(* Title: HOL/ex/CTL.thy

Author: Gertrud Bauer

*)

header {* CTL formulae *}

theory CTL

imports Main

begin

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

type_synonym 'a ctl = "'a set"

definition

imp :: "'a ctl => 'a ctl => 'a ctl" (infixr "->" 75) where

"p -> q = - p ∪ q"

lemma [intro!]: "p ∩ p -> q ⊆ q" unfolding imp_def by auto

lemma [intro!]: "p ⊆ (q -> p)" unfolding imp_def by 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"}.

*}

axiomatization \<M> :: "('a × 'a) set"

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}.

*}

definition

EX ("\<EX> _" [80] 90) where "\<EX> p = {s. ∃s'. (s, s') ∈ \<M> ∧ s' ∈ p}"

definition

EF ("\<EF> _" [80] 90) where "\<EF> p = lfp (λs. p ∪ \<EX> s)"

definition

EG ("\<EG> _" [80] 90) where "\<EG> p = gfp (λs. p ∩ \<EX> s)"

text {*

@{text "\<AX>"}, @{text "\<AF>"} and @{text "\<AG>"} are now defined

dually in terms of @{text "\<EX>"}, @{text "\<EF>"} and @{text

"\<EG>"}.

*}

definition

AX ("\<AX> _" [80] 90) where "\<AX> p = - \<EX> - p"

definition

AF ("\<AF> _" [80] 90) where "\<AF> p = - \<EG> - p"

definition

AG ("\<AG> _" [80] 90) where "\<AG> p = - \<EF> - p"

lemmas [simp] = EX_def EG_def AX_def EF_def AF_def AG_def

subsection {* Basic fixed point properties *}

text {*

First of all, we use the de-Morgan property of fixed points

*}

lemma lfp_gfp: "lfp f = - gfp (λs::'a set. - (f (- s)))"

proof

show "lfp f ⊆ - gfp (λs. - f (- s))"

proof

fix x assume l: "x ∈ lfp f"

show "x ∈ - gfp (λs. - f (- s))"

proof

assume "x ∈ gfp (λs. - f (- s))"

then obtain u where "x ∈ u" and "u ⊆ - f (- u)"

by (auto simp add: gfp_def)

then have "f (- u) ⊆ - u" by auto

then have "lfp f ⊆ - u" by (rule lfp_lowerbound)

from l and this have "x ∉ u" by auto

with `x ∈ u` show False by contradiction

qed

qed

show "- gfp (λs. - f (- s)) ⊆ lfp f"

proof (rule lfp_greatest)

fix u assume "f u ⊆ u"

then have "- u ⊆ - f u" by auto

then have "- u ⊆ - f (- (- u))" by simp

then have "- u ⊆ gfp (λs. - f (- s))" by (rule gfp_upperbound)

then show "- gfp (λs. - f (- s)) ⊆ u" by auto

qed

qed

lemma lfp_gfp': "- lfp f = gfp (λs::'a set. - (f (- s)))"

by (simp add: lfp_gfp)

lemma gfp_lfp': "- gfp f = lfp (λs::'a set. - (f (- s)))"

by (simp add: lfp_gfp)

text {*

in order to give dual fixed point representations of @{term "AF p"}

and @{term "AG p"}:

*}

lemma AF_lfp: "\<AF> p = lfp (λs. p ∪ \<AX> s)" by (simp add: lfp_gfp)

lemma AG_gfp: "\<AG> p = gfp (λs. p ∩ \<AX> s)" by (simp add: lfp_gfp)

lemma EF_fp: "\<EF> p = p ∪ \<EX> \<EF> p"

proof -

have "mono (λs. p ∪ \<EX> s)" by rule auto

then show ?thesis by (simp only: EF_def) (rule lfp_unfold)

qed

lemma AF_fp: "\<AF> p = p ∪ \<AX> \<AF> p"

proof -

have "mono (λs. p ∪ \<AX> s)" by rule auto

then show ?thesis by (simp only: AF_lfp) (rule lfp_unfold)

qed

lemma EG_fp: "\<EG> p = p ∩ \<EX> \<EG> p"

proof -

have "mono (λs. p ∩ \<EX> s)" by rule auto

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 "λs. p ∩ \<AX> s"}.

*}

lemma AG_fp: "\<AG> p = p ∩ \<AX> \<AG> p"

proof -

have "mono (λs. p ∩ \<AX> s)" by rule auto

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 "⊆" }, which is an instance of the overloaded

@{text "≤"} in Isabelle/HOL).

