(* ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
*)
header{*Greatest Fixed Points and the Knaster-Tarski Theorem*}
theory Gfp
imports Lfp
begin
constdefs
gfp :: "['a set=>'a set] => 'a set"
"gfp(f) == Union({u. u \<subseteq> f(u)})"
subsection{*Proof of Knaster-Tarski Theorem using @{term gfp}*}
text{*@{term "gfp f"} is the greatest lower bound of
the set @{term "{u. u \<subseteq> f(u)}"} *}
lemma gfp_upperbound: "[| X \<subseteq> f(X) |] ==> X \<subseteq> gfp(f)"
by (auto simp add: gfp_def)
lemma gfp_least: "[| !!u. u \<subseteq> f(u) ==> u\<subseteq>X |] ==> gfp(f) \<subseteq> X"
by (auto simp add: gfp_def)
lemma gfp_lemma2: "mono(f) ==> gfp(f) \<subseteq> f(gfp(f))"
by (rules intro: gfp_least subset_trans monoD gfp_upperbound)
lemma gfp_lemma3: "mono(f) ==> f(gfp(f)) \<subseteq> gfp(f)"
by (rules intro: gfp_lemma2 monoD gfp_upperbound)
lemma gfp_unfold: "mono(f) ==> gfp(f) = f(gfp(f))"
by (rules intro: equalityI gfp_lemma2 gfp_lemma3)
subsection{*Coinduction rules for greatest fixed points*}
text{*weak version*}
lemma weak_coinduct: "[| a: X; X \<subseteq> f(X) |] ==> a : gfp(f)"
by (rule gfp_upperbound [THEN subsetD], auto)
lemma weak_coinduct_image: "!!X. [| a : X; g`X \<subseteq> f (g`X) |] ==> g a : gfp f"
apply (erule gfp_upperbound [THEN subsetD])
apply (erule imageI)
done
lemma coinduct_lemma:
"[| X \<subseteq> f(X Un gfp(f)); mono(f) |] ==> X Un gfp(f) \<subseteq> f(X Un gfp(f))"
by (blast dest: gfp_lemma2 mono_Un)
text{*strong version, thanks to Coen and Frost*}
lemma coinduct: "[| mono(f); a: X; X \<subseteq> f(X Un gfp(f)) |] ==> a : gfp(f)"
by (blast intro: weak_coinduct [OF _ coinduct_lemma])
lemma gfp_fun_UnI2: "[| mono(f); a: gfp(f) |] ==> a: f(X Un gfp(f))"
by (blast dest: gfp_lemma2 mono_Un)
subsection{*Even Stronger Coinduction Rule, by Martin Coen*}
text{* Weakens the condition @{term "X \<subseteq> f(X)"} to one expressed using both
@{term lfp} and @{term gfp}*}
lemma coinduct3_mono_lemma: "mono(f) ==> mono(%x. f(x) Un X Un B)"
by (rules intro: subset_refl monoI Un_mono monoD)
lemma coinduct3_lemma:
"[| X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f))); mono(f) |]
==> lfp(%x. f(x) Un X Un gfp(f)) \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f)))"
apply (rule subset_trans)
apply (erule coinduct3_mono_lemma [THEN lfp_lemma3])
apply (rule Un_least [THEN Un_least])
apply (rule subset_refl, assumption)
apply (rule gfp_unfold [THEN equalityD1, THEN subset_trans], assumption)
apply (rule monoD, assumption)
apply (subst coinduct3_mono_lemma [THEN lfp_unfold], auto)
done
lemma coinduct3:
"[| mono(f); a:X; X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f))) |] ==> a : gfp(f)"
apply (rule coinduct3_lemma [THEN [2] weak_coinduct])
apply (rule coinduct3_mono_lemma [THEN lfp_unfold, THEN ssubst], auto)
done
text{*Definition forms of @{text gfp_unfold} and @{text coinduct},
to control unfolding*}
lemma def_gfp_unfold: "[| A==gfp(f); mono(f) |] ==> A = f(A)"
by (auto intro!: gfp_unfold)
lemma def_coinduct:
"[| A==gfp(f); mono(f); a:X; X \<subseteq> f(X Un A) |] ==> a: A"
by (auto intro!: coinduct)
(*The version used in the induction/coinduction package*)
lemma def_Collect_coinduct:
"[| A == gfp(%w. Collect(P(w))); mono(%w. Collect(P(w)));
a: X; !!z. z: X ==> P (X Un A) z |] ==>
a : A"
apply (erule def_coinduct, auto)
done
lemma def_coinduct3:
"[| A==gfp(f); mono(f); a:X; X \<subseteq> f(lfp(%x. f(x) Un X Un A)) |] ==> a: A"
by (auto intro!: coinduct3)
text{*Monotonicity of @{term gfp}!*}
lemma gfp_mono: "[| !!Z. f(Z)\<subseteq>g(Z) |] ==> gfp(f) \<subseteq> gfp(g)"
by (rule gfp_upperbound [THEN gfp_least], blast)
ML
{*
val gfp_def = thm "gfp_def";
val gfp_upperbound = thm "gfp_upperbound";
val gfp_least = thm "gfp_least";
val gfp_unfold = thm "gfp_unfold";
val weak_coinduct = thm "weak_coinduct";
val weak_coinduct_image = thm "weak_coinduct_image";
val coinduct = thm "coinduct";
val gfp_fun_UnI2 = thm "gfp_fun_UnI2";
val coinduct3 = thm "coinduct3";
val def_gfp_unfold = thm "def_gfp_unfold";
val def_coinduct = thm "def_coinduct";
val def_Collect_coinduct = thm "def_Collect_coinduct";
val def_coinduct3 = thm "def_coinduct3";
val gfp_mono = thm "gfp_mono";
*}
end