
diff -r f65444d29e74 -r 78cc255e286f thys2/Journal/Paper.tex
--- a/thys2/Journal/Paper.tex	Tue Nov 02 13:57:59 2021 +0000
+++ b/thys2/Journal/Paper.tex	Tue Nov 02 19:42:52 2021 +0000
@@ -59,9 +59,9 @@
 \endisadelimdocument
 %
 \begin{isamarkuptext}%
-This works builds on previous work by Ausaf and Urban using 
+This paper builds on previous work by Ausaf and Urban using 
 regular expression'd bit-coded derivatives to do lexing that 
-is both fast and satisfied the POSIX specification.
+is both fast and satisfies the POSIX specification.
 In their work, a bit-coded algorithm introduced by Sulzmann and Lu
 was formally verified in Isabelle, by a very clever use of
 flex function and retrieve to carefully mimic the way a value is 
@@ -81,45 +81,15 @@
 
 However what is not certain is whether we can add simplification
 to the bit-coded algorithm, without breaking the correct lexing output.
-This might sound trivial in the case of producing a YES/NO answer,
-but once we require a lexing output to be produced (which is required
-in applications like compiler front-end, malicious attack domain extraction, 
-etc.), it is not straightforward if we still extract what is needed according
-to the POSIX standard.
-
-By simplification, we mean specifically the following rules:
-
-\begin{center}
-  \begin{tabular}{lcl}
-  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ AZERO\ r\isactrlsub {\isadigit{2}}\ {\isasymleadsto}\ AZERO}}}\\
-  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ AZERO\ {\isasymleadsto}\ AZERO}}}\\
-  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ {\isacharparenleft}{\kern0pt}AONE\ bs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymleadsto}\ fuse\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ bs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}}\\
-  \isa{\mbox{}\inferrule{\mbox{bs\ {\isasymleadsto}\ r\isactrlsub {\isadigit{1}}}}{\mbox{ASEQ\ bs\ bs\ r{\isadigit{3}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isasymleadsto}\ ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ r{\isadigit{3}}{\isachardot}{\kern0pt}{\isadigit{0}}}}}\\
-  \isa{\mbox{}\inferrule{\mbox{bs\ {\isasymleadsto}\ r\isactrlsub {\isadigit{2}}}}{\mbox{ASEQ\ bs\ r{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ bs\ {\isasymleadsto}\ ASEQ\ bs\ r{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ r\isactrlsub {\isadigit{2}}}}}\\
-  \isa{\mbox{}\inferrule{\mbox{bs\ {\isasymleadsto}\ r\isactrlsub {\isadigit{1}}}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}bs{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs{\isadigit{2}}{\isachardot}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs{\isadigit{2}}{\isachardot}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}}}}\\
-  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}\ {\isacharat}{\kern0pt}\ AZERO\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}rsb{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}\ {\isacharat}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}}}}\\
-  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ AALTs\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ rs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}rsb{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharat}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}}}}\\
-  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ rs}}}\\
-  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}}}}\\
-  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ AZERO}}}\\
-  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ fuse\ bs\ r\isactrlsub {\isadigit{1}}}}}\\
-  \isa{\mbox{}\inferrule{\mbox{bs\mbox{$^\downarrow$}\ {\isacharequal}{\kern0pt}\ r\isactrlsub {\isadigit{2}}\mbox{$^\downarrow$}}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}bs{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsb\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsc{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}bs{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsb\ {\isacharat}{\kern0pt}\ rsc{\isacharparenright}{\kern0pt}}}}\\
 
 
-  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{Empty\ {\isacharcolon}{\kern0pt}\ \isactrlbold {\isadigit{1}}}}} & 
-  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{Char\ c\ {\isacharcolon}{\kern0pt}\ c}}}\\[4mm]
-  \isa{\mbox{}\inferrule{\mbox{v\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}{\mbox{Left\ v\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}}}} &
-  \isa{\mbox{}\inferrule{\mbox{v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}{\mbox{Right\ v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{2}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}}\\[4mm]
-  \isa{\mbox{}\inferrule{\mbox{v\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}\\\ \mbox{v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{2}}}}{\mbox{Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}}}}  &
-  \isa{\mbox{}\inferrule{\mbox{{\isasymforall}v{\isasymin}vs{\isachardot}{\kern0pt}\ v\ {\isacharcolon}{\kern0pt}\ r\ {\isasymand}\ {\isacharbar}{\kern0pt}v{\isacharbar}{\kern0pt}\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}}}{\mbox{Stars\ vs\ {\isacharcolon}{\kern0pt}\ r\isactrlsup {\isasymstar}}}}
-
-
-  \end{tabular}
-\end{center}
-
-
-And these can be made compact by the following simplification function:
-
+The reason that we do need to add a simplification phase
+after each derivative step of  $\textit{blexer}$ is
+because it produces intermediate
+regular expressions that can grow exponentially.
