--- a/thys/Journal/Paper.thy Wed Jul 19 14:55:46 2017 +0100
+++ b/thys/Journal/Paper.thy Fri Aug 11 20:29:01 2017 +0100
@@ -15,6 +15,11 @@
declare [[show_question_marks = false]]
+syntax (latex output)
+ "_Collect" :: "pttrn => bool => 'a set" ("(1{_ \<^raw:\mbox{\boldmath$\mid$}> _})")
+ "_CollectIn" :: "pttrn => 'a set => bool => 'a set" ("(1{_ \<in> _ |e _})")
+
+
abbreviation
"der_syn r c \<equiv> der c r"
@@ -26,12 +31,14 @@
"nprec v1 v2 \<equiv> \<not>(v1 :\<sqsubset>val v2)"
+
+
notation (latex output)
If ("(\<^raw:\textrm{>if\<^raw:}> (_)/ \<^raw:\textrm{>then\<^raw:}> (_)/ \<^raw:\textrm{>else\<^raw:}> (_))" 10) and
Cons ("_\<^raw:\mbox{$\,$}>::\<^raw:\mbox{$\,$}>_" [75,73] 73) and
ZERO ("\<^bold>0" 78) and
- ONE ("\<^bold>1" 78) and
+ ONE ("\<^bold>1" 1000) and
CHAR ("_" [1000] 80) and
ALT ("_ + _" [77,77] 78) and
SEQ ("_ \<cdot> _" [77,77] 78) and
@@ -53,8 +60,10 @@
mkeps ("mkeps _" [79] 76) and
length ("len _" [73] 73) and
intlen ("len _" [73] 73) and
+ set ("_" [73] 73) and
- Prf ("_ : _" [75,75] 75) and
+ Prf ("_ : _" [75,75] 75) and
+ CPrf ("_ \<^raw:\mbox{\textbf{\textlengthmark}}> _" [75,75] 75) and
Posix ("'(_, _') \<rightarrow> _" [63,75,75] 75) and
lexer ("lexer _ _" [78,78] 77) and
@@ -84,6 +93,13 @@
definition
"match r s \<equiv> nullable (ders s r)"
+
+lemma CV_STAR_ONE_empty:
+ shows "CV (STAR ONE) [] = {Stars []}"
+by(auto simp add: CV_def elim: CPrf.cases intro: CPrf.intros)
+
+
+
(*
comments not implemented
@@ -94,6 +110,8 @@
(*>*)
+
+
section {* Introduction *}
@@ -153,7 +171,7 @@
not match an empty string unless this is the only match for the repetition.\smallskip
\item[$\bullet$] \emph{Empty String Rule:} An empty string shall be considered to
-be longer than no match at all.
+be longer than no match at all.\marginpar{Explain its purpose}
\end{itemize}
\noindent Consider for example a regular expression @{text "r\<^bsub>key\<^esub>"} for recognising keywords
@@ -165,7 +183,9 @@
by the Longest Match Rule a single identifier token, not a keyword
followed by an identifier. For @{text "if"} we obtain by the Priority
Rule a keyword token, not an identifier token---even if @{text "r\<^bsub>id\<^esub>"}
-matches also.
+matches also. By the Star Rule we know @{text "(r\<^bsub>key\<^esub> + r\<^bsub>id\<^esub>)\<^sup>\<star>"} matches @{text "iffoo"}, respectively @{text "if"}, in exactly one
+`iteration' of the star.
+
One limitation of Brzozowski's matcher is that it only generates a
YES/NO answer for whether a string is being matched by a regular
@@ -185,7 +205,8 @@
algorithm. This proof idea is inspired by work of Frisch and Cardelli
\cite{Frisch2004} on a GREEDY regular expression matching
algorithm. However, we were not able to establish transitivity and
-totality for the ``order relation'' by Sulzmann and Lu. ??In Section
+totality for the ``order relation'' by Sulzmann and Lu. \marginpar{We probably drop this section}
+??In Section
\ref{argu} we identify some inherent problems with their approach (of
which some of the proofs are not published in \cite{Sulzmann2014});
perhaps more importantly, we give a simple inductive (and
@@ -230,6 +251,8 @@
informal proof contains gaps, and possible fixes are not fully worked out.}
Our specification of a POSIX value consists of a simple inductive definition
that given a string and a regular expression uniquely determines this value.
+We also show that our definition is equivalent to an ordering
+of values based on positions by Okui and Suzuki \cite{OkuiSuzuki2013}.
Derivatives as calculated by Brzozowski's method are usually more complex
regular expressions than the initial one; various optimisations are
possible. We prove the correctness when simplifications of @{term "ALT ZERO
@@ -267,13 +290,10 @@
recursive function @{term L} with the six clauses:
\begin{center}
- \begin{tabular}{l@ {\hspace{3mm}}rcl}
+ \begin{tabular}{l@ {\hspace{4mm}}rcl}
(1) & @{thm (lhs) L.simps(1)} & $\dn$ & @{thm (rhs) L.simps(1)}\\
(2) & @{thm (lhs) L.simps(2)} & $\dn$ & @{thm (rhs) L.simps(2)}\\
(3) & @{thm (lhs) L.simps(3)} & $\dn$ & @{thm (rhs) L.simps(3)}\\
- \end{tabular}
- \hspace{14mm}
- \begin{tabular}{l@ {\hspace{3mm}}rcl}
(4) & @{thm (lhs) L.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) L.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
(5) & @{thm (lhs) L.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) L.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\
(6) & @{thm (lhs) L.simps(6)} & $\dn$ & @{thm (rhs) L.simps(6)}\\
@@ -325,14 +345,12 @@
@{thm (lhs) nullable.simps(3)} & $\dn$ & @{thm (rhs) nullable.simps(3)}\\
@{thm (lhs) nullable.simps(4)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) nullable.simps(4)[of "r\<^sub>1" "r\<^sub>2"]}\\
@{thm (lhs) nullable.simps(5)[of "r\<^sub>1" "r\<^sub>2"]} & $\dn$ & @{thm (rhs) nullable.simps(5)[of "r\<^sub>1" "r\<^sub>2"]}\\
- @{thm (lhs) nullable.simps(6)} & $\dn$ & @{thm (rhs) nullable.simps(6)}\medskip\\
-
- %\end{tabular}
- %\end{center}
+ @{thm (lhs) nullable.simps(6)} & $\dn$ & @{thm (rhs) nullable.simps(6)}%\medskip\\
+ \end{tabular}
+ \end{center}
- %\begin{center}
- %\begin{tabular}{lcl}
-
+ \begin{center}
+ \begin{tabular}{lcl}
@{thm (lhs) der.simps(1)} & $\dn$ & @{thm (rhs) der.simps(1)}\\
@{thm (lhs) der.simps(2)} & $\dn$ & @{thm (rhs) der.simps(2)}\\
@{thm (lhs) der.simps(3)} & $\dn$ & @{thm (rhs) der.simps(3)}\\
@@ -384,7 +402,7 @@
text {*
- The clever idea by Sulzmann and Lu \cite{Sulzmann2014} is to define
+ The clever idea by Sulzmann and Lu \cite{Sulzmann2014} is to use
values for encoding \emph{how} a regular expression matches a string
and then define a function on values that mirrors (but inverts) the
construction of the derivative on regular expressions. \emph{Values}
@@ -422,7 +440,7 @@
\end{center}
\noindent Sulzmann and Lu also define inductively an inhabitation relation
- that associates values to regular expressions:
+ that associates values to regular expressions
\begin{center}
\begin{tabular}{c}
@@ -436,7 +454,10 @@
\end{tabular}
\end{center}
- \noindent Note that no values are associated with the regular expression
+ \noindent
+ where in the clause for @{const "Stars"} we use the notation @{term "v \<in> set vs"}
+ for indicating that @{text v} is a member in the list @{text vs}.
+ Note that no values are associated with the regular expression
@{term ZERO}, and that the only value associated with the regular
expression @{term ONE} is @{term Void}, pronounced (if one must) as @{text
"Void"}. It is routine to establish how values ``inhabiting'' a regular
@@ -446,10 +467,58 @@
@{thm L_flat_Prf}
\end{proposition}
- In general there is more than one value associated with a regular
- expression. In case of POSIX matching the problem is to calculate the
- unique value that satisfies the (informal) POSIX rules from the
- Introduction. Graphically the POSIX value calculation algorithm by
+ \noindent
+ Given a regular expression @{text r} and a string @{text s}, we can define the
+ set of all \emph{Lexical Values} inhabited by @{text r} with the underlying string
+ being @{text s} by
+
+ \begin{center}
+ @{thm LV_def}
+ \end{center}
+
+ \noindent However, later on it will sometimes be necessary to
+ restrict the set of lexical values to a subset called
+ \emph{Canonical Values}. The idea of canonical values is that they
+ satisfy the Star Rule (see Introduction) where the $^\star$ does not
+ match the empty string unless this is the only match for the
+ repetition. One way to define canonical values formally is to use a
+ stronger inhabitation relation, written @{term "\<Turnstile> DUMMY : DUMMY"}, which has the same rules as @{term
+ "\<turnstile> DUMMY : DUMMY"} shown above, except that the rule for
+ @{term Stars} has
+ the additional side-condition of flattened values not being the
+ empty string, namely
+
+ \begin{center}
+ @{thm [mode=Rule] CPrf.intros(6)}
+ \end{center}
+
+ \noindent
+ With this we can define
+
+ \begin{center}
+ @{thm CV_def}
+ \end{center}
+
+ \noindent
+ Clearly we have @{thm LV_CV_subset}.
