diff -r a730a5a0bab9 -r 89e6605c4ca4 thys/Journal/Paper.thy
--- a/thys/Journal/Paper.thy	Tue Jul 23 21:21:49 2019 +0100
+++ b/thys/Journal/Paper.thy	Mon Jul 29 09:37:20 2019 +0100
@@ -132,14 +132,14 @@
 
 
 
-section {* Introduction *}
+section \<open>Introduction\<close>
 
 
-text {*
+text \<open>
 
 Brzozowski \cite{Brzozowski1964} introduced the notion of the {\em
-derivative} @{term "der c r"} of a regular expression @{text r} w.r.t.\
-a character~@{text c}, and showed that it gave a simple solution to the
+derivative} @{term "der c r"} of a regular expression \<open>r\<close> w.r.t.\
+a character~\<open>c\<close>, and showed that it gave a simple solution to the
 problem of matching a string @{term s} with a regular expression @{term
 r}: if the derivative of @{term r} w.r.t.\ (in succession) all the
 characters of the string matches the empty string, then @{term r}
@@ -175,8 +175,7 @@
 into a sequence of tokens, POSIX is the more natural disambiguation
 strategy for what programmers consider basic syntactic building blocks
 in their programs.  These building blocks are often specified by some
-regular expressions, say @{text "r\<^bsub>key\<^esub>"} and @{text
-"r\<^bsub>id\<^esub>"} for recognising keywords and identifiers,
+regular expressions, say \<open>r\<^bsub>key\<^esub>\<close> and \<open>r\<^bsub>id\<^esub>\<close> for recognising keywords and identifiers,
 respectively. There are a few underlying (informal) rules behind
 tokenising a string in a POSIX \cite{POSIX} fashion:
 
@@ -196,23 +195,22 @@
 be longer than no match at all.
 \end{itemize}
 
-\noindent Consider for example a regular expression @{text
-"r\<^bsub>key\<^esub>"} for recognising keywords such as @{text "if"},
-@{text "then"} and so on; and @{text "r\<^bsub>id\<^esub>"}
+\noindent Consider for example a regular expression \<open>r\<^bsub>key\<^esub>\<close> for recognising keywords such as \<open>if\<close>,
+\<open>then\<close> and so on; and \<open>r\<^bsub>id\<^esub>\<close>
 recognising identifiers (say, a single character followed by
 characters or numbers).  Then we can form the regular expression
-@{text "(r\<^bsub>key\<^esub> + r\<^bsub>id\<^esub>)\<^sup>\<star>"}
-and use POSIX matching to tokenise strings, say @{text "iffoo"} and
-@{text "if"}.  For @{text "iffoo"} we obtain by the Longest Match Rule
+\<open>(r\<^bsub>key\<^esub> + r\<^bsub>id\<^esub>)\<^sup>\<star>\<close>
+and use POSIX matching to tokenise strings, say \<open>iffoo\<close> and
+\<open>if\<close>.  For \<open>iffoo\<close> we obtain 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. 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. The
+identifier. For \<open>if\<close> we obtain by the Priority Rule a keyword
+token, not an identifier token---even if \<open>r\<^bsub>id\<^esub>\<close>
+matches also. By the Star Rule we know \<open>(r\<^bsub>key\<^esub> +
+r\<^bsub>id\<^esub>)\<^sup>\<star>\<close> matches \<open>iffoo\<close>,
+respectively \<open>if\<close>, in exactly one `iteration' of the star. The
 Empty String Rule is for cases where, for example, the regular expression 
-@{text "(a\<^sup>\<star>)\<^sup>\<star>"} matches against the
-string @{text "bc"}. Then the longest initial matched substring is the
+\<open>(a\<^sup>\<star>)\<^sup>\<star>\<close> matches against the
+string \<open>bc\<close>. Then the longest initial matched substring is the
 empty string, which is matched by both the whole regular expression
 and the parenthesised subexpression.
 
@@ -225,25 +223,24 @@
 expression matches a string, values encode the information of
 \emph{how} the string is matched by the regular expression---that is,
 which part of the string is matched by which part of the regular
-expression. For this consider again the string @{text "xy"} and
-the regular expression \mbox{@{text "(x + (y + xy))\<^sup>\<star>"}}
+expression. For this consider again the string \<open>xy\<close> and
+the regular expression \mbox{\<open>(x + (y + xy))\<^sup>\<star>\<close>}
 (this time fully parenthesised). We can view this regular expression
-as tree and if the string @{text xy} is matched by two Star
-`iterations', then the @{text x} is matched by the left-most
-alternative in this tree and the @{text y} by the right-left alternative. This
+as tree and if the string \<open>xy\<close> is matched by two Star
+`iterations', then the \<open>x\<close> is matched by the left-most
+alternative in this tree and the \<open>y\<close> by the right-left alternative. This
 suggests to record this matching as
 
 \begin{center}
 @{term "Stars [Left(Char x), Right(Left(Char y))]"}
 \end{center}
 
-\noindent where @{const Stars}, @{text Left}, @{text Right} and @{text
-Char} are constructors for values. @{text Stars} records how many
-iterations were used; @{text Left}, respectively @{text Right}, which
+\noindent where @{const Stars}, \<open>Left\<close>, \<open>Right\<close> and \<open>Char\<close> are constructors for values. \<open>Stars\<close> records how many
+iterations were used; \<open>Left\<close>, respectively \<open>Right\<close>, which
 alternative is used. This `tree view' leads naturally to the idea that
 regular expressions act as types and values as inhabiting those types
 (see, for example, \cite{HosoyaVouillonPierce2005}).  The value for
-matching @{text "xy"} in a single `iteration', i.e.~the POSIX value,
+matching \<open>xy\<close> in a single `iteration', i.e.~the POSIX value,
 would look as follows
 
 \begin{center}
@@ -316,11 +313,11 @@
 
 We extend our results to ??? Bitcoded version??
 
-*}
+\<close>
 
-section {* Preliminaries *}
+section \<open>Preliminaries\<close>
 
-text {* \noindent Strings in Isabelle/HOL are lists of characters with
+text \<open>\noindent Strings in Isabelle/HOL are lists of characters with
 the empty string being represented by the empty list, written @{term
 "[]"}, and list-cons being written as @{term "DUMMY # DUMMY"}. Often
 we use the usual bracket notation for lists also for strings; for
@@ -333,7 +330,7 @@
 inductive datatype:
 
   \begin{center}
-  @{text "r :="}
+  \<open>r :=\<close>
   @{const "ZERO"} $\mid$
   @{const "ONE"} $\mid$
   @{term "CHAR c"} $\mid$
@@ -365,8 +362,8 @@
   DUMMY"} for the concatenation of two languages (it is also list-append for
   strings). We use the star-notation for regular expressions and for
   languages (in the last clause above). The star for languages is defined
-  inductively by two clauses: @{text "(i)"} the empty string being in
-  the star of a language and @{text "(ii)"} if @{term "s\<^sub>1"} is in a
+  inductively by two clauses: \<open>(i)\<close> the empty string being in
+  the star of a language and \<open>(ii)\<close> if @{term "s\<^sub>1"} is in a
   language and @{term "s\<^sub>2"} in the star of this language, then also @{term
   "s\<^sub>1 @ s\<^sub>2"} is in the star of this language. It will also be convenient
   to use the following notion of a \emph{semantic derivative} (or \emph{left
@@ -459,11 +456,11 @@
   \cite{Sulzmann2014} is to append another phase to this algorithm in order
   to calculate a [lexical] value. We will explain the details next.
 
