# Theory Gfp

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theory Gfp
imports Lfp
`(*  Title:      CCL/Gfp.thy    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory    Copyright   1992  University of Cambridge*)header {* Greatest fixed points *}theory Gfpimports Lfpbegindefinition  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 blastlemma gfp_least: "[| !!u. u <= f(u) ==> u<=A |] ==> gfp(f) <= A"  unfolding gfp_def by blastlemma 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(%x. f(x) Un A Un B)"  by (rule monoI) (blast dest: monoD)lemma coinduct3_lemma:  assumes prem: "A <= f(lfp(%x. f(x) Un A Un gfp(f)))"    and mono: "mono(f)"  shows "lfp(%x. f(x) Un A Un gfp(f)) <= f(lfp(%x. 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)  donelemma coinduct3:  assumes 1: "a:A"    and 2: "A <= f(lfp(%x. 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  donesubsection {* Definition forms of @{text "gfp_Tarski"}, to control unfolding *}lemma def_gfp_Tarski: "[| h==gfp(f);  mono(f) |] ==> h = f(h)"  apply unfold  apply (erule gfp_Tarski)  donelemma def_coinduct: "[| h==gfp(f);  a:A;  A <= f(A) |] ==> a: h"  apply unfold  apply (erule coinduct)  apply assumption  donelemma 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  donelemma def_coinduct3: "[| h==gfp(f);  a:A;  A <= f(lfp(%x. 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)  doneend`