+ The main point of canonical values is that for every regular expression @{text r} and every
+ string @{text s}, the set @{term "CV r s"} is finite.
+
+ \begin{lemma}
+ @{thm CV_finite}
+ \end{lemma}
+
+ \noindent This finiteness property does not generally hold for lexical values where
+ for example @{term "LV (STAR ONE) []"} contains infinitely many
+ values, but @{thm CV_STAR_ONE_empty}. However, if a regular
+ expression @{text r} matches a string @{text s}, then in general the
+ set @{term "CV r s"} is not just a
+ singleton set. In case of POSIX matching the problem is to
+ calculate the unique value that satisfies the (informal) POSIX rules
+ from the Introduction. It will turn out that this POSIX value is in fact a
+ canonical value.
+
+ Graphically the POSIX value calculation algorithm by
Sulzmann and Lu can be illustrated by the picture in Figure~\ref{Sulz}
where the path from the left to the right involving @{term derivatives}/@{const
nullable} is the first phase of the algorithm (calculating successive
@@ -614,7 +683,7 @@
\end{proof}
Having defined the @{const mkeps} and @{text inj} function we can extend
- \Brz's matcher so that a [lexical] value is constructed (assuming the
+ \Brz's matcher so that a value is constructed (assuming the
regular expression matches the string). The clauses of the Sulzmann and Lu lexer are
\begin{center}
@@ -633,19 +702,25 @@
functional programming language and also in Isabelle/HOL. In the remaining
part of this section we prove that this algorithm is correct.
- The well-known idea of POSIX matching is informally defined by the longest
- match and priority rule (see Introduction); as correctly argued in \cite{Sulzmann2014}, this
+ The well-known idea of POSIX matching is informally defined by some
+ rules such as the longest match and priority rule (see
+ Introduction); as correctly argued in \cite{Sulzmann2014}, this
needs formal specification. Sulzmann and Lu define an ``ordering
- relation'' between values and argue
- that there is a maximum value, as given by the derivative-based algorithm.
- In contrast, we shall introduce a simple inductive definition that
- specifies directly what a \emph{POSIX value} is, incorporating the
- POSIX-specific choices into the side-conditions of our rules. Our
- definition is inspired by the matching relation given by Vansummeren
- \cite{Vansummeren2006}. The relation we define is ternary and written as
- \mbox{@{term "s \<in> r \<rightarrow> v"}}, relating strings, regular expressions and
- values.
+ relation'' between values and argue that there is a maximum value,
+ as given by the derivative-based algorithm. In contrast, we shall
+ introduce a simple inductive definition that specifies directly what
+ a \emph{POSIX value} is, incorporating the POSIX-specific choices
+ into the side-conditions of our rules. Our definition is inspired by
+ the matching relation given by Vansummeren
+ \cite{Vansummeren2006}. The relation we define is ternary and
+ written as \mbox{@{term "s \<in> r \<rightarrow> v"}}, relating
+ strings, regular expressions and values; the inductive rules are given in
+ Figure~\ref{POSIXrules}.
+ We can prove that given a string @{term s} and regular expression @{term
+ r}, the POSIX value @{term v} is uniquely determined by @{term "s \<in> r \<rightarrow> v"}.
+
%
+ \begin{figure}[t]
\begin{center}
\begin{tabular}{c}
@{thm[mode=Axiom] Posix.intros(1)}@{text "P"}@{term "ONE"} \qquad
@@ -668,10 +743,10 @@
{@{thm (concl) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]}}$@{text "P\<star>"}
\end{tabular}
\end{center}
+ \caption{Our inductive definition of POSIX values.}\label{POSIXrules}
+ \end{figure}
- \noindent
- We can prove that given a string @{term s} and regular expression @{term
- r}, the POSIX value @{term v} is uniquely determined by @{term "s \<in> r \<rightarrow> v"}.
+
\begin{theorem}\mbox{}\smallskip\\\label{posixdeterm}
\begin{tabular}{ll}
@@ -687,7 +762,7 @@
\end{proof}
\noindent
- We claim that our @{term "s \<in> r \<rightarrow> v"} relation captures the idea behind the two
+ We claim that our @{term "s \<in> r \<rightarrow> v"} relation captures the idea behind the four
informal POSIX rules shown in the Introduction: Consider for example the
rules @{text "P+L"} and @{text "P+R"} where the POSIX value for a string
and an alternative regular expression, that is @{term "(s, ALT r\<^sub>1 r\<^sub>2)"},
@@ -718,8 +793,19 @@
@{term v} cannot be flattened to the empty string. In effect, we require
that in each ``iteration'' of the star, some non-empty substring needs to
be ``chipped'' away; only in case of the empty string we accept @{term
- "Stars []"} as the POSIX value.
+ "Stars []"} as the POSIX value. Indeed we can show that our POSIX value
+ is a canonical value which excludes those @{text Stars} containing values
+ that flatten to the empty string.
+ \begin{lemma}
+ @{thm [mode=IfThen] Posix_CV}
+ \end{lemma}
+
+ \begin{proof}
+ By routine induction on @{thm (prem 1) Posix_CV}.\qed
+ \end{proof}
+
+ \noindent
Next is the lemma that shows the function @{term "mkeps"} calculates
the POSIX value for the empty string and a nullable regular expression.
@@ -1117,17 +1203,11 @@
text {*
- Theorems:
-
- @{thm [mode=IfThen] Posix_CPT}
+ Theorem 1:
@{thm [mode=IfThen] Posix_PosOrd[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
- Corrollary from the last one
-
- @{thm [mode=IfThen] Posix_PosOrd_stronger[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
-
- Theorem
+ Theorem 2:
@{thm [mode=IfThen] PosOrd_Posix[where ?v1.0="v\<^sub>1"]}
*}
--- a/thys/Journal/document/root.tex Wed Jul 19 14:55:46 2017 +0100
+++ b/thys/Journal/document/root.tex Fri Aug 11 20:29:01 2017 +0100
@@ -12,6 +12,9 @@
%%\usepackage{stmaryrd}
\usepackage{url}
\usepackage{color}
+\usepackage[safe]{tipa}
+
+
\titlerunning{POSIX Lexing with Derivatives of Regular Expressions}
@@ -26,6 +29,8 @@
\renewcommand{\isasymemptyset}{$\varnothing$}
\renewcommand{\isacharunderscore}{\mbox{$\_\!\_$}}
\renewcommand{\isasymiota}{\makebox[0mm]{${}^{\prime}$}}
+\renewcommand{\isasymin}{\ensuremath{\,\in\,}}
+
\def\Brz{Brzozowski}
\def\der{\backslash}
@@ -36,8 +41,10 @@
\renewcommand{\thefootnote}{$\star$} \footnotetext[1]{This paper is a
revised and expanded version of \cite{AusafDyckhoffUrban2016}.