-*}
+\<close>
 
-section {* POSIX Regular Expression Matching\label{posixsec} *}
+section \<open>POSIX Regular Expression Matching\label{posixsec}\<close>
 
-text {* 
+text \<open>
 
   There have been many previous works that use values for encoding 
   \emph{how} a regular expression matches a string.
@@ -473,7 +470,7 @@
   are defined as the inductive datatype
 
   \begin{center}
-  @{text "v :="}
+  \<open>v :=\<close>
   @{const "Void"} $\mid$
   @{term "val.Char c"} $\mid$
   @{term "Left v"} $\mid$
@@ -532,8 +529,8 @@
   \end{center}
 
   \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}.  We require in this rule that every
+  notation @{term "v \<in> set vs"} for indicating that \<open>v\<close> is a
+  member in the list \<open>vs\<close>.  We require in this rule that every
   value in @{term vs} flattens to a non-empty string. The idea is that
   @{term "Stars"}-values satisfy the informal Star Rule (see Introduction)
   where the $^\star$ does not match the empty string unless this is
@@ -549,9 +546,9 @@
   \end{proposition}
 
   \noindent
-  Given a regular expression @{text r} and a string @{text s}, we define the 
-  set of all \emph{Lexical Values} inhabited by @{text r} with the underlying string 
-  being @{text s}:\footnote{Okui and Suzuki refer to our lexical values 
+  Given a regular expression \<open>r\<close> and a string \<open>s\<close>, we define the 
+  set of all \emph{Lexical Values} inhabited by \<open>r\<close> with the underlying string 
+  being \<open>s\<close>:\footnote{Okui and Suzuki refer to our lexical values 
   as \emph{canonical values} in \cite{OkuiSuzuki2010}. The notion of \emph{non-problematic
   values} by Cardelli and Frisch \cite{Frisch2004} is related, but not identical
   to our lexical values.}
@@ -573,7 +570,7 @@
   infinitely many values, but according to our more restricted
   definition only a single value, namely @{thm LV_STAR_ONE_empty}.
 
-  If a regular expression @{text r} matches a string @{text s}, then
+  If a regular expression \<open>r\<close> matches a string \<open>s\<close>, then
   generally the set @{term "LV r s"} is not just a singleton set.  In
   case of POSIX matching the problem is to calculate the unique lexical value
   that satisfies the (informal) POSIX rules from the Introduction.
@@ -582,9 +579,9 @@
   path from the left to the right involving @{term
   derivatives}/@{const nullable} is the first phase of the algorithm
   (calculating successive \Brz's derivatives) and @{const
-  mkeps}/@{text inj}, the path from right to left, the second
+  mkeps}/\<open>inj\<close>, the path from right to left, the second
   phase. This picture shows the steps required when a regular
-  expression, say @{text "r\<^sub>1"}, matches the string @{term
+  expression, say \<open>r\<^sub>1\<close>, matches the string @{term
   "[a,b,c]"}. We first build the three derivatives (according to
   @{term a}, @{term b} and @{term c}). We then use @{const nullable}
   to find out whether the resulting derivative regular expression
@@ -609,11 +606,11 @@
 \node (v4) [below=of r4]{@{term "v\<^sub>4"}};
 \draw[->,line width=1mm](r4) -- (v4);
 \node (v3) [left=of v4] {@{term "v\<^sub>3"}};
-\draw[->,line width=1mm](v4)--(v3) node[below,midway] {@{text "inj r\<^sub>3 c"}};
+\draw[->,line width=1mm](v4)--(v3) node[below,midway] {\<open>inj r\<^sub>3 c\<close>};
 \node (v2) [left=of v3]{@{term "v\<^sub>2"}};
-\draw[->,line width=1mm](v3)--(v2) node[below,midway] {@{text "inj r\<^sub>2 b"}};
+\draw[->,line width=1mm](v3)--(v2) node[below,midway] {\<open>inj r\<^sub>2 b\<close>};
 \node (v1) [left=of v2] {@{term "v\<^sub>1"}};
-\draw[->,line width=1mm](v2)--(v1) node[below,midway] {@{text "inj r\<^sub>1 a"}};
+\draw[->,line width=1mm](v2)--(v1) node[below,midway] {\<open>inj r\<^sub>1 a\<close>};
 \draw (r4) node[anchor=north west] {\;\raisebox{-8mm}{@{term "mkeps"}}};
 \end{tikzpicture}
 \end{center}
@@ -647,8 +644,7 @@
   makes some subtle choices leading to a POSIX value: for example if an
   alternative regular expression, say @{term "ALT r\<^sub>1 r\<^sub>2"}, can
   match the empty string and furthermore @{term "r\<^sub>1"} can match the
-  empty string, then we return a @{text Left}-value. The @{text
-  Right}-value will only be returned if @{term "r\<^sub>1"} cannot match the empty
+  empty string, then we return a \<open>Left\<close>-value. The \<open>Right\<close>-value will only be returned if @{term "r\<^sub>1"} cannot match the empty
   string.
 
   The most interesting idea from Sulzmann and Lu \cite{Sulzmann2014} is
@@ -690,25 +686,25 @@
   might be instructive to look first at the three sequence cases (clauses
   \textit{(4)} -- \textit{(6)}). In each case we need to construct an ``injected value'' for
   @{term "SEQ r\<^sub>1 r\<^sub>2"}. This must be a value of the form @{term
-  "Seq DUMMY DUMMY"}\,. Recall the clause of the @{text derivative}-function
+  "Seq DUMMY DUMMY"}\,. Recall the clause of the \<open>derivative\<close>-function
   for sequence regular expressions:
 
   \begin{center}
   @{thm (lhs) der.simps(5)[of c "r\<^sub>1" "r\<^sub>2"]} $\dn$ @{thm (rhs) der.simps(5)[of c "r\<^sub>1" "r\<^sub>2"]}
   \end{center}
 
-  \noindent Consider first the @{text "else"}-branch where the derivative is @{term
+  \noindent Consider first the \<open>else\<close>-branch where the derivative is @{term
   "SEQ (der c r\<^sub>1) r\<^sub>2"}. The corresponding value must therefore
   be of the form @{term "Seq v\<^sub>1 v\<^sub>2"}, which matches the left-hand
-  side in clause~\textit{(4)} of @{term inj}. In the @{text "if"}-branch the derivative is an
+  side in clause~\textit{(4)} of @{term inj}. In the \<open>if\<close>-branch the derivative is an
   alternative, namely @{term "ALT (SEQ (der c r\<^sub>1) r\<^sub>2) (der c
-  r\<^sub>2)"}. This means we either have to consider a @{text Left}- or
-  @{text Right}-value. In case of the @{text Left}-value we know further it
+  r\<^sub>2)"}. This means we either have to consider a \<open>Left\<close>- or
+  \<open>Right\<close>-value. In case of the \<open>Left\<close>-value we know further it
   must be a value for a sequence regular expression. Therefore the pattern
   we match in the clause \textit{(5)} is @{term "Left (Seq v\<^sub>1 v\<^sub>2)"},
   while in \textit{(6)} it is just @{term "Right v\<^sub>2"}. One more interesting
   point is in the right-hand side of clause \textit{(6)}: since in this case the
-  regular expression @{text "r\<^sub>1"} does not ``contribute'' to
+  regular expression \<open>r\<^sub>1\<close> does not ``contribute'' to
   matching the string, that means it only matches the empty string, we need to
   call @{const mkeps} in order to construct a value for how @{term "r\<^sub>1"}
   can match this empty string. A similar argument applies for why we can
@@ -728,7 +724,7 @@
   value has a prepended character @{term c}; the second part shows that
   the underlying string of an @{const mkeps}-value is always the empty
   string (given the regular expression is nullable since otherwise
-  @{text mkeps} might not be defined).
+  \<open>mkeps\<close> might not be defined).
 
   \begin{lemma}\mbox{}\smallskip\\\label{Prf_injval_flat}
   \begin{tabular}{ll}
@@ -743,16 +739,16 @@
   an induction on @{term r}. There are no interesting cases.\qed
   \end{proof}
 
