author bauerg Thu, 28 Apr 2005 17:08:08 +0200 changeset 15871 e524119dbf19 parent 15870 4320bce5873f child 15872 8336ff711d80
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--- a/src/HOL/CTL/CTL.thy	Thu Apr 28 12:04:34 2005 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,310 +0,0 @@
-
-theory CTL = Main:
-
-section {* CTL formulae *}
-
-text {*
-  We formalize basic concepts of Computational Tree Logic (CTL)
-  \cite{McMillan-PhDThesis,McMillan-LectureNotes} within the
-  simply-typed set theory of HOL.
-
-  By using the common technique of shallow embedding'', a CTL
-  formula is identified with the corresponding set of states where it
-  holds.  Consequently, CTL operations such as negation, conjunction,
-  disjunction simply become complement, intersection, union of sets.
-  We only require a separate operation for implication, as point-wise
-  inclusion is usually not encountered in plain set-theory.
-*}
-
-lemmas [intro!] = Int_greatest Un_upper2 Un_upper1 Int_lower1 Int_lower2
-
-types 'a ctl = "'a set"
-constdefs
-  imp :: "'a ctl \<Rightarrow> 'a ctl \<Rightarrow> 'a ctl"    (infixr "\<rightarrow>" 75)
-  "p \<rightarrow> q \<equiv> - p \<union> q"
-
-lemma [intro!]: "p \<inter> p \<rightarrow> q \<subseteq> q" by (unfold imp_def) auto
-lemma [intro!]: "p \<subseteq> (q \<rightarrow> p)" by (unfold imp_def) rule
-
-
-text {*
-  \smallskip The CTL path operators are more interesting; they are
-  based on an arbitrary, but fixed model @{text \<M>}, which is simply
-  a transition relation over states @{typ "'a"}.
-*}
-
-consts model :: "('a \<times> 'a) set"    ("\<M>")
-
-text {*
-  The operators @{text \<EX>}, @{text \<EF>}, @{text \<EG>} are taken
-  as primitives, while @{text \<AX>}, @{text \<AF>}, @{text \<AG>} are
-  defined as derived ones.  The formula @{text "\<EX> p"} holds in a
-  state @{term s}, iff there is a successor state @{term s'} (with
-  respect to the model @{term \<M>}), such that @{term p} holds in
-  @{term s'}.  The formula @{text "\<EF> p"} holds in a state @{term
-  s}, iff there is a path in @{text \<M>}, starting from @{term s},
-  such that there exists a state @{term s'} on the path, such that
-  @{term p} holds in @{term s'}.  The formula @{text "\<EG> p"} holds
-  in a state @{term s}, iff there is a path, starting from @{term s},
-  such that for all states @{term s'} on the path, @{term p} holds in
-  @{term s'}.  It is easy to see that @{text "\<EF> p"} and @{text
-  "\<EG> p"} may be expressed using least and greatest fixed points
-  \cite{McMillan-PhDThesis}.
-*}
-
-constdefs
-  EX :: "'a ctl \<Rightarrow> 'a ctl"    ("\<EX> _"  90)    "\<EX> p \<equiv> {s. \<exists>s'. (s, s') \<in> \<M> \<and> s' \<in> p}"
-  EF :: "'a ctl \<Rightarrow> 'a ctl"    ("\<EF> _"  90)    "\<EF> p \<equiv> lfp (\<lambda>s. p \<union> \<EX> s)"
-  EG :: "'a ctl \<Rightarrow> 'a ctl"    ("\<EG> _"  90)    "\<EG> p \<equiv> gfp (\<lambda>s. p \<inter> \<EX> s)"
-
-text {*
-  @{text "\<AX>"}, @{text "\<AF>"} and @{text "\<AG>"} are now defined
-  dually in terms of @{text "\<EX>"}, @{text "\<EF>"} and @{text
-  "\<EG>"}.