Compared with that paper we give a second definition for POSIX
- values and prove that it is equivalent to the original one. This
- definition is based on an ordering of values and very similar to the
+ values introduced by Okui Suzuki \cite{OkuiSuzuki2013} and prove that it is
+ equivalent to our original one. This
+ second definition is based on an ordering of values and very similar to,
+ but not equivalent with, the
definition given by Sulzmann and Lu~\cite{Sulzmann2014}. We also
extend our results to additional constructors of regular
expressions.} \renewcommand{\thefootnote}{\arabic{footnote}}
@@ -54,7 +61,6 @@
\maketitle
\begin{abstract}
-
Brzozowski introduced the notion of derivatives for regular
expressions. They can be used for a very simple regular expression
matching algorithm. Sulzmann and Lu cleverly extended this algorithm
@@ -68,10 +74,10 @@
of an optimised version of the POSIX matching algorithm. In the
second part we show that $(iii)$ our inductive definition of a POSIX
value is equivalent to an alternative definition by Okui and Suzuki
-which identifies a POSIX value as least element according to an
+which identifies POSIX values as least elements according to an
ordering of values. The advantage of the definition based on the
-ordering is that it implements more directly the POSIX
-longest-leftmost matching semantics.\smallskip
+ordering is that it implements more directly the informal rules from the
+POSIX standard.\smallskip
{\bf Keywords:} POSIX matching, Derivatives of Regular Expressions,
Isabelle/HOL
--- a/thys/Paper/document/root.tex Wed Jul 19 14:55:46 2017 +0100
+++ b/thys/Paper/document/root.tex Fri Aug 11 20:29:01 2017 +0100
@@ -1,4 +1,5 @@
\documentclass[runningheads]{llncs}
+\usepackage{stix}
\usepackage{times}
\usepackage{isabelle}
\usepackage{isabellesym}
@@ -13,6 +14,8 @@
\usepackage{url}
\usepackage{color}
+
+
\titlerunning{POSIX Lexing with Derivatives of Regular Expressions}
\urlstyle{rm}
--- a/thys/Positions.thy Wed Jul 19 14:55:46 2017 +0100
+++ b/thys/Positions.thy Fri Aug 11 20:29:01 2017 +0100
@@ -31,10 +31,8 @@
lemma Pos_stars:
"Pos (Stars vs) = {[]} \<union> (\<Union>n < length vs. {n#ps | ps. ps \<in> Pos (vs ! n)})"
apply(induct vs)
-apply(simp)
-apply(simp add: insert_ident)
-apply(rule subset_antisym)
-using less_Suc_eq_0_disj by auto
+apply(auto simp add: insert_ident less_Suc_eq_0_disj)
+done
lemma Pos_empty:
shows "[] \<in> Pos v"
@@ -45,31 +43,25 @@
"intlen [] = 0"
| "intlen (x # xs) = 1 + intlen xs"
+lemma intlen_int:
+ shows "intlen xs = int (length xs)"
+by (induct xs)(simp_all)
+
lemma intlen_bigger:
shows "0 \<le> intlen xs"
by (induct xs)(auto)
lemma intlen_append:
shows "intlen (xs @ ys) = intlen xs + intlen ys"
-by (induct xs arbitrary: ys) (auto)
+by (simp add: intlen_int)
lemma intlen_length:
shows "intlen xs < intlen ys \<longleftrightarrow> length xs < length ys"
-apply(induct xs arbitrary: ys)
-apply (auto simp add: intlen_bigger not_less)
-apply (metis intlen.elims intlen_bigger le_imp_0_less)
-apply (smt Suc_lessI intlen.simps(2) length_Suc_conv nat_neq_iff)
-by (smt Suc_lessE intlen.simps(2) length_Suc_conv)
+by (simp add: intlen_int)
lemma intlen_length_eq:
shows "intlen xs = intlen ys \<longleftrightarrow> length xs = length ys"
-apply(induct xs arbitrary: ys)
-apply (auto simp add: intlen_bigger not_less)
-apply(case_tac ys)
-apply(simp_all)
-apply (smt intlen_bigger)
-apply (smt intlen.elims intlen_bigger length_Suc_conv)
-by (metis intlen.simps(2) length_Suc_conv)
+by (simp add: intlen_int)
definition pflat_len :: "val \<Rightarrow> nat list => int"
where
@@ -90,7 +82,7 @@
lemma pflat_len_Stars_simps:
assumes "n < length vs"
shows "pflat_len (Stars vs) (n#p) = pflat_len (vs!n) p"
-using assms
+using assms
apply(induct vs arbitrary: n p)
apply(auto simp add: less_Suc_eq_0_disj pflat_len_simps)
done
@@ -98,7 +90,8 @@
lemma pflat_len_outside:
assumes "p \<notin> Pos v1"
shows "pflat_len v1 p = -1 "
-using assms by (auto simp add: pflat_len_def)
+using assms by (simp add: pflat_len_def)
+
section {* Orderings *}
@@ -175,15 +168,10 @@
lemma PosOrd_shorterE:
assumes "v1 :\<sqsubset>val v2"
shows "length (flat v2) \<le> length (flat v1)"
-using assms
-apply(auto simp add: pflat_len_simps PosOrd_ex_def PosOrd_def)
-apply(case_tac p)
-apply(simp add: pflat_len_simps intlen_length)
-apply(simp)
-apply(drule_tac x="[]" in bspec)
-apply(simp add: Pos_empty)
-apply(simp add: pflat_len_simps le_less intlen_length_eq)
-done
+using assms unfolding PosOrd_ex_def PosOrd_def
+apply(auto simp add: pflat_len_def intlen_int split: if_splits)
+apply (metis Pos_empty Un_iff at.simps(1) eq_iff lex_simps(1) nat_less_le)
+by (metis Pos_empty UnI2 at.simps(1) lex_simps(2) lex_trichotomous linear)
lemma PosOrd_shorterI:
assumes "length (flat v2) < length (flat v1)"
@@ -206,8 +194,7 @@
unfolding PosOrd_ex_def
apply(rule_tac x="[0]" in exI)
using assms
-apply(auto simp add: PosOrd_def pflat_len_simps)
-apply(smt intlen_bigger)
+apply(auto simp add: PosOrd_def pflat_len_simps intlen_int)
done
lemma PosOrd_Left_eq:
@@ -547,34 +534,35 @@
by (metis PosOrd_SeqI1 PosOrd_shorterI WW1 antisym_conv3 append_eq_append_conv assms(2))
+
section {* The Posix Value is smaller than any other Value *}
lemma Posix_PosOrd:
- assumes "s \<in> r \<rightarrow> v1" "v2 \<in> CPT r s"
+ assumes "s \<in> r \<rightarrow> v1" "v2 \<in> CV r s"
shows "v1 :\<sqsubseteq>val v2"
using assms
proof (induct arbitrary: v2 rule: Posix.induct)
case (Posix_ONE v)
- have "v \<in> CPT ONE []" by fact
+ have "v \<in> CV ONE []" by fact
then have "v = Void"
- by (simp add: CPT_simps)
+ by (simp add: CV_simps)
then show "Void :\<sqsubseteq>val v"
by (simp add: PosOrd_ex_eq_def)
next
case (Posix_CHAR c v)
- have "v \<in> CPT (CHAR c) [c]" by fact
+ have "v \<in> CV (CHAR c) [c]" by fact
then have "v = Char c"
- by (simp add: CPT_simps)
+ by (simp add: CV_simps)
then show "Char c :\<sqsubseteq>val v"
by (simp add: PosOrd_ex_eq_def)
next
case (Posix_ALT1 s r1 v r2 v2)
have as1: "s \<in> r1 \<rightarrow> v" by fact
- have IH: "\<And>v2. v2 \<in> CPT r1 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
- have "v2 \<in> CPT (ALT r1 r2) s" by fact
+ have IH: "\<And>v2. v2 \<in> CV r1 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
+ have "v2 \<in> CV (ALT r1 r2) s" by fact
then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s"
- by(auto simp add: CPT_def prefix_list_def)
+ by(auto simp add: CV_def prefix_list_def)
then consider
(Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 = s"
| (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 = s"
@@ -582,8 +570,8 @@
then show "Left v :\<sqsubseteq>val v2"
proof(cases)
case (Left v3)
- have "v3 \<in> CPT r1 s" using Left(2,3)
- by (auto simp add: CPT_def prefix_list_def)
+ have "v3 \<in> CV r1 s" using Left(2,3)
+ by (auto simp add: CV_def prefix_list_def)
with IH have "v :\<sqsubseteq>val v3" by simp
moreover
have "flat v3 = flat v" using as1 Left(3)
@@ -603,10 +591,10 @@
case (Posix_ALT2 s r2 v r1 v2)
have as1: "s \<in> r2 \<rightarrow> v" by fact
have as2: "s \<notin> L r1" by fact
- have IH: "\<And>v2. v2 \<in> CPT r2 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
- have "v2 \<in> CPT (ALT r1 r2) s" by fact
+ have IH: "\<And>v2. v2 \<in> CV r2 s \<Longrightarrow> v :\<sqsubseteq>val v2" by fact
+ have "v2 \<in> CV (ALT r1 r2) s" by fact
then have "\<Turnstile> v2 : ALT r1 r2" "flat v2 = s"
- by(auto simp add: CPT_def prefix_list_def)
+ by(auto simp add: CV_def prefix_list_def)
then consider
(Left) v3 where "v2 = Left v3" "\<Turnstile> v3 : r1" "flat v3 = s"
| (Right) v3 where "v2 = Right v3" "\<Turnstile> v3 : r2" "flat v3 = s"
@@ -614,8 +602,8 @@
then show "Right v :\<sqsubseteq>val v2"
proof (cases)
case (Right v3)
- have "v3 \<in> CPT r2 s" using Right(2,3)
- by (auto simp add: CPT_def prefix_list_def)
+ have "v3 \<in> CV r2 s" using Right(2,3)
+ by (auto simp add: CV_def prefix_list_def)
with IH have "v :\<sqsubseteq>val v3" by simp
moreover
have "flat v3 = flat v" using as1 Right(3)
@@ -625,10 +613,10 @@
then show "Right v :\<sqsubseteq>val v2" unfolding Right .