-  Having defined the @{const mkeps} and @{text inj} function we can extend
+  Having defined the @{const mkeps} and \<open>inj\<close> function we can extend
   \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}
   \begin{tabular}{lcl}
   @{thm (lhs) lexer.simps(1)} & $\dn$ & @{thm (rhs) lexer.simps(1)}\\
-  @{thm (lhs) lexer.simps(2)} & $\dn$ & @{text "case"} @{term "lexer (der c r) s"} @{text of}\\
-                     & & \phantom{$|$} @{term "None"}  @{text "\<Rightarrow>"} @{term None}\\
-                     & & $|$ @{term "Some v"} @{text "\<Rightarrow>"} @{term "Some (injval r c v)"}                          
+  @{thm (lhs) lexer.simps(2)} & $\dn$ & \<open>case\<close> @{term "lexer (der c r) s"} \<open>of\<close>\\
+                     & & \phantom{$|$} @{term "None"}  \<open>\<Rightarrow>\<close> @{term None}\\
+                     & & $|$ @{term "Some v"} \<open>\<Rightarrow>\<close> @{term "Some (injval r c v)"}                          
   \end{tabular}
   \end{center}
 
@@ -784,24 +780,24 @@
   \begin{figure}[t]
   \begin{center}
   \begin{tabular}{c}
-  @{thm[mode=Axiom] Posix.intros(1)}@{text "P"}@{term "ONE"} \qquad
-  @{thm[mode=Axiom] Posix.intros(2)}@{text "P"}@{term "c"}\medskip\\
-  @{thm[mode=Rule] Posix.intros(3)[of "s" "r\<^sub>1" "v" "r\<^sub>2"]}@{text "P+L"}\qquad
-  @{thm[mode=Rule] Posix.intros(4)[of "s" "r\<^sub>2" "v" "r\<^sub>1"]}@{text "P+R"}\medskip\\
+  @{thm[mode=Axiom] Posix.intros(1)}\<open>P\<close>@{term "ONE"} \qquad
+  @{thm[mode=Axiom] Posix.intros(2)}\<open>P\<close>@{term "c"}\medskip\\
+  @{thm[mode=Rule] Posix.intros(3)[of "s" "r\<^sub>1" "v" "r\<^sub>2"]}\<open>P+L\<close>\qquad
+  @{thm[mode=Rule] Posix.intros(4)[of "s" "r\<^sub>2" "v" "r\<^sub>1"]}\<open>P+R\<close>\medskip\\
   $\mprset{flushleft}
    \inferrule
    {@{thm (prem 1) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]} \qquad
     @{thm (prem 2) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]} \\\\
     @{thm (prem 3) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]}}
-   {@{thm (concl) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]}}$@{text "PS"}\\
-  @{thm[mode=Axiom] Posix.intros(7)}@{text "P[]"}\medskip\\
+   {@{thm (concl) Posix.intros(5)[of "s\<^sub>1" "r\<^sub>1" "v\<^sub>1" "s\<^sub>2" "r\<^sub>2" "v\<^sub>2"]}}$\<open>PS\<close>\\
+  @{thm[mode=Axiom] Posix.intros(7)}\<open>P[]\<close>\medskip\\
   $\mprset{flushleft}
    \inferrule
    {@{thm (prem 1) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \qquad
     @{thm (prem 2) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \qquad
     @{thm (prem 3) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]} \\\\
     @{thm (prem 4) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]}}
-   {@{thm (concl) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]}}$@{text "P\<star>"}
+   {@{thm (concl) Posix.intros(6)[of "s\<^sub>1" "r" "v" "s\<^sub>2" "vs"]}}$\<open>P\<star>\<close>
   \end{tabular}
   \end{center}
   \caption{Our inductive definition of POSIX values.}\label{POSIXrules}
@@ -825,13 +821,12 @@
   \noindent
   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
+  rules \<open>P+L\<close> and \<open>P+R\<close> where the POSIX value for a string
   and an alternative regular expression, that is @{term "(s, ALT r\<^sub>1 r\<^sub>2)"},
-  is specified---it is always a @{text "Left"}-value, \emph{except} when the
+  is specified---it is always a \<open>Left\<close>-value, \emph{except} when the
   string to be matched is not in the language of @{term "r\<^sub>1"}; only then it
-  is a @{text Right}-value (see the side-condition in @{text "P+R"}).
-  Interesting is also the rule for sequence regular expressions (@{text
-  "PS"}). The first two premises state that @{term "v\<^sub>1"} and @{term "v\<^sub>2"}
+  is a \<open>Right\<close>-value (see the side-condition in \<open>P+R\<close>).
+  Interesting is also the rule for sequence regular expressions (\<open>PS\<close>). The first two premises state that @{term "v\<^sub>1"} and @{term "v\<^sub>2"}
   are the POSIX values for @{term "(s\<^sub>1, r\<^sub>1)"} and @{term "(s\<^sub>2, r\<^sub>2)"}
   respectively. Consider now the third premise and note that the POSIX value
   of this rule should match the string \mbox{@{term "s\<^sub>1 @ s\<^sub>2"}}. According to the
@@ -841,21 +836,20 @@
   \emph{exist} an @{term "s\<^sub>3"} and @{term "s\<^sub>4"} such that @{term "s\<^sub>2"}
   can be split up into a non-empty string @{term "s\<^sub>3"} and a possibly empty
   string @{term "s\<^sub>4"}. Moreover the longer string @{term "s\<^sub>1 @ s\<^sub>3"} can be
-  matched by @{text "r\<^sub>1"} and the shorter @{term "s\<^sub>4"} can still be
+  matched by \<open>r\<^sub>1\<close> and the shorter @{term "s\<^sub>4"} can still be
   matched by @{term "r\<^sub>2"}. In this case @{term "s\<^sub>1"} would \emph{not} be the
   longest initial split of \mbox{@{term "s\<^sub>1 @ s\<^sub>2"}} and therefore @{term "Seq v\<^sub>1
   v\<^sub>2"} cannot be a POSIX value for @{term "(s\<^sub>1 @ s\<^sub>2, SEQ r\<^sub>1 r\<^sub>2)"}. 
   The main point is that our side-condition ensures the Longest 
   Match Rule is satisfied.
 
-  A similar condition is imposed on the POSIX value in the @{text
-  "P\<star>"}-rule. Also there we want that @{term "s\<^sub>1"} is the longest initial
+  A similar condition is imposed on the POSIX value in the \<open>P\<star>\<close>-rule. Also there we want that @{term "s\<^sub>1"} is the longest initial
   split of @{term "s\<^sub>1 @ s\<^sub>2"} and furthermore the corresponding value
   @{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. Indeed we can show that our POSIX values
-  are lexical values which exclude those @{text Stars} that contain subvalues 
+  are lexical values which exclude those \<open>Stars\<close> that contain subvalues 
   that flatten to the empty string.
 
   \begin{lemma}\label{LVposix}
@@ -879,7 +873,7 @@
   \end{proof}
 
   \noindent
-  The central lemma for our POSIX relation is that the @{text inj}-function
+  The central lemma for our POSIX relation is that the \<open>inj\<close>-function
   preserves POSIX values.
 
   \begin{lemma}\label{Posix2}
@@ -887,17 +881,17 @@
   \end{lemma}
 
   \begin{proof}
-  By induction on @{text r}. We explain two cases.
+  By induction on \<open>r\<close>. We explain two cases.
 