*}

lemma AG_fp_1: "\<AG> p ⊆ p"

proof -

note AG_fp also have "p ∩ \<AX> \<AG> p ⊆ p" by auto

finally show ?thesis .

qed

lemma AG_fp_2: "\<AG> p ⊆ \<AX> \<AG> p"

proof -

note AG_fp also have "p ∩ \<AX> \<AG> p ⊆ \<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 "λs. p ∩ AX s"} is

smaller than @{term "AG p"}. A post-fixed point is a set of states

@{term q} such that @{term "q ⊆ p ∩ AX q"}. This leads to the

following co-induction principle for @{term "AG p"}.

*}

lemma AG_I: "q ⊆ p ∩ \<AX> q ==> q ⊆ \<AG> p"

by (simp only: AG_gfp) (rule gfp_upperbound)

subsection {* 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 ∩ q) = \<AX> p ∩ \<AX> q" by auto

lemma AX_mono: "p ⊆ q ==> \<AX> p ⊆ \<AX> q" by auto

lemma AG_mono: "p ⊆ q ==> \<AG> p ⊆ \<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 "⊆"} with monotonicity).

*}

lemma AG_AX: "\<AG> p ⊆ \<AX> p"

proof -

have "\<AG> p ⊆ \<AX> \<AG> p" by (rule AG_fp_2)

also have "\<AG> p ⊆ 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 ⊆ \<AG> p" by (rule AG_fp_1)

next

show "\<AG> p ⊆ \<AG> \<AG> p"

proof (rule AG_I)

have "\<AG> p ⊆ \<AG> p" ..

moreover have "\<AG> p ⊆ \<AX> \<AG> p" by (rule AG_fp_2)

ultimately show "\<AG> p ⊆ \<AG> p ∩ \<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 ∩ \<AG> (p -> \<AX> p) = \<AG> p"

proof

show "p ∩ \<AG> (p -> \<AX> p) ⊆ \<AG> p" (is "?lhs ⊆ _")

proof (rule AG_I)

show "?lhs ⊆ p ∩ \<AX> ?lhs"

proof

show "?lhs ⊆ p" ..

show "?lhs ⊆ \<AX> ?lhs"

proof -

{

have "\<AG> (p -> \<AX> p) ⊆ p -> \<AX> p" by (rule AG_fp_1)

also have "p ∩ p -> \<AX> p ⊆ \<AX> p" ..

finally have "?lhs ⊆ \<AX> p" by auto

}

moreover

{

have "p ∩ \<AG> (p -> \<AX> p) ⊆ \<AG> (p -> \<AX> p)" ..

also have "… ⊆ \<AX> …" by (rule AG_fp_2)

finally have "?lhs ⊆ \<AX> \<AG> (p -> \<AX> p)" .

}

ultimately have "?lhs ⊆ \<AX> p ∩ \<AX> \<AG> (p -> \<AX> p)" ..

also have "… = \<AX> ?lhs" by (simp only: AX_int)

finally show ?thesis .

qed

qed

qed

next

show "\<AG> p ⊆ p ∩ \<AG> (p -> \<AX> p)"

proof

show "\<AG> p ⊆ p" by (rule AG_fp_1)

show "\<AG> p ⊆ \<AG> (p -> \<AX> p)"

proof -

have "\<AG> p = \<AG> \<AG> p" by (simp only: AG_AG)

also have "\<AG> p ⊆ \<AX> p" by (rule AG_AX) moreover note AG_mono

also have "\<AX> p ⊆ (p -> \<AX> p)" .. moreover note AG_mono

finally show ?thesis .

qed

qed

qed

subsection {* 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 ∩ \<AX> \<AG> \<AX> p" by (rule AG_fp)

also have "… = \<AX> (p ∩ \<AG> \<AX> p)" by (simp only: AX_int)

also have "p ∩ \<AG> \<AX> p = \<AG> p" (is "?lhs = _")

proof

have "\<AX> p ⊆ p -> \<AX> p" ..

also have "p ∩ \<AG> (p -> \<AX> p) = \<AG> p" by (rule AG_induct)

also note Int_mono AG_mono

ultimately show "?lhs ⊆ \<AG> p" by fast

next

have "\<AG> p ⊆ p" by (rule AG_fp_1)

moreover

{

have "\<AG> p = \<AG> \<AG> p" by (simp only: AG_AG)

also have "\<AG> p ⊆ \<AX> p" by (rule AG_AX)

also note AG_mono

ultimately have "\<AG> p ⊆ \<AG> \<AX> p" .

}

ultimately show "\<AG> p ⊆ ?lhs" ..

qed

finally show ?thesis .

qed

end