-*}
-
-constdefs
-  AX :: "'a ctl \<Rightarrow> 'a ctl"    ("\<AX> _"  90)    "\<AX> p \<equiv> - \<EX> - p"
-  AF :: "'a ctl \<Rightarrow> 'a ctl"    ("\<AF> _"  90)    "\<AF> p \<equiv> - \<EG> - p"
-  AG :: "'a ctl \<Rightarrow> 'a ctl"    ("\<AG> _"  90)    "\<AG> p \<equiv> - \<EF> - p"
-
-lemmas [simp] = EX_def EG_def AX_def EF_def AF_def AG_def
-
-
-section {* Basic fixed point properties *}
-
-text {*
-  First of all, we use the de-Morgan property of fixed points
-*}
-
-lemma lfp_gfp: "lfp f = - gfp (\<lambda>s . - (f (- s)))"
-proof
-  show "lfp f \<subseteq> - gfp (\<lambda>s. - f (- s))"
-  proof
-    fix x assume l: "x \<in> lfp f"
-    show "x \<in> - gfp (\<lambda>s. - f (- s))"
-    proof
-      assume "x \<in> gfp (\<lambda>s. - f (- s))"
-      then obtain u where "x \<in> u" and "u \<subseteq> - f (- u)" by (unfold gfp_def) auto
-      then have "f (- u) \<subseteq> - u" by auto
-      then have "lfp f \<subseteq> - u" by (rule lfp_lowerbound)
-      from l and this have "x \<notin> u" by auto
-      then show False by contradiction
-    qed
-  qed
-  show "- gfp (\<lambda>s. - f (- s)) \<subseteq> lfp f"
-  proof (rule lfp_greatest)
-    fix u assume "f u \<subseteq> u"
-    then have "- u \<subseteq> - f u" by auto
-    then have "- u \<subseteq> - f (- (- u))" by simp
-    then have "- u \<subseteq> gfp (\<lambda>s. - f (- s))" by (rule gfp_upperbound)
-    then show "- gfp (\<lambda>s. - f (- s)) \<subseteq> u" by auto
-  qed
-qed
-
-lemma lfp_gfp': "- lfp f = gfp (\<lambda>s. - (f (- s)))"
-
-lemma gfp_lfp': "- gfp f = lfp (\<lambda>s. - (f (- s)))"
-
-text {*
-  in order to give dual fixed point representations of @{term "AF p"}
-  and @{term "AG p"}:
-*}
-
-lemma AF_lfp: "\<AF> p = lfp (\<lambda>s. p \<union> \<AX> s)" by (simp add: lfp_gfp)
-lemma AG_gfp: "\<AG> p = gfp (\<lambda>s. p \<inter> \<AX> s)" by (simp add: lfp_gfp)
-
-lemma EF_fp: "\<EF> p = p \<union> \<EX> \<EF> p"
-proof -
-  have "mono (\<lambda>s. p \<union> \<EX> s)" by rule (auto simp add: EX_def)
-  then show ?thesis by (simp only: EF_def) (rule lfp_unfold)
-qed
-
-lemma AF_fp: "\<AF> p = p \<union> \<AX> \<AF> p"
-proof -
-  have "mono (\<lambda>s. p \<union> \<AX> s)" by rule (auto simp add: AX_def EX_def)
-  then show ?thesis by (simp only: AF_lfp) (rule lfp_unfold)
-qed
-
-lemma EG_fp: "\<EG> p = p \<inter> \<EX> \<EG> p"
-proof -
-  have "mono (\<lambda>s. p \<inter> \<EX> s)" by rule (auto simp add: EX_def)
-  then show ?thesis by (simp only: EG_def) (rule gfp_unfold)
-qed
-
-text {*
-  From the greatest fixed point definition of @{term "\<AG> p"}, we
-  derive as a consequence of the Knaster-Tarski theorem on the one
-  hand that @{term "\<AG> p"} is a fixed point of the monotonic
-  function @{term "\<lambda>s. p \<inter> \<AX> s"}.
-*}
-
-lemma AG_fp: "\<AG> p = p \<inter> \<AX> \<AG> p"
-proof -
-  have "mono (\<lambda>s. p \<inter> \<AX> s)" by rule (auto simp add: AX_def EX_def)
-  then show ?thesis by (simp only: AG_gfp) (rule gfp_unfold)
-qed
-
-text {*
-  This fact may be split up into two inequalities (merely using
-  transitivity of @{text "\<subseteq>" }, which is an instance of the overloaded
-  @{text "\<le>"} in Isabelle/HOL).
-*}
-
-lemma AG_fp_1: "\<AG> p \<subseteq> p"
-proof -
-  note AG_fp also have "p \<inter> \<AX> \<AG> p \<subseteq> p" by auto
-  finally show ?thesis .
-qed
-
-text {**}
-
-lemma AG_fp_2: "\<AG> p \<subseteq> \<AX> \<AG> p"
-proof -
-  note AG_fp also have "p \<inter> \<AX> \<AG> p \<subseteq> \<AX> \<AG> p" by auto
-  finally show ?thesis .
-qed
-
-text {*
-  On the other hand, we have from the Knaster-Tarski fixed point
-  theorem that any other post-fixed point of @{term "\<lambda>s. p \<inter> AX s"} is
-  smaller than @{term "AG p"}.  A post-fixed point is a set of states
-  @{term q} such that @{term "q \<subseteq> p \<inter> AX q"}.  This leads to the
-  following co-induction principle for @{term "AG p"}.
-*}
-
-lemma AG_I: "q \<subseteq> p \<inter> \<AX> q \<Longrightarrow> q \<subseteq> \<AG> p"
-  by (simp only: AG_gfp) (rule gfp_upperbound)
-
-
-section {* The tree induction principle \label{sec:calc-ctl-tree-induct} *}
-
-text {*
-  With the most basic facts available, we are now able to establish a
-  few more interesting results, leading to the \emph{tree induction}
-  principle for @{text AG} (see below).  We will use some elementary
-  monotonicity and distributivity rules.