   \begin{itemize}
   \item[$\bullet$] Case @{term "r = ALT r\<^sub>1 r\<^sub>2"}. There are
-  two subcases, namely @{text "(a)"} \mbox{@{term "v = Left v'"}} and @{term
-  "s \<in> der c r\<^sub>1 \<rightarrow> v'"}; and @{text "(b)"} @{term "v = Right v'"}, @{term
-  "s \<notin> L (der c r\<^sub>1)"} and @{term "s \<in> der c r\<^sub>2 \<rightarrow> v'"}. In @{text "(a)"} we
+  two subcases, namely \<open>(a)\<close> \mbox{@{term "v = Left v'"}} and @{term
+  "s \<in> der c r\<^sub>1 \<rightarrow> v'"}; and \<open>(b)\<close> @{term "v = Right v'"}, @{term
+  "s \<notin> L (der c r\<^sub>1)"} and @{term "s \<in> der c r\<^sub>2 \<rightarrow> v'"}. In \<open>(a)\<close> we
   know @{term "s \<in> der c r\<^sub>1 \<rightarrow> v'"}, from which we can infer @{term "(c # s)
   \<in> r\<^sub>1 \<rightarrow> injval r\<^sub>1 c v'"} by induction hypothesis and hence @{term "(c #
   s) \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> injval (ALT r\<^sub>1 r\<^sub>2) c (Left v')"} as needed. Similarly
-  in subcase @{text "(b)"} where, however, in addition we have to use
+  in subcase \<open>(b)\<close> where, however, in addition we have to use
   Proposition~\ref{derprop}(2) in order to infer @{term "c # s \<notin> L r\<^sub>1"} from @{term
   "s \<notin> L (der c r\<^sub>1)"}.\smallskip
 
@@ -905,13 +899,13 @@
   
   \begin{quote}
   \begin{description}
-  \item[@{text "(a)"}] @{term "v = Left (Seq v\<^sub>1 v\<^sub>2)"} and @{term "nullable r\<^sub>1"} 
-  \item[@{text "(b)"}] @{term "v = Right v\<^sub>1"} and @{term "nullable r\<^sub>1"} 
-  \item[@{text "(c)"}] @{term "v = Seq v\<^sub>1 v\<^sub>2"} and @{term "\<not> nullable r\<^sub>1"} 
+  \item[\<open>(a)\<close>] @{term "v = Left (Seq v\<^sub>1 v\<^sub>2)"} and @{term "nullable r\<^sub>1"} 
+  \item[\<open>(b)\<close>] @{term "v = Right v\<^sub>1"} and @{term "nullable r\<^sub>1"} 
+  \item[\<open>(c)\<close>] @{term "v = Seq v\<^sub>1 v\<^sub>2"} and @{term "\<not> nullable r\<^sub>1"} 
   \end{description}
   \end{quote}
 
-  \noindent For @{text "(a)"} we know @{term "s\<^sub>1 \<in> der c r\<^sub>1 \<rightarrow> v\<^sub>1"} and
+  \noindent For \<open>(a)\<close> we know @{term "s\<^sub>1 \<in> der c r\<^sub>1 \<rightarrow> v\<^sub>1"} and
   @{term "s\<^sub>2 \<in> r\<^sub>2 \<rightarrow> v\<^sub>2"} as well as
   %
   \[@{term "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s\<^sub>2 \<and> s\<^sub>1 @ s\<^sub>3 \<in> L (der c r\<^sub>1) \<and> s\<^sub>4 \<in> L r\<^sub>2)"}\]
@@ -920,12 +914,12 @@
   %
   \[@{term "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s\<^sub>2 \<and> (c # s\<^sub>1) @ s\<^sub>3 \<in> L r\<^sub>1 \<and> s\<^sub>4 \<in> L r\<^sub>2)"}\]
 
-  \noindent We can use the induction hypothesis for @{text "r\<^sub>1"} to obtain
+  \noindent We can use the induction hypothesis for \<open>r\<^sub>1\<close> to obtain
   @{term "(c # s\<^sub>1) \<in> r\<^sub>1 \<rightarrow> injval r\<^sub>1 c v\<^sub>1"}. Putting this all together allows us to infer
-  @{term "((c # s\<^sub>1) @ s\<^sub>2) \<in> SEQ r\<^sub>1 r\<^sub>2 \<rightarrow> Seq (injval r\<^sub>1 c v\<^sub>1) v\<^sub>2"}. The case @{text "(c)"}
+  @{term "((c # s\<^sub>1) @ s\<^sub>2) \<in> SEQ r\<^sub>1 r\<^sub>2 \<rightarrow> Seq (injval r\<^sub>1 c v\<^sub>1) v\<^sub>2"}. The case \<open>(c)\<close>
   is similar.
 
-  For @{text "(b)"} we know @{term "s \<in> der c r\<^sub>2 \<rightarrow> v\<^sub>1"} and 
+  For \<open>(b)\<close> we know @{term "s \<in> der c r\<^sub>2 \<rightarrow> v\<^sub>1"} and 
   @{term "s\<^sub>1 @ s\<^sub>2 \<notin> L (SEQ (der c r\<^sub>1) r\<^sub>2)"}. From the former
   we have @{term "(c # s) \<in> r\<^sub>2 \<rightarrow> (injval r\<^sub>2 c v\<^sub>1)"} by induction hypothesis
   for @{term "r\<^sub>2"}. From the latter we can infer
@@ -979,11 +973,11 @@
   In the next section we show that our specification coincides with
   another one given by Okui and Suzuki using a different technique.
 
-*}
+\<close>
 
-section {* Ordering of Values according to Okui and Suzuki*}
+section \<open>Ordering of Values according to Okui and Suzuki\<close>
 
-text {*
+text \<open>
   
   While in the previous section we have defined POSIX values directly
   in terms of a ternary relation (see inference rules in Figure~\ref{POSIXrules}),
@@ -1014,54 +1008,51 @@
   \end{center}
 
   \noindent
-  At position @{text "[0,1]"} of this value is the
-  subvalue @{text "Char y"} and at position @{text "[1]"} the
+  At position \<open>[0,1]\<close> of this value is the
+  subvalue \<open>Char y\<close> and at position \<open>[1]\<close> the
   subvalue @{term "Char z"}.  At the `root' position, or empty list
-  @{term "[]"}, is the whole value @{term v}. Positions such as @{text
-  "[0,1,0]"} or @{text "[2]"} are outside of @{text
-  v}. If it exists, the subvalue of @{term v} at a position @{text
-  p}, written @{term "at v p"}, can be recursively defined by
+  @{term "[]"}, is the whole value @{term v}. Positions such as \<open>[0,1,0]\<close> or \<open>[2]\<close> are outside of \<open>v\<close>. If it exists, the subvalue of @{term v} at a position \<open>p\<close>, written @{term "at v p"}, can be recursively defined by
   
   \begin{center}
   \begin{tabular}{r@ {\hspace{0mm}}lcl}
-  @{term v} &  @{text "\<downharpoonleft>\<^bsub>[]\<^esub>"} & @{text "\<equiv>"}& @{thm (rhs) at.simps(1)}\\
-  @{term "Left v"} & @{text "\<downharpoonleft>\<^bsub>0::ps\<^esub>"} & @{text "\<equiv>"}& @{thm (rhs) at.simps(2)}\\
-  @{term "Right v"} & @{text "\<downharpoonleft>\<^bsub>1::ps\<^esub>"} & @{text "\<equiv>"} & 
+  @{term v} &  \<open>\<downharpoonleft>\<^bsub>[]\<^esub>\<close> & \<open>\<equiv>\<close>& @{thm (rhs) at.simps(1)}\\
+  @{term "Left v"} & \<open>\<downharpoonleft>\<^bsub>0::ps\<^esub>\<close> & \<open>\<equiv>\<close>& @{thm (rhs) at.simps(2)}\\
+  @{term "Right v"} & \<open>\<downharpoonleft>\<^bsub>1::ps\<^esub>\<close> & \<open>\<equiv>\<close> & 
   @{thm (rhs) at.simps(3)[simplified Suc_0_fold]}\\
-  @{term "Seq v\<^sub>1 v\<^sub>2"} & @{text "\<downharpoonleft>\<^bsub>0::ps\<^esub>"} & @{text "\<equiv>"} & 
+  @{term "Seq v\<^sub>1 v\<^sub>2"} & \<open>\<downharpoonleft>\<^bsub>0::ps\<^esub>\<close> & \<open>\<equiv>\<close> & 
   @{thm (rhs) at.simps(4)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} \\
-  @{term "Seq v\<^sub>1 v\<^sub>2"} & @{text "\<downharpoonleft>\<^bsub>1::ps\<^esub>"}
-  & @{text "\<equiv>"} & 
+  @{term "Seq v\<^sub>1 v\<^sub>2"} & \<open>\<downharpoonleft>\<^bsub>1::ps\<^esub>\<close>
+  & \<open>\<equiv>\<close> & 
   @{thm (rhs) at.simps(5)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2", simplified Suc_0_fold]} \\
-  @{term "Stars vs"} & @{text "\<downharpoonleft>\<^bsub>n::ps\<^esub>"} & @{text "\<equiv>"}& @{thm (rhs) at.simps(6)}\\
+  @{term "Stars vs"} & \<open>\<downharpoonleft>\<^bsub>n::ps\<^esub>\<close> & \<open>\<equiv>\<close>& @{thm (rhs) at.simps(6)}\\
   \end{tabular} 
   \end{center}
 