+For example, the regular expression $(a+aa)^*$ after taking
+derivative against just 10 $a$s will have size 8192.
+%TODO: add figure for this?
 \begin{center}
   \begin{tabular}{lcl}
   \isa{bsimp\ {\isacharparenleft}{\kern0pt}ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs\ {\isacharparenleft}{\kern0pt}bsimp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bsimp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
@@ -129,12 +99,76 @@
 \end{tabular}
 \end{center}
 
+
+This might sound trivial in the case of producing a YES/NO answer,
+but once we require a lexing output to be produced (which is required
+in applications like compiler front-end, malicious attack domain extraction, 
+etc.), it is not straightforward if we still extract what is needed according
+to the POSIX standard.
+
+
+
+
+
+By simplification, we mean specifically the following rules:
+
+\begin{center}
+  \begin{tabular}{lcl}
+  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ AZERO\ r\isactrlsub {\isadigit{2}}\ {\isasymleadsto}\ AZERO}}}\\
+  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ AZERO\ {\isasymleadsto}\ AZERO}}}\\
+  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ {\isacharparenleft}{\kern0pt}AONE\ bs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymleadsto}\ fuse\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ bs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}}\\
+  \isa{\mbox{}\inferrule{\mbox{r\isactrlsub {\isadigit{1}}\ {\isasymleadsto}\ r\isactrlsub {\isadigit{2}}}}{\mbox{ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{3}}\ {\isasymleadsto}\ ASEQ\ bs\ r\isactrlsub {\isadigit{2}}\ r\isactrlsub {\isadigit{3}}}}}\\
+  \isa{\mbox{}\inferrule{\mbox{r\isactrlsub {\isadigit{3}}\ {\isasymleadsto}\ r\isactrlsub {\isadigit{4}}}}{\mbox{ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{3}}\ {\isasymleadsto}\ ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{4}}}}}\\
+  \isa{\mbox{}\inferrule{\mbox{r\ {\isasymleadsto}\ r{\isacharprime}{\kern0pt}}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r{\isacharprime}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}}}\\
+  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub a\ {\isacharat}{\kern0pt}\ AZERO\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}rs\isactrlsub b{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub a\ {\isacharat}{\kern0pt}\ rs\isactrlsub b{\isacharparenright}{\kern0pt}}}}\\
+  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub a\ {\isacharat}{\kern0pt}\ AALTs\ bs\isactrlsub {\isadigit{1}}\ rs\isactrlsub {\isadigit{1}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}rs\isactrlsub b{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub a\ {\isacharat}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ rs\isactrlsub b{\isacharparenright}{\kern0pt}}}}\\
+  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ rs}}}\\
+  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}}}}\\
+  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ AZERO}}}\\
+  \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ fuse\ bs\ r\isactrlsub {\isadigit{1}}}}}\\
+  \isa{\mbox{}\inferrule{\mbox{a\isactrlsub {\isadigit{1}}\mbox{$^\downarrow$}\ {\isacharequal}{\kern0pt}\ a\isactrlsub {\isadigit{2}}\mbox{$^\downarrow$}}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub a\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs\isactrlsub b\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a\isactrlsub {\isadigit{2}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs\isactrlsub c{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs\isactrlsub a\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}a\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs\isactrlsub b\ {\isacharat}{\kern0pt}\ rs\isactrlsub c{\isacharparenright}{\kern0pt}}}}\\
+
+  \end{tabular}
+\end{center}
+
+
+And these can be made compact by the following simplification function:
+
+where the function $\mathit{bsimp_AALTs}$
+
 The core idea of the proof is that two regular expressions,
 if "isomorphic" up to a finite number of rewrite steps, will
-remain so when we take derivative on both of them.
+remain "isomorphic" when we take the same sequence of
+derivatives on both of them.
