatest fixed points›

theory Gfp
imports Lfp
begin

definition
gfp :: "['a set⇒'a set] ⇒ 'a set" where ― ‹greatest fixed point›
"gfp(f) == Union({u. u <= f(u)})"

(* gfp(f) is the least upper bound of {u. u <= f(u)} *)

lemma gfp_upperbound: "A <= f(A) ⟹ A <= gfp(f)"
unfolding gfp_def by blast

lemma gfp_least: "(⋀u. u <= f(u) ⟹ u <= A) ⟹ gfp(f) <= A"
unfolding gfp_def by blast

lemma gfp_lemma2: "mono(f) ⟹ gfp(f) <= f(gfp(f))"
by (rule gfp_least, rule subset_trans, assumption, erule monoD,
rule gfp_upperbound, assumption)

lemma gfp_lemma3: "mono(f) ⟹ f(gfp(f)) <= gfp(f)"
by (rule gfp_upperbound, frule monoD, rule gfp_lemma2, assumption+)

lemma gfp_Tarski: "mono(f) ⟹ gfp(f) = f(gfp(f))"
by (rule equalityI gfp_lemma2 gfp_lemma3 | assumption)+

(*** Coinduction rules for greatest fixed points ***)

(*weak version*)
lemma coinduct: "⟦a: A;  A <= f(A)⟧ ⟹ a : gfp(f)"
by (blast dest: gfp_upperbound)

lemma coinduct2_lemma: "⟦A <= f(A) Un gfp(f); mono(f)⟧ ⟹ A Un gfp(f) <= f(A Un gfp(f))"
apply (rule subset_trans)
prefer 2
apply (erule mono_Un)
apply (rule subst, erule gfp_Tarski)
apply (erule Un_least)
apply (rule Un_upper2)
done

(*strong version, thanks to Martin Coen*)
lemma coinduct2: "⟦a: A; A <= f(A) Un gfp(f); mono(f)⟧ ⟹ a : gfp(f)"
apply (rule coinduct)
prefer 2
apply (erule coinduct2_lemma, assumption)
apply blast
done

(***  Even Stronger version of coinduct  [by Martin Coen]
- instead of the condition  A <= f(A)
consider  A <= (f(A) Un f(f(A)) ...) Un gfp(A) ***)

lemma coinduct3_mono_lemma: "mono(f) ⟹ mono(λx. f(x) Un A Un B)"
by (rule monoI) (blast dest: monoD)

lemma coinduct3_lemma:
assumes prem: "A <= f(lfp(λx. f(x) Un A Un gfp(f)))"
and mono: "mono(f)"
shows "lfp(λx. f(x) Un A Un gfp(f)) <= f(lfp(λx. f(x) Un A Un gfp(f)))"
apply (rule subset_trans)
apply (rule mono [THEN coinduct3_mono_lemma, THEN lfp_lemma3])
apply (rule Un_least [THEN Un_least])
apply (rule subset_refl)
apply (rule prem)
apply (rule mono [THEN gfp_Tarski, THEN equalityD1, THEN subset_trans])
apply (rule mono [THEN monoD])
apply (subst mono [THEN coinduct3_mono_lemma, THEN lfp_Tarski])
apply (rule Un_upper2)
done

lemma coinduct3:
assumes 1: "a:A"
and 2: "A <= f(lfp(λx. f(x) Un A Un gfp(f)))"
and 3: "mono(f)"
shows "a : gfp(f)"
apply (rule coinduct)
prefer 2
apply (rule coinduct3_lemma [OF 2 3])
apply (subst lfp_Tarski [OF coinduct3_mono_lemma, OF 3])
using 1 apply blast
done

subsection ‹Definition forms of ‹gfp_Tarski›, to control unfolding›

lemma def_gfp_Tarski: "⟦h == gfp(f); mono(f)⟧ ⟹ h = f(h)"
apply unfold
apply (erule gfp_Tarski)
done

lemma def_coinduct: "⟦h == gfp(f); a:A; A <= f(A)⟧ ⟹ a: h"
apply unfold
apply (erule coinduct)
apply assumption
done

lemma def_coinduct2: "⟦h == gfp(f); a:A; A <= f(A) Un h; mono(f)⟧ ⟹ a: h"
apply unfold
apply (erule coinduct2)
apply assumption
apply assumption
done

lemma def_coinduct3: "⟦h == gfp(f); a:A; A <= f(lfp(λx. f(x) Un A Un h)); mono(f)⟧ ⟹ a: h"
apply unfold
apply (erule coinduct3)
apply assumption
apply assumption
done

(*Monotonicity of gfp!*)
lemma gfp_mono: "⟦mono(f); ⋀Z. f(Z) <= g(Z)⟧ ⟹ gfp(f) <= gfp(g)"
apply (rule gfp_upperbound)
apply (rule subset_trans)
apply (rule gfp_lemma2)
apply assumption
apply (erule meta_spec)
done

end
```