   \noindent In the last clause we use Isabelle's notation @{term "vs ! n"} for the
-  @{text n}th element in a list.  The set of positions inside a value @{text v},
+  \<open>n\<close>th element in a list.  The set of positions inside a value \<open>v\<close>,
   written @{term "Pos v"}, is given by 
 
   \begin{center}
   \begin{tabular}{lcl}
-  @{thm (lhs) Pos.simps(1)} & @{text "\<equiv>"} & @{thm (rhs) Pos.simps(1)}\\
-  @{thm (lhs) Pos.simps(2)} & @{text "\<equiv>"} & @{thm (rhs) Pos.simps(2)}\\
-  @{thm (lhs) Pos.simps(3)} & @{text "\<equiv>"} & @{thm (rhs) Pos.simps(3)}\\
-  @{thm (lhs) Pos.simps(4)} & @{text "\<equiv>"} & @{thm (rhs) Pos.simps(4)}\\
+  @{thm (lhs) Pos.simps(1)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(1)}\\
+  @{thm (lhs) Pos.simps(2)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(2)}\\
+  @{thm (lhs) Pos.simps(3)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(3)}\\
+  @{thm (lhs) Pos.simps(4)} & \<open>\<equiv>\<close> & @{thm (rhs) Pos.simps(4)}\\
   @{thm (lhs) Pos.simps(5)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}
-  & @{text "\<equiv>"} 
+  & \<open>\<equiv>\<close> 
   & @{thm (rhs) Pos.simps(5)[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]}\\
-  @{thm (lhs) Pos_stars} & @{text "\<equiv>"} & @{thm (rhs) Pos_stars}\\
+  @{thm (lhs) Pos_stars} & \<open>\<equiv>\<close> & @{thm (rhs) Pos_stars}\\
   \end{tabular}
   \end{center}
 
   \noindent 
-  whereby @{text len} in the last clause stands for the length of a list. Clearly
+  whereby \<open>len\<close> in the last clause stands for the length of a list. Clearly
   for every position inside a value there exists a subvalue at that position.
  
 
   To help understanding the ordering of Okui and Suzuki, consider again 
   the earlier value
-  @{text v} and compare it with the following @{text w}:
+  \<open>v\<close> and compare it with the following \<open>w\<close>:
 
   \begin{center}
   \begin{tabular}{l}
@@ -1070,16 +1061,16 @@
   \end{tabular}
   \end{center}
 
-  \noindent Both values match the string @{text "xyz"}, that means if
+  \noindent Both values match the string \<open>xyz\<close>, that means if
   we flatten these values at their respective root position, we obtain
-  @{text "xyz"}. However, at position @{text "[0]"}, @{text v} matches
-  @{text xy} whereas @{text w} matches only the shorter @{text x}. So
-  according to the Longest Match Rule, we should prefer @{text v},
-  rather than @{text w} as POSIX value for string @{text xyz} (and
+  \<open>xyz\<close>. However, at position \<open>[0]\<close>, \<open>v\<close> matches
+  \<open>xy\<close> whereas \<open>w\<close> matches only the shorter \<open>x\<close>. So
+  according to the Longest Match Rule, we should prefer \<open>v\<close>,
+  rather than \<open>w\<close> as POSIX value for string \<open>xyz\<close> (and
   corresponding regular expression). In order to
   formalise this idea, Okui and Suzuki introduce a measure for
-  subvalues at position @{text p}, called the \emph{norm} of @{text v}
-  at position @{text p}. We can define this measure in Isabelle as an
+  subvalues at position \<open>p\<close>, called the \emph{norm} of \<open>v\<close>
+  at position \<open>p\<close>. We can define this measure in Isabelle as an
   integer as follows
   
   \begin{center}
@@ -1087,10 +1078,10 @@
   \end{center}
 
   \noindent where we take the length of the flattened value at
-  position @{text p}, provided the position is inside @{text v}; if
-  not, then the norm is @{text "-1"}. The default for outside
+  position \<open>p\<close>, provided the position is inside \<open>v\<close>; if
+  not, then the norm is \<open>-1\<close>. The default for outside
   positions is crucial for the POSIX requirement of preferring a
-  @{text Left}-value over a @{text Right}-value (if they can match the
+  \<open>Left\<close>-value over a \<open>Right\<close>-value (if they can match the
   same string---see the Priority Rule from the Introduction). For this
   consider
 
@@ -1098,17 +1089,17 @@
   @{term "v == Left (Char x)"} \qquad and \qquad @{term "w == Right (Char x)"}
   \end{center}
 
-  \noindent Both values match @{text x}. At position @{text "[0]"}
-  the norm of @{term v} is @{text 1} (the subvalue matches @{text x}),
-  but the norm of @{text w} is @{text "-1"} (the position is outside
-  @{text w} according to how we defined the `inside' positions of
-  @{text Left}- and @{text Right}-values).  Of course at position
-  @{text "[1]"}, the norms @{term "pflat_len v [1]"} and @{term
+  \noindent Both values match \<open>x\<close>. At position \<open>[0]\<close>
+  the norm of @{term v} is \<open>1\<close> (the subvalue matches \<open>x\<close>),
+  but the norm of \<open>w\<close> is \<open>-1\<close> (the position is outside
+  \<open>w\<close> according to how we defined the `inside' positions of
+  \<open>Left\<close>- and \<open>Right\<close>-values).  Of course at position
+  \<open>[1]\<close>, the norms @{term "pflat_len v [1]"} and @{term
   "pflat_len w [1]"} are reversed, but the point is that subvalues
   will be analysed according to lexicographically ordered
-  positions. According to this ordering, the position @{text "[0]"}
-  takes precedence over @{text "[1]"} and thus also @{text v} will be 
-  preferred over @{text w}.  The lexicographic ordering of positions, written
+  positions. According to this ordering, the position \<open>[0]\<close>
+  takes precedence over \<open>[1]\<close> and thus also \<open>v\<close> will be 
+  preferred over \<open>w\<close>.  The lexicographic ordering of positions, written
   @{term "DUMMY \<sqsubset>lex DUMMY"}, can be conveniently formalised
   by three inference rules
 