-*}
-
-lemma AX_int: "\<AX> (p \<inter> q) = \<AX> p \<inter> \<AX> q" by auto
-lemma AX_mono: "p \<subseteq> q \<Longrightarrow> \<AX> p \<subseteq> \<AX> q" by auto
-lemma AG_mono: "p \<subseteq> q \<Longrightarrow> \<AG> p \<subseteq> \<AG> q"
-  by (simp only: AG_gfp, rule gfp_mono) auto
-
-text {*
-  The formula @{term "AG p"} implies @{term "AX p"} (we use
-  substitution of @{text "\<subseteq>"} with monotonicity).
-*}
-
-lemma AG_AX: "\<AG> p \<subseteq> \<AX> p"
-proof -
-  have "\<AG> p \<subseteq> \<AX> \<AG> p" by (rule AG_fp_2)
-  also have "\<AG> p \<subseteq> p" by (rule AG_fp_1) moreover note AX_mono
-  finally show ?thesis .
-qed
-
-text {*
-  Furthermore we show idempotency of the @{text "\<AG>"} operator.
-  The proof is a good example of how accumulated facts may get
-  used to feed a single rule step.
-*}
-
-lemma AG_AG: "\<AG> \<AG> p = \<AG> p"
-proof
-  show "\<AG> \<AG> p \<subseteq> \<AG> p" by (rule AG_fp_1)
-next
-  show "\<AG> p \<subseteq> \<AG> \<AG> p"
-  proof (rule AG_I)
-    have "\<AG> p \<subseteq> \<AG> p" ..
-    moreover have "\<AG> p \<subseteq> \<AX> \<AG> p" by (rule AG_fp_2)
-    ultimately show "\<AG> p \<subseteq> \<AG> p \<inter> \<AX> \<AG> p" ..
-  qed
-qed
-
-text {*
-  \smallskip We now give an alternative characterization of the @{text
-  "\<AG>"} operator, which describes the @{text "\<AG>"} operator in
-  an operational'' way by tree induction: In a state holds @{term
-  "AG p"} iff in that state holds @{term p}, and in all reachable
-  states @{term s} follows from the fact that @{term p} holds in
-  @{term s}, that @{term p} also holds in all successor states of
-  @{term s}.  We use the co-induction principle @{thm [source] AG_I}
-  to establish this in a purely algebraic manner.
-*}
-
-theorem AG_induct: "p \<inter> \<AG> (p \<rightarrow> \<AX> p) = \<AG> p"
-proof
-  show "p \<inter> \<AG> (p \<rightarrow> \<AX> p) \<subseteq> \<AG> p"  (is "?lhs \<subseteq> _")
-  proof (rule AG_I)
-    show "?lhs \<subseteq> p \<inter> \<AX> ?lhs"
-    proof
-      show "?lhs \<subseteq> p" ..
-      show "?lhs \<subseteq> \<AX> ?lhs"
-      proof -
-	{
-	  have "\<AG> (p \<rightarrow> \<AX> p) \<subseteq> p \<rightarrow> \<AX> p" by (rule AG_fp_1)
-          also have "p \<inter> p \<rightarrow> \<AX> p \<subseteq> \<AX> p" ..
-          finally have "?lhs \<subseteq> \<AX> p" by auto
-	}
-	moreover
-	{
-	  have "p \<inter> \<AG> (p \<rightarrow> \<AX> p) \<subseteq> \<AG> (p \<rightarrow> \<AX> p)" ..
-          also have "\<dots> \<subseteq> \<AX> \<dots>" by (rule AG_fp_2)
-          finally have "?lhs \<subseteq> \<AX> \<AG> (p \<rightarrow> \<AX> p)" .
-	}
-	ultimately have "?lhs \<subseteq> \<AX> p \<inter> \<AX> \<AG> (p \<rightarrow> \<AX> p)" ..
-	also have "\<dots> = \<AX> ?lhs" by (simp only: AX_int)
-	finally show ?thesis .
-      qed
-    qed
-  qed
-next
-  show "\<AG> p \<subseteq> p \<inter> \<AG> (p \<rightarrow> \<AX> p)"
-  proof
-    show "\<AG> p \<subseteq> p" by (rule AG_fp_1)
-    show "\<AG> p \<subseteq> \<AG> (p \<rightarrow> \<AX> p)"
-    proof -
-      have "\<AG> p = \<AG> \<AG> p" by (simp only: AG_AG)
-      also have "\<AG> p \<subseteq> \<AX> p" by (rule AG_AX) moreover note AG_mono
-      also have "\<AX> p \<subseteq> (p \<rightarrow> \<AX> p)" .. moreover note AG_mono
-      finally show ?thesis .