next
case (Left v3)
- have "v3 \<in> CPT r1 s" using Left(2,3) as2
- by (auto simp add: CPT_def prefix_list_def)
+ have "v3 \<in> CV r1 s" using Left(2,3) as2
+ by (auto simp add: CV_def prefix_list_def)
then have "flat v3 = flat v \<and> \<Turnstile> v3 : r1" using as1 Left(3)
- by (simp add: Posix1(2) CPT_def)
+ by (simp add: Posix1(2) CV_def)
then have "False" using as1 as2 Left
by (auto simp add: Posix1(2) L_flat_Prf1 Prf_CPrf)
then show "Right v :\<sqsubseteq>val v2" by simp
@@ -637,22 +625,22 @@
case (Posix_SEQ s1 r1 v1 s2 r2 v2 v3)
have "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" by fact+
then have as1: "s1 = flat v1" "s2 = flat v2" by (simp_all add: Posix1(2))
- have IH1: "\<And>v3. v3 \<in> CPT r1 s1 \<Longrightarrow> v1 :\<sqsubseteq>val v3" by fact
- have IH2: "\<And>v3. v3 \<in> CPT r2 s2 \<Longrightarrow> v2 :\<sqsubseteq>val v3" by fact
+ have IH1: "\<And>v3. v3 \<in> CV r1 s1 \<Longrightarrow> v1 :\<sqsubseteq>val v3" by fact
+ have IH2: "\<And>v3. v3 \<in> CV r2 s2 \<Longrightarrow> v2 :\<sqsubseteq>val v3" by fact
have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r1 \<and> s\<^sub>4 \<in> L r2)" by fact
- have "v3 \<in> CPT (SEQ r1 r2) (s1 @ s2)" by fact
+ have "v3 \<in> CV (SEQ r1 r2) (s1 @ s2)" by fact
then obtain v3a v3b where eqs:
"v3 = Seq v3a v3b" "\<Turnstile> v3a : r1" "\<Turnstile> v3b : r2"
"flat v3a @ flat v3b = s1 @ s2"
- by (force simp add: prefix_list_def CPT_def elim: CPrf.cases)
+ by (force simp add: prefix_list_def CV_def elim: CPrf.cases)
with cond have "flat v3a \<sqsubseteq>pre s1" unfolding prefix_list_def
by (smt L_flat_Prf1 Prf_CPrf append_eq_append_conv2 append_self_conv)
then have "flat v3a \<sqsubset>spre s1 \<or> (flat v3a = s1 \<and> flat v3b = s2)" using eqs
by (simp add: sprefix_list_def append_eq_conv_conj)
then have q2: "v1 :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat v3b = s2)"
using PosOrd_spreI as1(1) eqs by blast
- then have "v1 :\<sqsubset>val v3a \<or> (v3a \<in> CPT r1 s1 \<and> v3b \<in> CPT r2 s2)" using eqs(2,3)
- by (auto simp add: CPT_def)
+ then have "v1 :\<sqsubset>val v3a \<or> (v3a \<in> CV r1 s1 \<and> v3b \<in> CV r2 s2)" using eqs(2,3)
+ by (auto simp add: CV_def)
then have "v1 :\<sqsubset>val v3a \<or> (v1 :\<sqsubseteq>val v3a \<and> v2 :\<sqsubseteq>val v3b)" using IH1 IH2 by blast
then have "Seq v1 v2 :\<sqsubseteq>val Seq v3a v3b" using eqs q2 as1
unfolding PosOrd_ex_eq_def by (auto simp add: PosOrd_SeqI1 PosOrd_SeqI2)
@@ -661,17 +649,17 @@
case (Posix_STAR1 s1 r v s2 vs v3)
have "s1 \<in> r \<rightarrow> v" "s2 \<in> STAR r \<rightarrow> Stars vs" by fact+
then have as1: "s1 = flat v" "s2 = flat (Stars vs)" by (auto dest: Posix1(2))
- have IH1: "\<And>v3. v3 \<in> CPT r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
- have IH2: "\<And>v3. v3 \<in> CPT (STAR r) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
+ have IH1: "\<And>v3. v3 \<in> CV r s1 \<Longrightarrow> v :\<sqsubseteq>val v3" by fact
+ have IH2: "\<And>v3. v3 \<in> CV (STAR r) s2 \<Longrightarrow> Stars vs :\<sqsubseteq>val v3" by fact
have cond: "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> L r \<and> s\<^sub>4 \<in> L (STAR r))" by fact
have cond2: "flat v \<noteq> []" by fact
- have "v3 \<in> CPT (STAR r) (s1 @ s2)" by fact
+ have "v3 \<in> CV (STAR r) (s1 @ s2)" by fact
then consider
(NonEmpty) v3a vs3 where "v3 = Stars (v3a # vs3)"
"\<Turnstile> v3a : r" "\<Turnstile> Stars vs3 : STAR r"
"flat (Stars (v3a # vs3)) = s1 @ s2"
| (Empty) "v3 = Stars []"
- unfolding CPT_def
+ unfolding CV_def
apply(auto)
apply(erule CPrf.cases)
apply(simp_all)
@@ -690,8 +678,8 @@
by (simp add: sprefix_list_def append_eq_conv_conj)
then have q2: "v :\<sqsubset>val v3a \<or> (flat v3a = s1 \<and> flat (Stars vs3) = s2)"
using PosOrd_spreI as1(1) NonEmpty(4) by blast
- then have "v :\<sqsubset>val v3a \<or> (v3a \<in> CPT r s1 \<and> Stars vs3 \<in> CPT (STAR r) s2)"
- using NonEmpty(2,3) by (auto simp add: CPT_def)
+ then have "v :\<sqsubset>val v3a \<or> (v3a \<in> CV r s1 \<and> Stars vs3 \<in> CV (STAR r) s2)"
+ using NonEmpty(2,3) by (auto simp add: CV_def)
then have "v :\<sqsubset>val v3a \<or> (v :\<sqsubseteq>val v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)" using IH1 IH2 by blast
then have "v :\<sqsubset>val v3a \<or> (v = v3a \<and> Stars vs :\<sqsubseteq>val Stars vs3)"
unfolding PosOrd_ex_eq_def by auto
@@ -708,58 +696,26 @@
qed
next
case (Posix_STAR2 r v2)
- have "v2 \<in> CPT (STAR r) []" by fact
+ have "v2 \<in> CV (STAR r) []" by fact
then have "v2 = Stars []"
- unfolding CPT_def by (auto elim: CPrf.cases)
+ unfolding CV_def by (auto elim: CPrf.cases)
then show "Stars [] :\<sqsubseteq>val v2"
by (simp add: PosOrd_ex_eq_def)
qed
-lemma Posix_PosOrd_stronger:
- assumes "s \<in> r \<rightarrow> v1" "v2 \<in> CPTpre r s"
- shows "v1 :\<sqsubseteq>val v2"
-proof -
- from assms(2) have "v2 \<in> CPT r s \<or> flat v2 \<sqsubset>spre s"
- unfolding CPTpre_def CPT_def sprefix_list_def prefix_list_def by auto
- moreover
- { assume "v2 \<in> CPT r s"
- with assms(1)
- have "v1 :\<sqsubseteq>val v2" by (rule Posix_PosOrd)
- }
- moreover
- { assume "flat v2 \<sqsubset>spre s"
- then have "flat v2 \<sqsubset>spre flat v1" using assms(1)
- using Posix1(2) by blast
- then have "v1 :\<sqsubseteq>val v2"
- by (simp add: PosOrd_ex_eq_def PosOrd_spreI)
- }
- ultimately show "v1 :\<sqsubseteq>val v2" by blast
-qed
lemma Posix_PosOrd_reverse:
assumes "s \<in> r \<rightarrow> v1"
- shows "\<not>(\<exists>v2 \<in> CPTpre r s. v2 :\<sqsubset>val v1)"
+ shows "\<not>(\<exists>v2 \<in> CV r s. v2 :\<sqsubset>val v1)"
using assms
-by (metis Posix_PosOrd_stronger less_irrefl PosOrd_def
+by (metis Posix_PosOrd less_irrefl PosOrd_def
PosOrd_ex_eq_def PosOrd_ex_def PosOrd_trans)
-lemma test2:
- assumes "\<forall>v \<in> set vs. flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []"
- shows "(Stars vs) \<in> CPT (STAR r) (flat (Stars vs))"
-using assms
-apply(induct vs)
-apply(auto simp add: CPT_def)
-apply(rule CPrf.intros)
-apply(simp)
-apply(rule CPrf.intros)
-apply(simp_all)
-by (metis (no_types, lifting) CPT_def Posix_CPT mem_Collect_eq)
-
lemma PosOrd_Posix_Stars:
assumes "\<forall>v \<in> set vs. flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []"
- and "\<not>(\<exists>vs2 \<in> PT (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val (Stars vs))"
+ and "\<not>(\<exists>vs2 \<in> LV (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val (Stars vs))"
shows "(flat (Stars vs)) \<in> (STAR r) \<rightarrow> Stars vs"
using assms
proof(induct vs)
@@ -769,24 +725,24 @@
next
case (Cons v vs)
have IH: "\<lbrakk>\<forall>v\<in>set vs. flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> [];
- \<not> (\<exists>vs2\<in>PT (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs)\<rbrakk>
+ \<not> (\<exists>vs2\<in>LV (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs)\<rbrakk>
\<Longrightarrow> flat (Stars vs) \<in> STAR r \<rightarrow> Stars vs" by fact
have as2: "\<forall>v\<in>set (v # vs). flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []" by fact
- have as3: "\<not> (\<exists>vs2\<in>PT (STAR r) (flat (Stars (v # vs))). vs2 :\<sqsubset>val Stars (v # vs))" by fact
+ have as3: "\<not> (\<exists>vs2\<in>LV (STAR r) (flat (Stars (v # vs))). vs2 :\<sqsubset>val Stars (v # vs))" by fact
have "flat v \<in> r \<rightarrow> v" using as2 by simp
moreover
have "flat (Stars vs) \<in> STAR r \<rightarrow> Stars vs"
proof (rule IH)
show "\<forall>v\<in>set vs. flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []" using as2 by simp
next
- show "\<not> (\<exists>vs2\<in>PT (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs)" using as3
+ show "\<not> (\<exists>vs2\<in>LV (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs)" using as3
apply(auto)
- apply(subst (asm) (2) PT_def)
+ apply(subst (asm) (2) LV_def)
apply(auto)
apply(erule Prf.cases)
apply(simp_all)
apply(drule_tac x="Stars (v # vs)" in bspec)
- apply(simp add: PT_def CPT_def)
+ apply(simp add: LV_def CV_def)
using Posix_Prf Prf.intros(6) calculation
apply(rule_tac Prf.intros)
apply(simp add:)
@@ -810,7 +766,7 @@
apply(simp_all)
apply(clarify)
apply(drule_tac x="Stars (va#vs)" in bspec)
- apply(auto simp add: PT_def)[1]
+ apply(auto simp add: LV_def)[1]
apply(rule Prf.intros)
apply(simp)
by (simp add: PosOrd_StarsI PosOrd_shorterI)
@@ -823,69 +779,69 @@
section {* The Smallest Value is indeed the Posix Value *}
text {*
- The next lemma seems to require PT instead of CPT in the Star-case.