@@ -1123,18 +1114,18 @@
 
   With the norm and lexicographic order in place,
   we can state the key definition of Okui and Suzuki
-  \cite{OkuiSuzuki2010}: a value @{term "v\<^sub>1"} is \emph{smaller at position @{text p}} than
+  \cite{OkuiSuzuki2010}: a value @{term "v\<^sub>1"} is \emph{smaller at position \<open>p\<close>} than
   @{term "v\<^sub>2"}, written @{term "v\<^sub>1 \<sqsubset>val p v\<^sub>2"}, 
-  if and only if  $(i)$ the norm at position @{text p} is
+  if and only if  $(i)$ the norm at position \<open>p\<close> is
   greater in @{term "v\<^sub>1"} (that is the string @{term "flat (at v\<^sub>1 p)"} is longer 
   than @{term "flat (at v\<^sub>2 p)"}) and $(ii)$ all subvalues at 
   positions that are inside @{term "v\<^sub>1"} or @{term "v\<^sub>2"} and that are
-  lexicographically smaller than @{text p}, we have the same norm, namely
+  lexicographically smaller than \<open>p\<close>, we have the same norm, namely
 
  \begin{center}
  \begin{tabular}{c}
  @{thm (lhs) PosOrd_def[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2"]} 
- @{text "\<equiv>"} 
+ \<open>\<equiv>\<close> 
  $\begin{cases}
  (i) & @{term "pflat_len v\<^sub>1 p > pflat_len v\<^sub>2 p"}   \quad\text{and}\smallskip \\
  (ii) & @{term "(\<forall>q \<in> Pos v\<^sub>1 \<union> Pos v\<^sub>2. q \<sqsubset>lex p --> pflat_len v\<^sub>1 q = pflat_len v\<^sub>2 q)"}
@@ -1142,11 +1133,9 @@
  \end{tabular}
  \end{center}
 
- \noindent The position @{text p} in this definition acts as the
-  \emph{first distinct position} of @{text "v\<^sub>1"} and @{text
-  "v\<^sub>2"}, where both values match strings of different length
-  \cite{OkuiSuzuki2010}.  Since at @{text p} the values @{text
-  "v\<^sub>1"} and @{text "v\<^sub>2"} match different strings, the
+ \noindent The position \<open>p\<close> in this definition acts as the
+  \emph{first distinct position} of \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close>, where both values match strings of different length
+  \cite{OkuiSuzuki2010}.  Since at \<open>p\<close> the values \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close> match different strings, the
   ordering is irreflexive. Derived from the definition above
   are the following two orderings:
   
@@ -1168,11 +1157,8 @@
  @{thm [mode=IfThen] PosOrd_trans[where ?v1.0="v\<^sub>1" and ?v2.0="v\<^sub>2" and ?v3.0="v\<^sub>3"]} 
  \end{lemma}
 
- \begin{proof} From the assumption we obtain two positions @{text p}
- and @{text q}, where the values @{text "v\<^sub>1"} and @{text
- "v\<^sub>2"} (respectively @{text "v\<^sub>2"} and @{text
- "v\<^sub>3"}) are `distinct'.  Since @{text
- "\<prec>\<^bsub>lex\<^esub>"} is trichotomous, we need to consider
+ \begin{proof} From the assumption we obtain two positions \<open>p\<close>
+ and \<open>q\<close>, where the values \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close> (respectively \<open>v\<^sub>2\<close> and \<open>v\<^sub>3\<close>) are `distinct'.  Since \<open>\<prec>\<^bsub>lex\<^esub>\<close> is trichotomous, we need to consider
  three cases, namely @{term "p = q"}, @{term "p \<sqsubset>lex q"} and
  @{term "q \<sqsubset>lex p"}. Let us look at the first case.  Clearly
  @{term "pflat_len v\<^sub>2 p < pflat_len v\<^sub>1 p"} and @{term
@@ -1184,7 +1170,7 @@
  v\<^sub>1"}, then we can infer from the first assumption that @{term
  "pflat_len v\<^sub>1 p' = pflat_len v\<^sub>2 p'"}.  But this means
  that @{term "p'"} must be in @{term "Pos v\<^sub>2"} too (the norm
- cannot be @{text "-1"} given @{term "p' \<in> Pos v\<^sub>1"}).  
+ cannot be \<open>-1\<close> given @{term "p' \<in> Pos v\<^sub>1"}).  
  Hence we can use the second assumption and
  infer @{term "pflat_len v\<^sub>2 p' = pflat_len v\<^sub>3 p'"},
  which concludes this case with @{term "v\<^sub>1 :\<sqsubset>val
@@ -1193,15 +1179,15 @@
 
  \noindent 
  The proof for $\preccurlyeq$ is similar and omitted.
- It is also straightforward to show that @{text "\<prec>"} and
+ It is also straightforward to show that \<open>\<prec>\<close> and
  $\preccurlyeq$ are partial orders.  Okui and Suzuki furthermore show that they
  are linear orderings for lexical values \cite{OkuiSuzuki2010} of a given
  regular expression and given string, but we have not formalised this in Isabelle. It is
  not essential for our results. What we are going to show below is
- that for a given @{text r} and @{text s}, the orderings have a unique
+ that for a given \<open>r\<close> and \<open>s\<close>, the orderings have a unique
  minimal element on the set @{term "LV r s"}, which is the POSIX value
  we defined in the previous section. We start with two properties that
- show how the length of a flattened value relates to the @{text "\<prec>"}-ordering.
+ show how the length of a flattened value relates to the \<open>\<prec>\<close>-ordering.
 
  \begin{proposition}\mbox{}\smallskip\\\label{ordlen}
  \begin{tabular}{@ {}ll}
@@ -1259,8 +1245,7 @@
 
   \noindent One might prefer that statements \textit{(4)} and \textit{(5)} 
   (respectively \textit{(6)} and \textit{(7)})
-  are combined into a single \textit{iff}-statement (like the ones for @{text
-  Left} and @{text Right}). Unfortunately this cannot be done easily: such
+  are combined into a single \textit{iff}-statement (like the ones for \<open>Left\<close> and \<open>Right\<close>). Unfortunately this cannot be done easily: such
   a single statement would require an additional assumption about the
   two values @{term "Seq v\<^sub>1 v\<^sub>2"} and @{term "Seq w\<^sub>1 w\<^sub>2"}
   being inhabited by the same regular expression. The
@@ -1271,10 +1256,8 @@
  
 
   Next we establish how Okui and Suzuki's orderings relate to our
-  definition of POSIX values.  Given a @{text POSIX} value @{text "v\<^sub>1"}
-  for @{text r} and @{text s}, then any other lexical value @{text
-  "v\<^sub>2"} in @{term "LV r s"} is greater or equal than @{text
-  "v\<^sub>1"}, namely:
+  definition of POSIX values.  Given a \<open>POSIX\<close> value \<open>v\<^sub>1\<close>
+  for \<open>r\<close> and \<open>s\<close>, then any other lexical value \<open>v\<^sub>2\<close> in @{term "LV r s"} is greater or equal than \<open>v\<^sub>1\<close>, namely:
 
 
   \begin{theorem}\label{orderone}
@@ -1283,53 +1266,53 @@
 
   \begin{proof} By induction on our POSIX rules. By
   Theorem~\ref{posixdeterm} and the definition of @{const LV}, it is clear
-  that @{text "v\<^sub>1"} and @{text "v\<^sub>2"} have the same
+  that \<open>v\<^sub>1\<close> and \<open>v\<^sub>2\<close> have the same
   underlying string @{term s}.  The three base cases are
   straightforward: for example for @{term "v\<^sub>1 = Void"}, we have
   that @{term "v\<^sub>2 \<in> LV ONE []"} must also be of the form
   \mbox{@{term "v\<^sub>2 = Void"}}. Therefore we have @{term
   "v\<^sub>1 :\<sqsubseteq>val v\<^sub>2"}.  The inductive cases for
-  @{text r} being of the form @{term "ALT r\<^sub>1 r\<^sub>2"} and
+  \<open>r\<close> being of the form @{term "ALT r\<^sub>1 r\<^sub>2"} and
   @{term "SEQ r\<^sub>1 r\<^sub>2"} are as follows:
 