-    qed
-  qed
-qed
-
-
-section {* An application of tree induction \label{sec:calc-ctl-commute} *}
-
-text {*
-  Further interesting properties of CTL expressions may be
-  demonstrated with the help of tree induction; here we show that
-  @{text \<AX>} and @{text \<AG>} commute.
-*}
-
-theorem AG_AX_commute: "\<AG> \<AX> p = \<AX> \<AG> p"
-proof -
-  have "\<AG> \<AX> p = \<AX> p \<inter> \<AX> \<AG> \<AX> p" by (rule AG_fp)
-  also have "\<dots> = \<AX> (p \<inter> \<AG> \<AX> p)" by (simp only: AX_int)
-  also have "p \<inter> \<AG> \<AX> p = \<AG> p"  (is "?lhs = _")
-  proof
-    have "\<AX> p \<subseteq> p \<rightarrow> \<AX> p" ..
-    also have "p \<inter> \<AG> (p \<rightarrow> \<AX> p) = \<AG> p" by (rule AG_induct)
-    also note Int_mono AG_mono
-    ultimately show "?lhs \<subseteq> \<AG> p" by fast
-  next
-    have "\<AG> p \<subseteq> p" by (rule AG_fp_1)
-    moreover
-    {
-      have "\<AG> p = \<AG> \<AG> p" by (simp only: AG_AG)
-      also have "\<AG> p \<subseteq> \<AX> p" by (rule AG_AX)
-      also note AG_mono
-      ultimately have "\<AG> p \<subseteq> \<AG> \<AX> p" .
-    }
-    ultimately show "\<AG> p \<subseteq> ?lhs" ..
-  qed
-  finally show ?thesis .
-qed
-
-end
--- a/src/HOL/CTL/ROOT.ML	Thu Apr 28 12:04:34 2005 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,3 +0,0 @@
-
-use_thy "CTL";
-
--- a/src/HOL/CTL/document/root.bib	Thu Apr 28 12:04:34 2005 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,13 +0,0 @@
-
-@Misc{McMillan-LectureNotes,
-  author =	 {Ken McMillan},
-  title =	 {Lecture notes on verification of digital and hybrid systems},
-  note =	 {{NATO} summer school, \url{http://www-cad.eecs.berkeley.edu/~kenmcmil/tutorial/toc.html}}
-}
-
-@PhdThesis{McMillan-PhDThesis,
-  author = 	 {Ken McMillan},
-  title = 	 {Symbolic Model Checking: an approach to the state explosion problem},
-  school = 	 {Carnegie Mellon University},
-  year = 	 1992
-}
--- a/src/HOL/CTL/document/root.tex	Thu Apr 28 12:04:34 2005 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,32 +0,0 @@
-
-\documentclass[11pt,a4paper]{article}
-\usepackage{isabelle,isabellesym,pdfsetup}
-
-\urlstyle{rm}
-\isabellestyle{it}
-
-\newcommand{\isasymEX}{\isamath{\mathrm{EX}}}
-\newcommand{\isasymEF}{\isamath{\mathrm{EF}}}
-\newcommand{\isasymEG}{\isamath{\mathrm{EG}}}
-\newcommand{\isasymAX}{\isamath{\mathrm{AX}}}
-\newcommand{\isasymAF}{\isamath{\mathrm{AF}}}
-\newcommand{\isasymAG}{\isamath{\mathrm{AG}}}
-
-
-\begin{document}
-
-\title{Some properties of CTL}
-\author{Gertrud Bauer}
-\maketitle
-
-\tableofcontents
-\bigskip
-
-\parindent 0pt\parskip 0.5ex
-
-\input{session}
-
-\bibliographystyle{abbrv}
-\bibliography{root}
-
-\end{document}
--- a/src/HOL/IsaMakefile	Thu Apr 28 12:04:34 2005 +0200
+++ b/src/HOL/IsaMakefile	Thu Apr 28 17:08:08 2005 +0200
@@ -16,7 +16,6 @@
HOL-Bali \
HOL-Complex-ex \
HOL-Complex-Import \
-  HOL-CTL \
HOL-Extraction \
HOL-Complex-HahnBanach \
HOL-Hoare \
@@ -534,15 +533,6 @@
@$(ISATOOL) usedir -g true$(OUT)/HOL Bali

-## HOL-CTL
-
-HOL-CTL: HOL $(LOG)/HOL-CTL.gz - -$(LOG)/HOL-CTL.gz: $(OUT)/HOL \ - CTL/CTL.thy CTL/ROOT.ML CTL/document/root.tex CTL/document/root.bib - @$(ISATOOL) usedir $(OUT)/HOL CTL - - ## HOL-Extraction HOL-Extraction: HOL$(LOG)/HOL-Extraction.gz
@@ -701,7 +691,7 @@
$(LOG)/HOL-Lex.gz$(LOG)/HOL-Algebra.gz \
$(LOG)/HOL-Auth.gz$(LOG)/HOL-UNITY.gz \
$(LOG)/HOL-Modelcheck.gz$(LOG)/HOL-Lambda.gz \
-                $(LOG)/HOL-Bali.gz$(LOG)/HOL-CTL.gz \
+                $(LOG)/HOL-Bali.gz \$(LOG)/HOL-MicroJava.gz $(LOG)/HOL-NanoJava.gz \$(LOG)/HOL-IOA.gz $(LOG)/HOL-AxClasses \$(LOG)/HOL-Lattice \
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/CTL.thy	Thu Apr 28 17:08:08 2005 +0200
@@ -0,0 +1,317 @@
+
+(*  Title:      HOL/ex/CTL.thy
+    ID:         $Id$
+    Author:     Gertrud Bauer
+*)
+
+
+theory CTL = Main:
+
+
+
+text {*
+  We formalize basic concepts of Computational Tree Logic (CTL)
+  \cite{McMillan-PhDThesis,McMillan-LectureNotes} within the
+  simply-typed set theory of HOL.