+ The next lemma seems to require LV instead of CV in the Star-case.
*}
lemma PosOrd_Posix:
- assumes "v1 \<in> CPT r s" "\<forall>v\<^sub>2 \<in> PT r s. \<not> v\<^sub>2 :\<sqsubset>val v1"
+ assumes "v1 \<in> CV r s" "\<forall>v\<^sub>2 \<in> LV r s. \<not> v\<^sub>2 :\<sqsubset>val v1"
shows "s \<in> r \<rightarrow> v1"
using assms
proof(induct r arbitrary: s v1)
case (ZERO s v1)
- have "v1 \<in> CPT ZERO s" by fact
- then show "s \<in> ZERO \<rightarrow> v1" unfolding CPT_def
+ have "v1 \<in> CV ZERO s" by fact
+ then show "s \<in> ZERO \<rightarrow> v1" unfolding CV_def
by (auto elim: CPrf.cases)
next
case (ONE s v1)
- have "v1 \<in> CPT ONE s" by fact
- then show "s \<in> ONE \<rightarrow> v1" unfolding CPT_def
+ have "v1 \<in> CV ONE s" by fact
+ then show "s \<in> ONE \<rightarrow> v1" unfolding CV_def
by(auto elim!: CPrf.cases intro: Posix.intros)
next
case (CHAR c s v1)
- have "v1 \<in> CPT (CHAR c) s" by fact
- then show "s \<in> CHAR c \<rightarrow> v1" unfolding CPT_def
+ have "v1 \<in> CV (CHAR c) s" by fact
+ then show "s \<in> CHAR c \<rightarrow> v1" unfolding CV_def
by (auto elim!: CPrf.cases intro: Posix.intros)
next
case (ALT r1 r2 s v1)
- have IH1: "\<And>s v1. \<lbrakk>v1 \<in> CPT r1 s; \<forall>v2 \<in> PT r1 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r1 \<rightarrow> v1" by fact
- have IH2: "\<And>s v1. \<lbrakk>v1 \<in> CPT r2 s; \<forall>v2 \<in> PT r2 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r2 \<rightarrow> v1" by fact
- have as1: "\<forall>v2\<in>PT (ALT r1 r2) s. \<not> v2 :\<sqsubset>val v1" by fact
- have as2: "v1 \<in> CPT (ALT r1 r2) s" by fact
+ have IH1: "\<And>s v1. \<lbrakk>v1 \<in> CV r1 s; \<forall>v2 \<in> LV r1 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r1 \<rightarrow> v1" by fact
+ have IH2: "\<And>s v1. \<lbrakk>v1 \<in> CV r2 s; \<forall>v2 \<in> LV r2 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r2 \<rightarrow> v1" by fact
+ have as1: "\<forall>v2\<in>LV (ALT r1 r2) s. \<not> v2 :\<sqsubset>val v1" by fact
+ have as2: "v1 \<in> CV (ALT r1 r2) s" by fact
then consider
(Left) v1' where
"v1 = Left v1'" "s = flat v1'"
- "v1' \<in> CPT r1 s"
+ "v1' \<in> CV r1 s"
| (Right) v1' where
"v1 = Right v1'" "s = flat v1'"
- "v1' \<in> CPT r2 s"
- unfolding CPT_def by (auto elim: CPrf.cases)
+ "v1' \<in> CV r2 s"
+ unfolding CV_def by (auto elim: CPrf.cases)
then show "s \<in> ALT r1 r2 \<rightarrow> v1"
proof (cases)
case (Left v1')
- have "v1' \<in> CPT r1 s" using as2
- unfolding CPT_def Left by (auto elim: CPrf.cases)
+ have "v1' \<in> CV r1 s" using as2
+ unfolding CV_def Left by (auto elim: CPrf.cases)
moreover
- have "\<forall>v2 \<in> PT r1 s. \<not> v2 :\<sqsubset>val v1'" using as1
- unfolding PT_def Left using Prf.intros(2) PosOrd_Left_eq by force
+ have "\<forall>v2 \<in> LV r1 s. \<not> v2 :\<sqsubset>val v1'" using as1
+ unfolding LV_def Left using Prf.intros(2) PosOrd_Left_eq by force
ultimately have "s \<in> r1 \<rightarrow> v1'" using IH1 by simp
then have "s \<in> ALT r1 r2 \<rightarrow> Left v1'" by (rule Posix.intros)
then show "s \<in> ALT r1 r2 \<rightarrow> v1" using Left by simp
next
case (Right v1')
- have "v1' \<in> CPT r2 s" using as2
- unfolding CPT_def Right by (auto elim: CPrf.cases)
+ have "v1' \<in> CV r2 s" using as2
+ unfolding CV_def Right by (auto elim: CPrf.cases)
moreover
- have "\<forall>v2 \<in> PT r2 s. \<not> v2 :\<sqsubset>val v1'" using as1
- unfolding PT_def Right using Prf.intros(3) PosOrd_RightI by force
+ have "\<forall>v2 \<in> LV r2 s. \<not> v2 :\<sqsubset>val v1'" using as1
+ unfolding LV_def Right using Prf.intros(3) PosOrd_RightI by force
ultimately have "s \<in> r2 \<rightarrow> v1'" using IH2 by simp
moreover
{ assume "s \<in> L r1"
- then obtain v' where "v' \<in> PT r1 s"
- unfolding PT_def using L_flat_Prf2 by blast
- then have "Left v' \<in> PT (ALT r1 r2) s"
- unfolding PT_def by (auto intro: Prf.intros)
+ then obtain v' where "v' \<in> LV r1 s"
+ unfolding LV_def using L_flat_Prf2 by blast
+ then have "Left v' \<in> LV (ALT r1 r2) s"
+ unfolding LV_def by (auto intro: Prf.intros)
with as1 have "\<not> (Left v' :\<sqsubset>val Right v1') \<and> (flat v' = s)"
- unfolding PT_def Right by (auto)
+ unfolding LV_def Right by (auto)
then have False using PosOrd_Left_Right Right by blast
}
then have "s \<notin> L r1" by rule
@@ -894,36 +850,36 @@
qed
next
case (SEQ r1 r2 s v1)
- have IH1: "\<And>s v1. \<lbrakk>v1 \<in> CPT r1 s; \<forall>v2 \<in> PT r1 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r1 \<rightarrow> v1" by fact
- have IH2: "\<And>s v1. \<lbrakk>v1 \<in> CPT r2 s; \<forall>v2 \<in> PT r2 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r2 \<rightarrow> v1" by fact
- have as1: "\<forall>v2\<in>PT (SEQ r1 r2) s. \<not> v2 :\<sqsubset>val v1" by fact
- have as2: "v1 \<in> CPT (SEQ r1 r2) s" by fact
+ have IH1: "\<And>s v1. \<lbrakk>v1 \<in> CV r1 s; \<forall>v2 \<in> LV r1 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r1 \<rightarrow> v1" by fact
+ have IH2: "\<And>s v1. \<lbrakk>v1 \<in> CV r2 s; \<forall>v2 \<in> LV r2 s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r2 \<rightarrow> v1" by fact
+ have as1: "\<forall>v2\<in>LV (SEQ r1 r2) s. \<not> v2 :\<sqsubset>val v1" by fact
+ have as2: "v1 \<in> CV (SEQ r1 r2) s" by fact
then obtain
v1a v1b where eqs:
"v1 = Seq v1a v1b" "s = flat v1a @ flat v1b"
- "v1a \<in> CPT r1 (flat v1a)" "v1b \<in> CPT r2 (flat v1b)"
- unfolding CPT_def by(auto elim: CPrf.cases)
- have "\<forall>v2 \<in> PT r1 (flat v1a). \<not> v2 :\<sqsubset>val v1a"
+ "v1a \<in> CV r1 (flat v1a)" "v1b \<in> CV r2 (flat v1b)"
+ unfolding CV_def by(auto elim: CPrf.cases)
+ have "\<forall>v2 \<in> LV r1 (flat v1a). \<not> v2 :\<sqsubset>val v1a"
proof
fix v2
- assume "v2 \<in> PT r1 (flat v1a)"
- with eqs(2,4) have "Seq v2 v1b \<in> PT (SEQ r1 r2) s"
- by (simp add: CPT_def PT_def Prf.intros(1) Prf_CPrf)
+ assume "v2 \<in> LV r1 (flat v1a)"
+ with eqs(2,4) have "Seq v2 v1b \<in> LV (SEQ r1 r2) s"
+ by (simp add: CV_def LV_def Prf.intros(1) Prf_CPrf)
with as1 have "\<not> Seq v2 v1b :\<sqsubset>val Seq v1a v1b \<and> flat (Seq v2 v1b) = flat (Seq v1a v1b)"
- using eqs by (simp add: PT_def)
+ using eqs by (simp add: LV_def)
then show "\<not> v2 :\<sqsubset>val v1a"
using PosOrd_SeqI1 by blast
qed
then have "flat v1a \<in> r1 \<rightarrow> v1a" using IH1 eqs by simp
moreover
- have "\<forall>v2 \<in> PT r2 (flat v1b). \<not> v2 :\<sqsubset>val v1b"
+ have "\<forall>v2 \<in> LV r2 (flat v1b). \<not> v2 :\<sqsubset>val v1b"
proof
fix v2
- assume "v2 \<in> PT r2 (flat v1b)"
- with eqs(2,3,4) have "Seq v1a v2 \<in> PT (SEQ r1 r2) s"
- by (simp add: CPT_def PT_def Prf.intros Prf_CPrf)
