# HG changeset patch
# User avigad
# Date 1123073277 -7200
# Node ID 332c28b2844eb281cdf1f37a7aef3726e7005d28
# Parent  cffca870816ab5079129963ac832c17f583577ae
removed Lfp

diff -r cffca870816a -r 332c28b2844e src/HOL/Lfp.thy
--- a/src/HOL/Lfp.thy	Wed Aug 03 14:47:51 2005 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,107 +0,0 @@
-(*  Title:      HOL/Lfp.thy
-    ID:         $Id$
-    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   1992  University of Cambridge
-*)
-
-header{*Least Fixed Points and the Knaster-Tarski Theorem*}
-
-theory Lfp
-imports Product_Type
-begin
-
-constdefs
-  lfp :: "['a set \<Rightarrow> 'a set] \<Rightarrow> 'a set"
-    "lfp(f) == Inter({u. f(u) \<subseteq> u})"    --{*least fixed point*}
-
-
-
-subsection{*Proof of Knaster-Tarski Theorem using @{term lfp}*}
-
-
-text{*@{term "lfp f"} is the least upper bound of 
-      the set @{term "{u. f(u) \<subseteq> u}"} *}
-
-lemma lfp_lowerbound: "f(A) \<subseteq> A ==> lfp(f) \<subseteq> A"
-by (auto simp add: lfp_def)
-
-lemma lfp_greatest: "[| !!u. f(u) \<subseteq> u ==> A\<subseteq>u |] ==> A \<subseteq> lfp(f)"
-by (auto simp add: lfp_def)
-
-lemma lfp_lemma2: "mono(f) ==> f(lfp(f)) \<subseteq> lfp(f)"
-by (rules intro: lfp_greatest subset_trans monoD lfp_lowerbound)
-
-lemma lfp_lemma3: "mono(f) ==> lfp(f) \<subseteq> f(lfp(f))"
-by (rules intro: lfp_lemma2 monoD lfp_lowerbound)
-
-lemma lfp_unfold: "mono(f) ==> lfp(f) = f(lfp(f))"
-by (rules intro: equalityI lfp_lemma2 lfp_lemma3)
-
-subsection{*General induction rules for greatest fixed points*}
-
-lemma lfp_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)
- apply (rule subset_trans [OF Int_lower1 [THEN mono [THEN monoD]]
-                              mono [THEN lfp_lemma2]]) 
-apply (blast intro: indhyp) 
-done
-
-
-text{*Version of induction for binary relations*}
-lemmas lfp_induct2 =  lfp_induct [of "(a,b)", split_format (complete)]
-
-
-lemma lfp_ordinal_induct: 
-  assumes mono: "mono f"
-  shows "[| !!S. P S ==> P(f S); !!M. !S:M. P S ==> P(Union M) |] 
-         ==> P(lfp f)"
-apply(subgoal_tac "lfp f = Union{S. S \<subseteq> lfp f & P S}")
- apply (erule ssubst, simp) 
-apply(subgoal_tac "Union{S. S \<subseteq> lfp f & P S} \<subseteq> lfp f")
- prefer 2 apply blast
-apply(rule equalityI)
- prefer 2 apply assumption
-apply(drule mono [THEN monoD])
-apply (cut_tac mono [THEN lfp_unfold], simp)
-apply (rule lfp_lowerbound, auto) 
-done
-
-
-text{*Definition forms of @{text lfp_unfold} and @{text lfp_induct}, 
-    to control unfolding*}
-
-lemma def_lfp_unfold: "[| h==lfp(f);  mono(f) |] ==> h = f(h)"
-by (auto intro!: lfp_unfold)
-
-lemma def_lfp_induct: 
-    "[| A == lfp(f);  mono(f);   a:A;                    
-        !!x. [| x: f(A Int {x. P(x)}) |] ==> P(x)         
-     |] ==> P(a)"
-by (blast intro: lfp_induct)
-
-(*Monotonicity of lfp!*)
-lemma lfp_mono: "[| !!Z. f(Z)\<subseteq>g(Z) |] ==> lfp(f) \<subseteq> lfp(g)"
-by (rule lfp_lowerbound [THEN lfp_greatest], blast)
-
-
-ML
-{*
-val lfp_def = thm "lfp_def";
-val lfp_lowerbound = thm "lfp_lowerbound";
-val lfp_greatest = thm "lfp_greatest";
-val lfp_unfold = thm "lfp_unfold";
-val lfp_induct = thm "lfp_induct";
-val lfp_induct2 = thm "lfp_induct2";
-val lfp_ordinal_induct = thm "lfp_ordinal_induct";
-val def_lfp_unfold = thm "def_lfp_unfold";
-val def_lfp_induct = thm "def_lfp_induct";
-val lfp_mono = thm "lfp_mono";
-*}
-
-end