+
+  By using the common technique of shallow embedding'', a CTL
+  formula is identified with the corresponding set of states where it
+  holds.  Consequently, CTL operations such as negation, conjunction,
+  disjunction simply become complement, intersection, union of sets.
+  We only require a separate operation for implication, as point-wise
+  inclusion is usually not encountered in plain set-theory.
+*}
+
+lemmas [intro!] = Int_greatest Un_upper2 Un_upper1 Int_lower1 Int_lower2
+
+types 'a ctl = "'a set"
+constdefs
+  imp :: "'a ctl \<Rightarrow> 'a ctl \<Rightarrow> 'a ctl"    (infixr "\<rightarrow>" 75)
+  "p \<rightarrow> q \<equiv> - p \<union> q"
+
+lemma [intro!]: "p \<inter> p \<rightarrow> q \<subseteq> q" by (unfold imp_def) auto
+lemma [intro!]: "p \<subseteq> (q \<rightarrow> p)" by (unfold imp_def) rule
+
+
+text {*
+  \smallskip The CTL path operators are more interesting; they are
+  based on an arbitrary, but fixed model @{text \<M>}, which is simply
+  a transition relation over states @{typ "'a"}.
+*}
+
+consts model :: "('a \<times> 'a) set"    ("\<M>")
+
+text {*
+  The operators @{text \<EX>}, @{text \<EF>}, @{text \<EG>} are taken
+  as primitives, while @{text \<AX>}, @{text \<AF>}, @{text \<AG>} are
+  defined as derived ones.  The formula @{text "\<EX> p"} holds in a
+  state @{term s}, iff there is a successor state @{term s'} (with
+  respect to the model @{term \<M>}), such that @{term p} holds in
+  @{term s'}.  The formula @{text "\<EF> p"} holds in a state @{term
+  s}, iff there is a path in @{text \<M>}, starting from @{term s},
+  such that there exists a state @{term s'} on the path, such that
+  @{term p} holds in @{term s'}.  The formula @{text "\<EG> p"} holds
+  in a state @{term s}, iff there is a path, starting from @{term s},
+  such that for all states @{term s'} on the path, @{term p} holds in
+  @{term s'}.  It is easy to see that @{text "\<EF> p"} and @{text
+  "\<EG> p"} may be expressed using least and greatest fixed points
+  \cite{McMillan-PhDThesis}.
+*}
+
+constdefs
+  EX :: "'a ctl \<Rightarrow> 'a ctl"    ("\<EX> _"  90)    "\<EX> p \<equiv> {s. \<exists>s'. (s, s') \<in> \<M> \<and> s' \<in> p}"
+  EF :: "'a ctl \<Rightarrow> 'a ctl"    ("\<EF> _"  90)    "\<EF> p \<equiv> lfp (\<lambda>s. p \<union> \<EX> s)"
+  EG :: "'a ctl \<Rightarrow> 'a ctl"    ("\<EG> _"  90)    "\<EG> p \<equiv> gfp (\<lambda>s. p \<inter> \<EX> s)"
+
+text {*
+  @{text "\<AX>"}, @{text "\<AF>"} and @{text "\<AG>"} are now defined
+  dually in terms of @{text "\<EX>"}, @{text "\<EF>"} and @{text
+  "\<EG>"}.