+ assume "v2 \<in> LV r2 (flat v1b)"
+ with eqs(2,3,4) have "Seq v1a v2 \<in> LV (SEQ r1 r2) s"
+ by (simp add: CV_def LV_def Prf.intros Prf_CPrf)
with as1 have "\<not> Seq v1a v2 :\<sqsubset>val Seq v1a v1b \<and> flat v2 = flat v1b"
- using eqs by (simp add: PT_def)
+ using eqs by (simp add: LV_def)
then show "\<not> v2 :\<sqsubset>val v1b"
using PosOrd_SeqI2 by auto
qed
@@ -935,8 +891,8 @@
then obtain s3 s4 where q1: "s3 \<noteq> [] \<and> s3 @ s4 = flat v1b \<and> flat v1a @ s3 \<in> L r1 \<and> s4 \<in> L r2" by blast
then obtain vA vB where q2: "flat vA = flat v1a @ s3" "\<turnstile> vA : r1" "flat vB = s4" "\<turnstile> vB : r2"
using L_flat_Prf2 by blast
- then have "Seq vA vB \<in> PT (SEQ r1 r2) s" unfolding eqs using q1
- by (auto simp add: PT_def intro: Prf.intros)
+ then have "Seq vA vB \<in> LV (SEQ r1 r2) s" unfolding eqs using q1
+ by (auto simp add: LV_def intro: Prf.intros)
with as1 have "\<not> Seq vA vB :\<sqsubset>val Seq v1a v1b" unfolding eqs by auto
then have "\<not> vA :\<sqsubset>val v1a \<and> length (flat vA) > length (flat v1a)" using q1 q2 PosOrd_SeqI1 by auto
then show "False"
@@ -947,57 +903,57 @@
by (rule Posix.intros)
next
case (STAR r s v1)
- have IH: "\<And>s v1. \<lbrakk>v1 \<in> CPT r s; \<forall>v2\<in>PT r s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r \<rightarrow> v1" by fact
- have as1: "\<forall>v2\<in>PT (STAR r) s. \<not> v2 :\<sqsubset>val v1" by fact
- have as2: "v1 \<in> CPT (STAR r) s" by fact
+ have IH: "\<And>s v1. \<lbrakk>v1 \<in> CV r s; \<forall>v2\<in>LV r s. \<not> v2 :\<sqsubset>val v1\<rbrakk> \<Longrightarrow> s \<in> r \<rightarrow> v1" by fact
+ have as1: "\<forall>v2\<in>LV (STAR r) s. \<not> v2 :\<sqsubset>val v1" by fact
+ have as2: "v1 \<in> CV (STAR r) s" by fact
then obtain
vs where eqs:
"v1 = Stars vs" "s = flat (Stars vs)"
- "\<forall>v \<in> set vs. v \<in> CPT r (flat v)"
- unfolding CPT_def by (auto elim: CPrf.cases dest!: CPrf_stars)
+ "\<forall>v \<in> set vs. v \<in> CV r (flat v)"
+ unfolding CV_def by (auto elim: CPrf.cases)
have "\<forall>v\<in>set vs. flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []"
proof
fix v
assume a: "v \<in> set vs"
then obtain pre post where e: "vs = pre @ [v] @ post"
by (metis append_Cons append_Nil in_set_conv_decomp_first)
- then have q: "\<forall>v2\<in>PT (STAR r) s. \<not> v2 :\<sqsubset>val Stars (pre @ [v] @ post)"
+ then have q: "\<forall>v2\<in>LV (STAR r) s. \<not> v2 :\<sqsubset>val Stars (pre @ [v] @ post)"
using as1 unfolding eqs by simp
- have "\<forall>v2\<in>PT r (flat v). \<not> v2 :\<sqsubset>val v" unfolding eqs
+ have "\<forall>v2\<in>LV r (flat v). \<not> v2 :\<sqsubset>val v" unfolding eqs
proof (rule ballI, rule notI)
fix v2
assume w: "v2 :\<sqsubset>val v"
- assume "v2 \<in> PT r (flat v)"
- then have "Stars (pre @ [v2] @ post) \<in> PT (STAR r) s"
+ assume "v2 \<in> LV r (flat v)"
+ then have "Stars (pre @ [v2] @ post) \<in> LV (STAR r) s"
using as2 unfolding e eqs
- apply(auto simp add: CPT_def PT_def intro!: Prf.intros)[1]
- using CPrf_Stars_appendE CPrf_stars Prf_CPrf apply blast
- by (meson CPrf_Stars_appendE CPrf_stars Prf_CPrf list.set_intros(2))
+ apply(auto simp add: CV_def LV_def intro!: Prf.intros)[1]
+ using CPrf_Stars_appendE Prf_CPrf Prf_elims(6) list.set_intros apply blast
+ by (metis CPrf_Stars_appendE Prf_CPrf Prf_elims(6) list.set_intros(2) val.inject(5))
then have "\<not> Stars (pre @ [v2] @ post) :\<sqsubset>val Stars (pre @ [v] @ post)"
using q by simp
with w show "False"
- using PT_def \<open>v2 \<in> PT r (flat v)\<close> append_Cons flat.simps(7) mem_Collect_eq
+ using LV_def \<open>v2 \<in> LV r (flat v)\<close> append_Cons flat.simps(7) mem_Collect_eq
PosOrd_StarsI PosOrd_Stars_appendI by auto
qed
with IH
- show "flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []" using a as2 unfolding eqs
- using eqs(3) by (smt CPT_def CPrf_stars mem_Collect_eq)
+ show "flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []" using a as2 unfolding eqs CV_def
+ by (auto elim: CPrf.cases)
qed
moreover
- have "\<not> (\<exists>vs2\<in>PT (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs)"
+ have "\<not> (\<exists>vs2\<in>LV (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs)"
proof
- assume "\<exists>vs2 \<in> PT (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs"
+ assume "\<exists>vs2 \<in> LV (STAR r) (flat (Stars vs)). vs2 :\<sqsubset>val Stars vs"
then obtain vs2 where "\<turnstile> Stars vs2 : STAR r" "flat (Stars vs2) = flat (Stars vs)"
"Stars vs2 :\<sqsubset>val Stars vs"
- unfolding PT_def
- apply(auto elim: Prf.cases)
+ unfolding LV_def
+ apply(auto)
apply(erule Prf.cases)
apply(auto intro: Prf.intros)
done
then show "False" using as1 unfolding eqs
apply -
apply(drule_tac x="Stars vs2" in bspec)
- apply(auto simp add: PT_def)
+ apply(auto simp add: LV_def)
done
qed
ultimately have "flat (Stars vs) \<in> STAR r \<rightarrow> Stars vs"
--- a/thys/Spec.thy Wed Jul 19 14:55:46 2017 +0100
+++ b/thys/Spec.thy Fri Aug 11 20:29:01 2017 +0100
@@ -1,9 +1,8 @@
theory Spec
- imports Main
+ imports Main "~~/src/HOL/Library/Sublist"
begin
-
section {* Sequential Composition of Languages *}
definition
@@ -172,13 +171,15 @@
lemma ders_correctness:
shows "L (ders s r) = Ders s (L r)"
-apply(induct s arbitrary: r)
-apply(simp_all add: Ders_def der_correctness Der_def)
-done
+by (induct s arbitrary: r)
+ (simp_all add: Ders_def der_correctness Der_def)
+
section {* Lemmas about ders *}
+(* not really needed *)
+
lemma ders_ZERO:
shows "ders s (ZERO) = ZERO"
apply(induct s)
@@ -201,9 +202,8 @@
lemma ders_ALT:
shows "ders s (ALT r1 r2) = ALT (ders s r1) (ders s r2)"
-apply(induct s arbitrary: r1 r2)
-apply(simp_all)
-done
+by (induct s arbitrary: r1 r2)(simp_all)
+
section {* Values *}
@@ -229,8 +229,11 @@
| "flat (Stars []) = []"
| "flat (Stars (v#vs)) = (flat v) @ (flat (Stars vs))"
+abbreviation
+ "flats vs \<equiv> concat (map flat vs)"
+
lemma flat_Stars [simp]:
- "flat (Stars vs) = concat (map flat vs)"
+ "flat (Stars vs) = flats vs"
by (induct vs) (auto)
@@ -273,7 +276,7 @@
lemma Star_val:
assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<turnstile> v : r"
- shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<turnstile> v : r)"
+ shows "\<exists>vs. flats vs = concat ss \<and> (\<forall>v\<in>set vs. \<turnstile> v : r)"
using assms
apply(induct ss)
apply(auto)
@@ -313,7 +316,7 @@
have "s \<in> L (STAR r)" by fact
then obtain ss where "concat ss = s" "\<forall>s \<in> set ss. s \<in> L r"
using Star_string by auto
- then obtain vs where "concat (map flat vs) = s" "\<forall>v\<in>set vs. \<turnstile> v : r"
+ then obtain vs where "flats vs = s" "\<forall>v\<in>set vs. \<turnstile> v : r"