 
   \begin{itemize} 
 
-  \item[$\bullet$] Case @{text "P+L"} with @{term "s \<in> (ALT r\<^sub>1 r\<^sub>2)
+  \item[$\bullet$] Case \<open>P+L\<close> with @{term "s \<in> (ALT r\<^sub>1 r\<^sub>2)
   \<rightarrow> (Left w\<^sub>1)"}: In this case the value 
   @{term "v\<^sub>2"} is either of the
   form @{term "Left w\<^sub>2"} or @{term "Right w\<^sub>2"}. In the
   latter case we can immediately conclude with \mbox{@{term "v\<^sub>1
-  :\<sqsubseteq>val v\<^sub>2"}} since a @{text Left}-value with the
-  same underlying string @{text s} is always smaller than a
-  @{text Right}-value by Proposition~\ref{ordintros}\textit{(1)}.  
+  :\<sqsubseteq>val v\<^sub>2"}} since a \<open>Left\<close>-value with the
+  same underlying string \<open>s\<close> is always smaller than a
+  \<open>Right\<close>-value by Proposition~\ref{ordintros}\textit{(1)}.  
   In the former case we have @{term "w\<^sub>2
   \<in> LV r\<^sub>1 s"} and can use the induction hypothesis to infer
   @{term "w\<^sub>1 :\<sqsubseteq>val w\<^sub>2"}. Because @{term
   "w\<^sub>1"} and @{term "w\<^sub>2"} have the same underlying string
-  @{text s}, we can conclude with @{term "Left w\<^sub>1
+  \<open>s\<close>, we can conclude with @{term "Left w\<^sub>1
   :\<sqsubseteq>val Left w\<^sub>2"} using
   Proposition~\ref{ordintros}\textit{(2)}.\smallskip
 
-  \item[$\bullet$] Case @{text "P+R"} with @{term "s \<in> (ALT r\<^sub>1 r\<^sub>2)
+  \item[$\bullet$] Case \<open>P+R\<close> with @{term "s \<in> (ALT r\<^sub>1 r\<^sub>2)
   \<rightarrow> (Right w\<^sub>1)"}: This case similar to the previous
   case, except that we additionally know @{term "s \<notin> L
   r\<^sub>1"}. This is needed when @{term "v\<^sub>2"} is of the form
   \mbox{@{term "Left w\<^sub>2"}}. Since \mbox{@{term "flat v\<^sub>2 = flat
-  w\<^sub>2"} @{text "= s"}} and @{term "\<Turnstile> w\<^sub>2 :
+  w\<^sub>2"} \<open>= s\<close>} and @{term "\<Turnstile> w\<^sub>2 :
   r\<^sub>1"}, we can derive a contradiction for \mbox{@{term "s \<notin> L
   r\<^sub>1"}} using
   Proposition~\ref{inhabs}. So also in this case \mbox{@{term "v\<^sub>1
   :\<sqsubseteq>val v\<^sub>2"}}.\smallskip
 
-  \item[$\bullet$] Case @{text "PS"} with @{term "(s\<^sub>1 @
+  \item[$\bullet$] Case \<open>PS\<close> with @{term "(s\<^sub>1 @
   s\<^sub>2) \<in> (SEQ r\<^sub>1 r\<^sub>2) \<rightarrow> (Seq
   w\<^sub>1 w\<^sub>2)"}: We can assume @{term "v\<^sub>2 = Seq
   (u\<^sub>1) (u\<^sub>2)"} with @{term "\<Turnstile> u\<^sub>1 :
   r\<^sub>1"} and \mbox{@{term "\<Turnstile> u\<^sub>2 :
   r\<^sub>2"}}. We have @{term "s\<^sub>1 @ s\<^sub>2 = (flat
   u\<^sub>1) @ (flat u\<^sub>2)"}.  By the side-condition of the
-  @{text PS}-rule we know that either @{term "s\<^sub>1 = flat
+  \<open>PS\<close>-rule we know that either @{term "s\<^sub>1 = flat
   u\<^sub>1"} or that @{term "flat u\<^sub>1"} is a strict prefix of
   @{term "s\<^sub>1"}. In the latter case we can infer @{term
   "w\<^sub>1 :\<sqsubset>val u\<^sub>1"} by
@@ -1348,19 +1331,18 @@
 
   \end{itemize}
 
-  \noindent The case for @{text "P\<star>"} is similar to the @{text PS}-case and omitted.\qed
+  \noindent The case for \<open>P\<star>\<close> is similar to the \<open>PS\<close>-case and omitted.\qed
   \end{proof}
 
-  \noindent This theorem shows that our @{text POSIX} value for a
-  regular expression @{text r} and string @{term s} is in fact a
-  minimal element of the values in @{text "LV r s"}. By
+  \noindent This theorem shows that our \<open>POSIX\<close> value for a
+  regular expression \<open>r\<close> and string @{term s} is in fact a
+  minimal element of the values in \<open>LV r s\<close>. By
   Proposition~\ref{ordlen}\textit{(2)} we also know that any value in
-  @{text "LV r s'"}, with @{term "s'"} being a strict prefix, cannot be
-  smaller than @{text "v\<^sub>1"}. The next theorem shows the
+  \<open>LV r s'\<close>, with @{term "s'"} being a strict prefix, cannot be
+  smaller than \<open>v\<^sub>1\<close>. The next theorem shows the
   opposite---namely any minimal element in @{term "LV r s"} must be a
-  @{text POSIX} value. This can be established by induction on @{text
-  r}, but the proof can be drastically simplified by using the fact
-  from the previous section about the existence of a @{text POSIX} value
+  \<open>POSIX\<close> value. This can be established by induction on \<open>r\<close>, but the proof can be drastically simplified by using the fact
+  from the previous section about the existence of a \<open>POSIX\<close> value
   whenever a string @{term "s \<in> L r"}.
 
 
@@ -1372,7 +1354,7 @@
   If @{thm (prem 1) PosOrd_Posix[where ?v1.0="v\<^sub>1"]} then 
   @{term "s \<in> L r"} by Proposition~\ref{inhabs}. Hence by Theorem~\ref{lexercorrect}(2) 
   there exists a
-  @{text POSIX} value @{term "v\<^sub>P"} with @{term "s \<in> r \<rightarrow> v\<^sub>P"}
+  \<open>POSIX\<close> value @{term "v\<^sub>P"} with @{term "s \<in> r \<rightarrow> v\<^sub>P"}
   and by Lemma~\ref{LVposix} we also have \mbox{@{term "v\<^sub>P \<in> LV r s"}}.
   By Theorem~\ref{orderone} we therefore have 
   @{term "v\<^sub>P :\<sqsubseteq>val v\<^sub>1"}. If @{term "v\<^sub>P = v\<^sub>1"} then
@@ -1393,17 +1375,17 @@
 
 
   \noindent To sum up, we have shown that the (unique) minimal elements 
-  of the ordering by Okui and Suzuki are exactly the @{text POSIX}
+  of the ordering by Okui and Suzuki are exactly the \<open>POSIX\<close>
   values we defined inductively in Section~\ref{posixsec}. This provides
-  an independent confirmation that our ternary relation formalise the
+  an independent confirmation that our ternary relation formalises the
   informal POSIX rules. 
 