+*}
+
+constdefs
+  AX :: "'a ctl \<Rightarrow> 'a ctl"    ("\<AX> _"  90)    "\<AX> p \<equiv> - \<EX> - p"
+  AF :: "'a ctl \<Rightarrow> 'a ctl"    ("\<AF> _"  90)    "\<AF> p \<equiv> - \<EG> - p"
+  AG :: "'a ctl \<Rightarrow> 'a ctl"    ("\<AG> _"  90)    "\<AG> p \<equiv> - \<EF> - p"
+
+lemmas [simp] = EX_def EG_def AX_def EF_def AF_def AG_def
+
+
+section {* Basic fixed point properties *}
+
+text {*
+  First of all, we use the de-Morgan property of fixed points
+*}
+
+lemma lfp_gfp: "lfp f = - gfp (\<lambda>s . - (f (- s)))"
+proof
+  show "lfp f \<subseteq> - gfp (\<lambda>s. - f (- s))"
+  proof
+    fix x assume l: "x \<in> lfp f"
+    show "x \<in> - gfp (\<lambda>s. - f (- s))"
+    proof
+      assume "x \<in> gfp (\<lambda>s. - f (- s))"
+      then obtain u where "x \<in> u" and "u \<subseteq> - f (- u)" by (unfold gfp_def) auto
+      then have "f (- u) \<subseteq> - u" by auto
+      then have "lfp f \<subseteq> - u" by (rule lfp_lowerbound)
+      from l and this have "x \<notin> u" by auto
+      then show False by contradiction
+    qed
+  qed
+  show "- gfp (\<lambda>s. - f (- s)) \<subseteq> lfp f"
+  proof (rule lfp_greatest)
+    fix u assume "f u \<subseteq> u"
+    then have "- u \<subseteq> - f u" by auto
+    then have "- u \<subseteq> - f (- (- u))" by simp
+    then have "- u \<subseteq> gfp (\<lambda>s. - f (- s))" by (rule gfp_upperbound)
+    then show "- gfp (\<lambda>s. - f (- s)) \<subseteq> u" by auto
+  qed
+qed
+
+lemma lfp_gfp': "- lfp f = gfp (\<lambda>s. - (f (- s)))"
+
+lemma gfp_lfp': "- gfp f = lfp (\<lambda>s. - (f (- s)))"
+
+text {*
+  in order to give dual fixed point representations of @{term "AF p"}
+  and @{term "AG p"}:
+*}
+
+lemma AF_lfp: "\<AF> p = lfp (\<lambda>s. p \<union> \<AX> s)" by (simp add: lfp_gfp)
+lemma AG_gfp: "\<AG> p = gfp (\<lambda>s. p \<inter> \<AX> s)" by (simp add: lfp_gfp)
+
+lemma EF_fp: "\<EF> p = p \<union> \<EX> \<EF> p"
+proof -
+  have "mono (\<lambda>s. p \<union> \<EX> s)" by rule (auto simp add: EX_def)
+  then show ?thesis by (simp only: EF_def) (rule lfp_unfold)
+qed
+
+lemma AF_fp: "\<AF> p = p \<union> \<AX> \<AF> p"
+proof -
+  have "mono (\<lambda>s. p \<union> \<AX> s)" by rule (auto simp add: AX_def EX_def)
+  then show ?thesis by (simp only: AF_lfp) (rule lfp_unfold)
+qed
+
+lemma EG_fp: "\<EG> p = p \<inter> \<EX> \<EG> p"
+proof -
+  have "mono (\<lambda>s. p \<inter> \<EX> s)" by rule (auto simp add: EX_def)
+  then show ?thesis by (simp only: EG_def) (rule gfp_unfold)
+qed
+
+text {*
+  From the greatest fixed point definition of @{term "\<AG> p"}, we
+  derive as a consequence of the Knaster-Tarski theorem on the one
+  hand that @{term "\<AG> p"} is a fixed point of the monotonic
+  function @{term "\<lambda>s. p \<inter> \<AX> s"}.
+*}
+
+lemma AG_fp: "\<AG> p = p \<inter> \<AX> \<AG> p"
+proof -
+  have "mono (\<lambda>s. p \<inter> \<AX> s)" by rule (auto simp add: AX_def EX_def)
+  then show ?thesis by (simp only: AG_gfp) (rule gfp_unfold)
+qed
+
+text {*
+  This fact may be split up into two inequalities (merely using
+  transitivity of @{text "\<subseteq>" }, which is an instance of the overloaded
+  @{text "\<le>"} in Isabelle/HOL).
+*}
+
+lemma AG_fp_1: "\<AG> p \<subseteq> p"
+proof -
+  note AG_fp also have "p \<inter> \<AX> \<AG> p \<subseteq> p" by auto
+  finally show ?thesis .
+qed
+
+text {**}
+
+lemma AG_fp_2: "\<AG> p \<subseteq> \<AX> \<AG> p"
+proof -
+  note AG_fp also have "p \<inter> \<AX> \<AG> p \<subseteq> \<AX> \<AG> p" by auto
+  finally show ?thesis .
+qed
+
+text {*
+  On the other hand, we have from the Knaster-Tarski fixed point
+  theorem that any other post-fixed point of @{term "\<lambda>s. p \<inter> AX s"} is
+  smaller than @{term "AG p"}.  A post-fixed point is a set of states
+  @{term q} such that @{term "q \<subseteq> p \<inter> AX q"}.  This leads to the
+  following co-induction principle for @{term "AG p"}.