using IH Star_val by blast
then show "\<exists>v. \<turnstile> v : STAR r \<and> flat v = s"
using Prf.intros(6) flat_Stars by blast
@@ -331,8 +334,8 @@
shows "L(r) = {flat v | v. \<turnstile> v : r}"
using L_flat_Prf1 L_flat_Prf2 by blast
-section {* CPT and CPTpre *}
+section {* Canonical Values *}
inductive
CPrf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<Turnstile> _ : _" [100, 100] 100)
@@ -350,71 +353,153 @@
using assms
by (induct)(auto intro: Prf.intros)
-lemma CPrf_stars:
- assumes "\<Turnstile> Stars vs : STAR r"
- shows "\<forall>v \<in> set vs. flat v \<noteq> [] \<and> \<Turnstile> v : r"
-using assms
-apply(erule_tac CPrf.cases)
-apply(simp_all)
-done
-
lemma CPrf_Stars_appendE:
assumes "\<Turnstile> Stars (vs1 @ vs2) : STAR r"
shows "\<Turnstile> Stars vs1 : STAR r \<and> \<Turnstile> Stars vs2 : STAR r"
using assms
apply(erule_tac CPrf.cases)
-apply(auto intro: CPrf.intros elim: Prf.cases)
+apply(auto intro: CPrf.intros)
done
-definition PT :: "rexp \<Rightarrow> string \<Rightarrow> val set"
-where "PT r s \<equiv> {v. flat v = s \<and> \<turnstile> v : r}"
+
+section {* Sets of Lexical and Canonical Values *}
-definition
- "CPT r s = {v. flat v = s \<and> \<Turnstile> v : r}"
+definition
+ LV :: "rexp \<Rightarrow> string \<Rightarrow> val set"
+where "LV r s \<equiv> {v. \<turnstile> v : r \<and> flat v = s}"
definition
- "CPTpre r s = {v. \<exists>s'. flat v @ s' = s \<and> \<Turnstile> v : r}"
+ CV :: "rexp \<Rightarrow> string \<Rightarrow> val set"
+where "CV r s \<equiv> {v. \<Turnstile> v : r \<and> flat v = s}"
+
+lemma LV_CV_subset:
+ shows "CV r s \<subseteq> LV r s"
+unfolding CV_def LV_def by(auto simp add: Prf_CPrf)
+
+abbreviation
+ "Prefixes s \<equiv> {s'. prefixeq s' s}"
+
+abbreviation
+ "Suffixes s \<equiv> {s'. suffixeq s' s}"
+
+abbreviation
+ "SSuffixes s \<equiv> {s'. suffix s' s}"
-lemma CPT_CPTpre_subset:
- shows "CPT r s \<subseteq> CPTpre r s"
-by(auto simp add: CPT_def CPTpre_def)
+lemma Suffixes_cons [simp]:
+ shows "Suffixes (c # s) = Suffixes s \<union> {c # s}"
+by (auto simp add: suffixeq_def Cons_eq_append_conv)
+
+lemma CV_simps:
+ shows "CV ZERO s = {}"
+ and "CV ONE s = (if s = [] then {Void} else {})"
+ and "CV (CHAR c) s = (if s = [c] then {Char c} else {})"
+ and "CV (ALT r1 r2) s = Left ` CV r1 s \<union> Right ` CV r2 s"
+unfolding CV_def
+by (auto intro: CPrf.intros elim: CPrf.cases)
+
+lemma finite_Suffixes:
+ shows "finite (Suffixes s)"
+by (induct s) (simp_all)
-lemma CPT_simps:
- shows "CPT ZERO s = {}"
- and "CPT ONE s = (if s = [] then {Void} else {})"
- and "CPT (CHAR c) s = (if s = [c] then {Char c} else {})"
- and "CPT (ALT r1 r2) s = Left ` CPT r1 s \<union> Right ` CPT r2 s"
- and "CPT (SEQ r1 r2) s =
- {Seq v1 v2 | v1 v2. flat v1 @ flat v2 = s \<and> v1 \<in> CPT r1 (flat v1) \<and> v2 \<in> CPT r2 (flat v2)}"
- and "CPT (STAR r) s =
- Stars ` {vs. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. v \<in> CPT r (flat v) \<and> flat v \<noteq> [])}"
-apply -
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
+lemma finite_SSuffixes:
+ shows "finite (SSuffixes s)"
+proof -
+ have "SSuffixes s \<subseteq> Suffixes s"
+ unfolding suffix_def suffixeq_def by auto
+ then show "finite (SSuffixes s)"
+ using finite_Suffixes finite_subset by blast
+qed
+
+lemma finite_Prefixes:
+ shows "finite (Prefixes s)"
+proof -
+ have "finite (Suffixes (rev s))"
+ by (rule finite_Suffixes)
+ then have "finite (rev ` Suffixes (rev s))" by simp
+ moreover
+ have "rev ` (Suffixes (rev s)) = Prefixes s"
+ unfolding suffixeq_def prefixeq_def image_def
+ by (auto)(metis rev_append rev_rev_ident)+
+ ultimately show "finite (Prefixes s)" by simp
+qed
+
+lemma CV_SEQ_subset:
+ "CV (SEQ r1 r2) s \<subseteq> (\<lambda>(v1,v2). Seq v1 v2) ` ((\<Union>s' \<in> Prefixes s. CV r1 s') \<times> (\<Union>s' \<in> Suffixes s. CV r2 s'))"
+unfolding image_def CV_def prefixeq_def suffixeq_def
+by (auto elim: CPrf.cases)
+
+lemma CV_STAR_subset:
+ "CV (STAR r) s \<subseteq> {Stars []} \<union>
+ (\<lambda>(v,vs). Stars (v#vs)) ` ((\<Union>s' \<in> Prefixes s. CV r s') \<times> (\<Union>s2 \<in> SSuffixes s. Stars -` (CV (STAR r) s2)))"
+unfolding image_def CV_def prefixeq_def suffix_def
+apply(auto)
apply(erule CPrf.cases)
-apply(simp_all)[6]
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
-(* STAR case *)
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[6]
+apply(auto)
+apply(case_tac vs)
+apply(auto intro: CPrf.intros)
done
+lemma CV_STAR_finite:
+ assumes "\<forall>s. finite (CV r s)"
+ shows "finite (CV (STAR r) s)"
+proof(induct s rule: length_induct)
+ fix s::"char list"
+ assume "\<forall>s'. length s' < length s \<longrightarrow> finite (CV (STAR r) s')"
+ then have IH: "\<forall>s' \<in> SSuffixes s. finite (CV (STAR r) s')"
+ by (auto simp add: suffix_def)
+ def f \<equiv> "\<lambda>(v, vs). Stars (v # vs)"
+ def S1 \<equiv> "\<Union>s' \<in> Prefixes s. CV r s'"
+ def S2 \<equiv> "\<Union>s2 \<in> SSuffixes s. Stars -` (CV (STAR r) s2)"
+ have "finite S1" using assms
+ unfolding S1_def by (simp_all add: finite_Prefixes)
+ moreover
+ with IH have "finite S2" unfolding S2_def
+ by (auto simp add: finite_SSuffixes inj_on_def finite_vimageI)
+ ultimately
+ have "finite ({Stars []} \<union> f ` (S1 \<times> S2))" by simp
+ moreover
+ have "CV (STAR r) s \<subseteq> {Stars []} \<union> f ` (S1 \<times> S2)" unfolding S1_def S2_def f_def
+ by (rule CV_STAR_subset)
+ ultimately
+ show "finite (CV (STAR r) s)" by (simp add: finite_subset)
+qed
+
+
+lemma CV_finite:
+ shows "finite (CV r s)"
+proof(induct r arbitrary: s)
+ case (ZERO s)
+ show "finite (CV ZERO s)" by (simp add: CV_simps)
+next
+ case (ONE s)
+ show "finite (CV ONE s)" by (simp add: CV_simps)
+next
+ case (CHAR c s)
+ show "finite (CV (CHAR c) s)" by (simp add: CV_simps)
+next
+ case (ALT r1 r2 s)
+ then show "finite (CV (ALT r1 r2) s)" by (simp add: CV_simps)
+next
+ case (SEQ r1 r2 s)
+ def f \<equiv> "\<lambda>(v1, v2). Seq v1 v2"
+ def S1 \<equiv> "\<Union>s' \<in> Prefixes s. CV r1 s'"
+ def S2 \<equiv> "\<Union>s' \<in> Suffixes s. CV r2 s'"
+ have IHs: "\<And>s. finite (CV r1 s)" "\<And>s. finite (CV r2 s)" by fact+
+ then have "finite S1" "finite S2" unfolding S1_def S2_def
+ by (simp_all add: finite_Prefixes finite_Suffixes)
+ moreover
+ have "CV (SEQ r1 r2) s \<subseteq> f ` (S1 \<times> S2)"
+ unfolding f_def S1_def S2_def by (auto simp add: CV_SEQ_subset)
+ ultimately
+ show "finite (CV (SEQ r1 r2) s)"
+ by (simp add: finite_subset)
+next
+ case (STAR r s)
+ then show "finite (CV (STAR r) s)" by (simp add: CV_STAR_finite)
+qed
+
+
section {* Our POSIX Definition *}
@@ -531,12 +616,12 @@
Our POSIX value is a canonical value.