-*}
+\<close>
 
-section {* Bitcoded Lexing *}
+section \<open>Bitcoded Lexing\<close>
 
 
-text {*
+text \<open>
 
 Incremental calculation of the value. To simplify the proof we first define the function
 @{const flex} which calculates the ``iterated'' injection function. With this we can 
@@ -1502,7 +1484,7 @@
 
 @{thm [mode=IfThen] bder_retrieve}
 
-By induction on @{text r}
+By induction on \<open>r\<close>
 
 \begin{theorem}[Main Lemma]\mbox{}\\
 @{thm [mode=IfThen] MAIN_decode}
@@ -1518,11 +1500,11 @@
 @{thm blexer_correctness}
 \end{theorem}
 
-*}
+\<close>
 
-section {* Optimisations *}
+section \<open>Optimisations\<close>
 
-text {*
+text \<open>
 
   Derivatives as calculated by \Brz's method are usually more complex
   regular expressions than the initial one; the result is that the
@@ -1538,10 +1520,10 @@
 
   \begin{equation}\label{Simpl}
   \begin{array}{lcllcllcllcl}
-  @{term "ALT ZERO r"} & @{text "\<Rightarrow>"} & @{term r} \hspace{8mm}%\\
-  @{term "ALT r ZERO"} & @{text "\<Rightarrow>"} & @{term r} \hspace{8mm}%\\
-  @{term "SEQ ONE r"}  & @{text "\<Rightarrow>"} & @{term r} \hspace{8mm}%\\
-  @{term "SEQ r ONE"}  & @{text "\<Rightarrow>"} & @{term r}
+  @{term "ALT ZERO r"} & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
+  @{term "ALT r ZERO"} & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
+  @{term "SEQ ONE r"}  & \<open>\<Rightarrow>\<close> & @{term r} \hspace{8mm}%\\
+  @{term "SEQ r ONE"}  & \<open>\<Rightarrow>\<close> & @{term r}
   \end{array}
   \end{equation}
 
@@ -1558,15 +1540,15 @@
 
   \begin{center}
   \begin{tabular}{lcl}
-  @{thm (lhs) F_RIGHT.simps(1)} & $\dn$ & @{text "Right (f v)"}\\
-  @{thm (lhs) F_LEFT.simps(1)} & $\dn$ & @{text "Left (f v)"}\\
+  @{thm (lhs) F_RIGHT.simps(1)} & $\dn$ & \<open>Right (f v)\<close>\\
+  @{thm (lhs) F_LEFT.simps(1)} & $\dn$ & \<open>Left (f v)\<close>\\
   
-  @{thm (lhs) F_ALT.simps(1)} & $\dn$ & @{text "Right (f\<^sub>2 v)"}\\
-  @{thm (lhs) F_ALT.simps(2)} & $\dn$ & @{text "Left (f\<^sub>1 v)"}\\
+  @{thm (lhs) F_ALT.simps(1)} & $\dn$ & \<open>Right (f\<^sub>2 v)\<close>\\
+  @{thm (lhs) F_ALT.simps(2)} & $\dn$ & \<open>Left (f\<^sub>1 v)\<close>\\
   
-  @{thm (lhs) F_SEQ1.simps(1)} & $\dn$ & @{text "Seq (f\<^sub>1 ()) (f\<^sub>2 v)"}\\
-  @{thm (lhs) F_SEQ2.simps(1)} & $\dn$ & @{text "Seq (f\<^sub>1 v) (f\<^sub>2 ())"}\\
-  @{thm (lhs) F_SEQ.simps(1)} & $\dn$ & @{text "Seq (f\<^sub>1 v\<^sub>1) (f\<^sub>2 v\<^sub>2)"}\medskip\\
+  @{thm (lhs) F_SEQ1.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 ()) (f\<^sub>2 v)\<close>\\
+  @{thm (lhs) F_SEQ2.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 v) (f\<^sub>2 ())\<close>\\
+  @{thm (lhs) F_SEQ.simps(1)} & $\dn$ & \<open>Seq (f\<^sub>1 v\<^sub>1) (f\<^sub>2 v\<^sub>2)\<close>\medskip\\
   %\end{tabular}
   %
   %\begin{tabular}{lcl}
@@ -1580,7 +1562,7 @@
   \end{center}
 
   \noindent
-  The functions @{text "simp\<^bsub>Alt\<^esub>"} and @{text "simp\<^bsub>Seq\<^esub>"} encode the simplification rules
+  The functions \<open>simp\<^bsub>Alt\<^esub>\<close> and \<open>simp\<^bsub>Seq\<^esub>\<close> encode the simplification rules
   in \eqref{Simpl} and compose the rectification functions (simplifications can occur
   deep inside the regular expression). The main simplification function is then 
 
@@ -1602,17 +1584,17 @@
   \begin{tabular}{lcl}
   @{thm (lhs) slexer.simps(1)} & $\dn$ & @{thm (rhs) slexer.simps(1)}\\
   @{thm (lhs) slexer.simps(2)} & $\dn$ & 
-                         @{text "let (r\<^sub>s, f\<^sub>r) = simp (r "}$\backslash$@{text " c) in"}\\
-                     & & @{text "case"} @{term "slexer r\<^sub>s s"} @{text of}\\
-                     & & \phantom{$|$} @{term "None"}  @{text "\<Rightarrow>"} @{term None}\\
-                     & & $|$ @{term "Some v"} @{text "\<Rightarrow>"} @{text "Some (inj r c (f\<^sub>r v))"}                          
+                         \<open>let (r\<^sub>s, f\<^sub>r) = simp (r \<close>$\backslash$\<open> c) in\<close>\\
+                     & & \<open>case\<close> @{term "slexer r\<^sub>s s"} \<open>of\<close>\\
+                     & & \phantom{$|$} @{term "None"}  \<open>\<Rightarrow>\<close> @{term None}\\
+                     & & $|$ @{term "Some v"} \<open>\<Rightarrow>\<close> \<open>Some (inj r c (f\<^sub>r v))\<close>                          
   \end{tabular}
   \end{center}
 
   \noindent
   In the second clause we first calculate the derivative @{term "der c r"}
   and then simplify the result. This gives us a simplified derivative
-  @{text "r\<^sub>s"} and a rectification function @{text "f\<^sub>r"}. The lexer
+  \<open>r\<^sub>s\<close> and a rectification function \<open>f\<^sub>r\<close>. The lexer
   is then recursively called with the simplified derivative, but before
   we inject the character @{term c} into the value @{term v}, we need to rectify
   @{term v} (that is construct @{term "f\<^sub>r v"}). Before we can establish the correctness
@@ -1627,7 +1609,7 @@
   \end{tabular}
   \end{lemma}
 
-  \begin{proof} Both are by induction on @{text r}. There is no
+  \begin{proof} Both are by induction on \<open>r\<close>. There is no
   interesting case for the first statement. For the second statement,
   of interest are the @{term "r = ALT r\<^sub>1 r\<^sub>2"} and @{term "r = SEQ r\<^sub>1
   r\<^sub>2"} cases. In each case we have to analyse four subcases whether
@@ -1639,7 +1621,7 @@
   and by IH also (*) @{term "s \<in> r\<^sub>2 \<rightarrow> (snd (simp r\<^sub>2) v)"}. Given @{term "fst (simp r\<^sub>1) = ZERO"}
   we know @{term "L (fst (simp r\<^sub>1)) = {}"}. By the first statement
   @{term "L r\<^sub>1"} is the empty set, meaning (**) @{term "s \<notin> L r\<^sub>1"}.
-  Taking (*) and (**) together gives by the \mbox{@{text "P+R"}}-rule 
+  Taking (*) and (**) together gives by the \mbox{\<open>P+R\<close>}-rule 
   @{term "s \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> Right (snd (simp r\<^sub>2) v)"}. In turn this
   gives @{term "s \<in> ALT r\<^sub>1 r\<^sub>2 \<rightarrow> snd (simp (ALT r\<^sub>1 r\<^sub>2)) v"} as we need to show.
   The other cases are similar.\qed
@@ -1683,12 +1665,12 @@
 
   \end{proof} 
 
-*}
+\<close>
 
 
-section {* HERE *}
+section \<open>HERE\<close>
 
-text {*
+text \<open>
   \begin{center}
   \begin{tabular}{llcl}
   1) & @{thm (lhs) erase.simps(1)} & $\dn$ & @{thm (rhs) erase.simps(1)}\\
@@ -1751,7 +1733,7 @@
   We can move out the @{term "fuse  [Z]"} and then use the IH to show that left-hand side
   and right-hand side are equal. This completes the proof. 
   \end{proof}   
-*}
+\<close>