+*}
+
+lemma AG_I: "q \<subseteq> p \<inter> \<AX> q \<Longrightarrow> q \<subseteq> \<AG> p"
+  by (simp only: AG_gfp) (rule gfp_upperbound)
+
+
+section {* The tree induction principle \label{sec:calc-ctl-tree-induct} *}
+
+text {*
+  With the most basic facts available, we are now able to establish a
+  few more interesting results, leading to the \emph{tree induction}
+  principle for @{text AG} (see below).  We will use some elementary
+  monotonicity and distributivity rules.
+*}
+
+lemma AX_int: "\<AX> (p \<inter> q) = \<AX> p \<inter> \<AX> q" by auto
+lemma AX_mono: "p \<subseteq> q \<Longrightarrow> \<AX> p \<subseteq> \<AX> q" by auto
+lemma AG_mono: "p \<subseteq> q \<Longrightarrow> \<AG> p \<subseteq> \<AG> q"
+  by (simp only: AG_gfp, rule gfp_mono) auto
+
+text {*
+  The formula @{term "AG p"} implies @{term "AX p"} (we use
+  substitution of @{text "\<subseteq>"} with monotonicity).
+*}
+
+lemma AG_AX: "\<AG> p \<subseteq> \<AX> p"
+proof -
+  have "\<AG> p \<subseteq> \<AX> \<AG> p" by (rule AG_fp_2)
+  also have "\<AG> p \<subseteq> p" by (rule AG_fp_1) moreover note AX_mono
+  finally show ?thesis .
+qed
+
+text {*
+  Furthermore we show idempotency of the @{text "\<AG>"} operator.
+  The proof is a good example of how accumulated facts may get
+  used to feed a single rule step.
+*}
+
+lemma AG_AG: "\<AG> \<AG> p = \<AG> p"
+proof
+  show "\<AG> \<AG> p \<subseteq> \<AG> p" by (rule AG_fp_1)
+next
+  show "\<AG> p \<subseteq> \<AG> \<AG> p"
+  proof (rule AG_I)
+    have "\<AG> p \<subseteq> \<AG> p" ..
+    moreover have "\<AG> p \<subseteq> \<AX> \<AG> p" by (rule AG_fp_2)
+    ultimately show "\<AG> p \<subseteq> \<AG> p \<inter> \<AX> \<AG> p" ..
+  qed
+qed
+
+text {*
+  \smallskip We now give an alternative characterization of the @{text
+  "\<AG>"} operator, which describes the @{text "\<AG>"} operator in
+  an operational'' way by tree induction: In a state holds @{term
+  "AG p"} iff in that state holds @{term p}, and in all reachable
+  states @{term s} follows from the fact that @{term p} holds in
+  @{term s}, that @{term p} also holds in all successor states of
+  @{term s}.  We use the co-induction principle @{thm [source] AG_I}
+  to establish this in a purely algebraic manner.
+*}
+
+theorem AG_induct: "p \<inter> \<AG> (p \<rightarrow> \<AX> p) = \<AG> p"
+proof
+  show "p \<inter> \<AG> (p \<rightarrow> \<AX> p) \<subseteq> \<AG> p"  (is "?lhs \<subseteq> _")
+  proof (rule AG_I)
+    show "?lhs \<subseteq> p \<inter> \<AX> ?lhs"
+    proof
+      show "?lhs \<subseteq> p" ..
+      show "?lhs \<subseteq> \<AX> ?lhs"
+      proof -
+	{
+	  have "\<AG> (p \<rightarrow> \<AX> p) \<subseteq> p \<rightarrow> \<AX> p" by (rule AG_fp_1)
+          also have "p \<inter> p \<rightarrow> \<AX> p \<subseteq> \<AX> p" ..
+          finally have "?lhs \<subseteq> \<AX> p" by auto
+	}
+	moreover
+	{
+	  have "p \<inter> \<AG> (p \<rightarrow> \<AX> p) \<subseteq> \<AG> (p \<rightarrow> \<AX> p)" ..
+          also have "\<dots> \<subseteq> \<AX> \<dots>" by (rule AG_fp_2)
+          finally have "?lhs \<subseteq> \<AX> \<AG> (p \<rightarrow> \<AX> p)" .
+	}
+	ultimately have "?lhs \<subseteq> \<AX> p \<inter> \<AX> \<AG> (p \<rightarrow> \<AX> p)" ..
+	also have "\<dots> = \<AX> ?lhs" by (simp only: AX_int)
+	finally show ?thesis .