*}
-lemma Posix_CPT:
+lemma Posix_CV:
assumes "s \<in> r \<rightarrow> v"
- shows "v \<in> CPT r s"
+ shows "v \<in> CV r s"
using assms
apply(induct rule: Posix.induct)
-apply(auto simp add: CPT_def intro: CPrf.intros elim: CPrf.cases)
+apply(auto simp add: CV_def intro: CPrf.intros elim: CPrf.cases)
apply(rotate_tac 5)
apply(erule CPrf.cases)
apply(simp_all)
@@ -544,203 +629,17 @@
apply(simp_all)
done
-
-
-(*
-lemma CPTpre_STAR_finite:
- assumes "\<And>s. finite (CPT r s)"
- shows "finite (CPT (STAR r) s)"
-apply(induct s rule: length_induct)
-apply(case_tac xs)
-apply(simp)
-apply(simp add: CPT_simps)
-apply(auto)
-apply(rule finite_imageI)
-using assms
-thm finite_Un
-prefer 2
-apply(simp add: CPT_simps)
-apply(rule finite_imageI)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(simp)
-apply(rule_tac B="(\<lambda>(v, vs). Stars (v#vs)) ` {(v, vs). v \<in> CPTpre r (a#list) \<and> flat v \<noteq> [] \<and> Stars vs \<in> CPTpre (STAR r) (drop (length (flat v)) (a#list))}" in finite_subset)
-apply(auto)[1]
-apply(rule finite_imageI)
-apply(simp add: Collect_case_prod_Sigma)
-apply(rule finite_SigmaI)
-apply(rule assms)
-apply(case_tac "flat v = []")
-apply(simp)
-apply(drule_tac x="drop (length (flat v)) (a # list)" in spec)
-apply(simp)
-apply(auto)[1]
-apply(rule test)
-apply(simp)
-done
-
-lemma CPTpre_subsets:
- "CPTpre ZERO s = {}"
- "CPTpre ONE s \<subseteq> {Void}"
- "CPTpre (CHAR c) s \<subseteq> {Char c}"
- "CPTpre (ALT r1 r2) s \<subseteq> Left ` CPTpre r1 s \<union> Right ` CPTpre r2 s"
- "CPTpre (SEQ r1 r2) s \<subseteq> {Seq v1 v2 | v1 v2. v1 \<in> CPTpre r1 s \<and> v2 \<in> CPTpre r2 (drop (length (flat v1)) s)}"
- "CPTpre (STAR r) s \<subseteq> {Stars []} \<union>
- {Stars (v#vs) | v vs. v \<in> CPTpre r s \<and> flat v \<noteq> [] \<and> Stars vs \<in> CPTpre (STAR r) (drop (length (flat v)) s)}"
- "CPTpre (STAR r) [] = {Stars []}"
-apply(auto simp add: CPTpre_def)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(rule CPrf.intros)
-done
-
-
-lemma CPTpre_simps:
- shows "CPTpre ONE s = {Void}"
- and "CPTpre (CHAR c) (d#s) = (if c = d then {Char c} else {})"
- and "CPTpre (ALT r1 r2) s = Left ` CPTpre r1 s \<union> Right ` CPTpre r2 s"
- and "CPTpre (SEQ r1 r2) s =
- {Seq v1 v2 | v1 v2. v1 \<in> CPTpre r1 s \<and> v2 \<in> CPTpre r2 (drop (length (flat v1)) s)}"
-apply -
-apply(rule subset_antisym)
-apply(rule CPTpre_subsets)
-apply(auto simp add: CPTpre_def intro: "CPrf.intros")[1]
-apply(case_tac "c = d")
-apply(simp)
-apply(rule subset_antisym)
-apply(rule CPTpre_subsets)
-apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
-apply(simp)
-apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)
-apply(rule subset_antisym)
-apply(rule CPTpre_subsets)
-apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
-apply(rule subset_antisym)
-apply(rule CPTpre_subsets)
-apply(auto simp add: CPTpre_def intro: CPrf.intros)[1]
-done
-
-lemma CPT_simps:
- shows "CPT ONE s = (if s = [] then {Void} else {})"
- and "CPT (CHAR c) [d] = (if c = d then {Char c} else {})"
- and "CPT (ALT r1 r2) s = Left ` CPT r1 s \<union> Right ` CPT r2 s"
- and "CPT (SEQ r1 r2) s =
- {Seq v1 v2 | v1 v2 s1 s2. s1 @ s2 = s \<and> v1 \<in> CPT r1 s1 \<and> v2 \<in> CPT r2 s2}"
-apply -
-apply(rule subset_antisym)
-apply(auto simp add: CPT_def)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(auto simp add: CPT_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(auto simp add: CPT_def image_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-apply(clarify)
-apply blast
-apply(auto simp add: CPT_def image_def intro: CPrf.intros)[1]
-apply(erule CPrf.cases)
-apply(simp_all)[7]
-done
-
-lemma test:
- assumes "finite A"
- shows "finite {vs. Stars vs \<in> A}"
-using assms
-apply(induct A)
-apply(simp)
-apply(auto)
-apply(case_tac x)
-apply(simp_all)
-done
-
-lemma CPTpre_STAR_finite:
- assumes "\<And>s. finite (CPTpre r s)"
- shows "finite (CPTpre (STAR r) s)"
-apply(induct s rule: length_induct)
-apply(case_tac xs)
-apply(simp)
-apply(simp add: CPTpre_subsets)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(simp)
-apply(rule_tac B="(\<lambda>(v, vs). Stars (v#vs)) ` {(v, vs). v \<in> CPTpre r (a#list) \<and> flat v \<noteq> [] \<and> Stars vs \<in> CPTpre (STAR r) (drop (length (flat v)) (a#list))}" in finite_subset)
-apply(auto)[1]
-apply(rule finite_imageI)
-apply(simp add: Collect_case_prod_Sigma)
-apply(rule finite_SigmaI)
-apply(rule assms)
-apply(case_tac "flat v = []")
-apply(simp)
-apply(drule_tac x="drop (length (flat v)) (a # list)" in spec)
-apply(simp)
-apply(auto)[1]
-apply(rule test)
-apply(simp)
-done
-
-lemma CPTpre_finite:
- shows "finite (CPTpre r s)"
-apply(induct r arbitrary: s)
-apply(simp add: CPTpre_subsets)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(simp)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(simp)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(rule_tac B="(\<lambda>(v1, v2). Seq v1 v2) ` {(v1, v2). v1 \<in> CPTpre r1 s \<and> v2 \<in> CPTpre r2 (drop (length (flat v1)) s)}" in finite_subset)
-apply(auto)[1]
-apply(rule finite_imageI)
-apply(simp add: Collect_case_prod_Sigma)
-apply(rule finite_subset)
-apply(rule CPTpre_subsets)
-apply(simp)
-by (simp add: CPTpre_STAR_finite)
-
-
-lemma CPT_finite:
- shows "finite (CPT r s)"
-apply(rule finite_subset)
-apply(rule CPT_CPTpre_subset)
-apply(rule CPTpre_finite)
-done
-*)
-
lemma test2:
assumes "\<forall>v \<in> set vs. flat v \<in> r \<rightarrow> v \<and> flat v \<noteq> []"
- shows "(Stars vs) \<in> CPT (STAR r) (flat (Stars vs))"
+ shows "(Stars vs) \<in> CV (STAR r) (flat (Stars vs))"
using assms
apply(induct vs)
-apply(auto simp add: CPT_def)
+apply(auto simp add: CV_def)
apply(rule CPrf.intros)
apply(simp)
apply(rule CPrf.intros)
apply(simp_all)
-by (metis (no_types, lifting) CPT_def Posix_CPT mem_Collect_eq)
+by (metis (no_types, lifting) CV_def Posix_CV mem_Collect_eq)
end
\ No newline at end of file