+      qed
+    qed
+  qed
+next
+  show "\<AG> p \<subseteq> p \<inter> \<AG> (p \<rightarrow> \<AX> p)"
+  proof
+    show "\<AG> p \<subseteq> p" by (rule AG_fp_1)
+    show "\<AG> p \<subseteq> \<AG> (p \<rightarrow> \<AX> p)"
+    proof -
+      have "\<AG> p = \<AG> \<AG> p" by (simp only: AG_AG)
+      also have "\<AG> p \<subseteq> \<AX> p" by (rule AG_AX) moreover note AG_mono
+      also have "\<AX> p \<subseteq> (p \<rightarrow> \<AX> p)" .. moreover note AG_mono
+      finally show ?thesis .
+    qed
+  qed
+qed
+
+
+section {* An application of tree induction \label{sec:calc-ctl-commute} *}
+
+text {*
+  Further interesting properties of CTL expressions may be
+  demonstrated with the help of tree induction; here we show that
+  @{text \<AX>} and @{text \<AG>} commute.
+*}
+
+theorem AG_AX_commute: "\<AG> \<AX> p = \<AX> \<AG> p"
+proof -
+  have "\<AG> \<AX> p = \<AX> p \<inter> \<AX> \<AG> \<AX> p" by (rule AG_fp)
+  also have "\<dots> = \<AX> (p \<inter> \<AG> \<AX> p)" by (simp only: AX_int)
+  also have "p \<inter> \<AG> \<AX> p = \<AG> p"  (is "?lhs = _")
+  proof
+    have "\<AX> p \<subseteq> p \<rightarrow> \<AX> p" ..
+    also have "p \<inter> \<AG> (p \<rightarrow> \<AX> p) = \<AG> p" by (rule AG_induct)
+    also note Int_mono AG_mono
+    ultimately show "?lhs \<subseteq> \<AG> p" by fast
+  next
+    have "\<AG> p \<subseteq> p" by (rule AG_fp_1)
+    moreover
+    {
+      have "\<AG> p = \<AG> \<AG> p" by (simp only: AG_AG)
+      also have "\<AG> p \<subseteq> \<AX> p" by (rule AG_AX)
+      also note AG_mono
+      ultimately have "\<AG> p \<subseteq> \<AG> \<AX> p" .
+    }
+    ultimately show "\<AG> p \<subseteq> ?lhs" ..
+  qed
+  finally show ?thesis .
+qed
+
+end
--- a/src/HOL/ex/ROOT.ML	Thu Apr 28 12:04:34 2005 +0200
+++ b/src/HOL/ex/ROOT.ML	Thu Apr 28 17:08:08 2005 +0200
@@ -22,6 +22,7 @@
time_use_thy "NatSum";
time_use_thy "Intuitionistic";
time_use_thy "Classical";
+time_use_thy "CTL";
time_use_thy "mesontest2";
time_use_thy "PresburgerEx";
time_use_thy "BT";
@@ -30,6 +31,7 @@
time_use_thy "MergeSort";
time_use_thy "Puzzle";

+
time_use_thy "Lagrange";

time_use_thy "set";
--- a/src/HOL/ex/document/root.bib	Thu Apr 28 12:04:34 2005 +0200
+++ b/src/HOL/ex/document/root.bib	Thu Apr 28 17:08:08 2005 +0200
@@ -1,3 +1,5 @@
+
+

@TechReport{Gordon:1985:HOL,
author =       {M. J. C. Gordon},
@@ -105,3 +107,16 @@
publisher	= {Springer},
year		= 2002,
note		= {LNCS 2283}}
+
+@Misc{McMillan-LectureNotes,
+  author =	 {Ken McMillan},
+  title =	 {Lecture notes on verification of digital and hybrid systems},
+  note =	 {{NATO} summer school, \url{http://www-cad.eecs.berkeley.edu/~kenmcmil/tutorial/toc.html}}
+}
+
+@PhdThesis{McMillan-PhDThesis,
+  author = 	 {Ken McMillan},
+  title = 	 {Symbolic Model Checking: an approach to the state explosion problem},
+  school = 	 {Carnegie Mellon University},
+  year = 	 1992
+}
\ No newline at end of file
--- a/src/HOL/ex/document/root.tex	Thu Apr 28 12:04:34 2005 +0200
+++ b/src/HOL/ex/document/root.tex	Thu Apr 28 17:08:08 2005 +0200
@@ -11,6 +11,15 @@
\urlstyle{rm}
\isabellestyle{it}

+\newcommand{\isasymEX}{\isamath{\mathrm{EX}}}
+\newcommand{\isasymEF}{\isamath{\mathrm{EF}}}
+\newcommand{\isasymEG}{\isamath{\mathrm{EG}}}
+\newcommand{\isasymAX}{\isamath{\mathrm{AX}}}
+\newcommand{\isasymAF}{\isamath{\mathrm{AF}}}
+\newcommand{\isasymAG}{\isamath{\mathrm{AG}}}
+
+
+
\begin{document}

\title{Miscellaneous HOL Examples}