 This can be expressed by the following rewrite relation lemma:
 \begin{lemma}
 \isa{{\isacharparenleft}{\kern0pt}r\mbox{$\bbslash$}s{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp\ r\ s}
+\end{lemma}
+
+This isomorphic relation implies a property that leads to the 
+correctness result: 
+if two (nullable) regular expressions are "rewritable" in many steps
+from one another, 
+then a call to function $\textit{bmkeps}$ gives the same
+bit-sequence :
+\begin{lemma}
+\isa{{\normalsize{}If\,}\ \mbox{r{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ r{\isadigit{2}}{\isachardot}{\kern0pt}{\isadigit{0}}}\ {\normalsize \,and\,}\ \mbox{nullable\mbox{$_b$}\ r{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}}\ {\normalsize \,then\,}\ mkeps\mbox{$_b$}\ r{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharequal}{\kern0pt}\ mkeps\mbox{$_b$}\ r{\isadigit{2}}{\isachardot}{\kern0pt}{\isadigit{0}}{\isachardot}{\kern0pt}}
+\end{lemma}
+
+Given the same bit-sequence, the decode function
+will give out the same value, which is the output
+of both lexers:
+\begin{lemma}
+\isa{lexer\mbox{$_b$}\ r\ s\ {\isasymequiv}\ \textrm{if}\ nullable\mbox{$_b$}\ {\isacharparenleft}{\kern0pt}r\mbox{$^\uparrow$}\mbox{$\bbslash$}s{\isacharparenright}{\kern0pt}\ \textrm{then}\ decode\ {\isacharparenleft}{\kern0pt}mkeps\mbox{$_b$}\ {\isacharparenleft}{\kern0pt}r\mbox{$^\uparrow$}\mbox{$\bbslash$}s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r\ \textrm{else}\ None}
+\end{lemma}
+
+\begin{lemma}
+\isa{blexer{\isacharunderscore}{\kern0pt}simp\ r\ s\ {\isasymequiv}\ \textrm{if}\ nullable\mbox{$_b$}\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}r\mbox{$^\uparrow$}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}\ \textrm{then}\ decode\ {\isacharparenleft}{\kern0pt}mkeps\mbox{$_b$}\ {\isacharparenleft}{\kern0pt}bders{\isacharunderscore}{\kern0pt}simp\ {\isacharparenleft}{\kern0pt}r\mbox{$^\uparrow$}{\isacharparenright}{\kern0pt}\ s{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ r\ \textrm{else}\ None}
+\end{lemma}
+
+And that yields the correctness result:
+\begin{lemma}
+\isa{blexer{\isacharunderscore}{\kern0pt}simp\ r\ s\ {\isacharequal}{\kern0pt}\ lexer\ r\ s}
 \end{lemma}%
 \end{isamarkuptext}\isamarkuptrue%
 %
@@ -1745,45 +1779,45 @@
 %:%136=210%:%
 %:%137=211%:%
 %:%138=212%:%
-%:%147=217%:%
-%:%159=223%:%
-%:%160=224%:%
-%:%161=225%:%
-%:%162=226%:%
-%:%162=227%:%
-%:%163=228%:%
-%:%164=229%:%
-%:%165=230%:%
-%:%166=231%:%
-%:%167=232%:%
-%:%168=233%:%
-%:%169=234%:%
-%:%170=235%:%
-%:%171=236%:%
-%:%172=237%:%
-%:%173=238%:%
-%:%174=239%:%
-%:%175=240%:%
-%:%176=241%:%
-%:%177=242%:%
-%:%178=243%:%
-%:%179=244%:%
-%:%180=245%:%
-%:%181=246%:%
-%:%182=247%:%
-%:%183=248%:%
-%:%184=249%:%
-%:%185=250%:%
-%:%186=251%:%
-%:%187=252%:%
-%:%188=253%:%
-%:%189=254%:%
-%:%190=255%:%
-%:%191=256%:%
-%:%192=257%:%
-%:%193=258%:%
-%:%194=259%:%
-%:%195=260%:%
+%:%139=213%:%
+%:%140=214%:%
+%:%141=215%:%
+%:%142=216%:%
+%:%143=217%:%
+%:%144=218%:%
+%:%145=219%:%
+%:%146=220%:%
+%:%147=221%:%
+%:%148=222%:%
+%:%149=223%:%
+%:%150=224%:%
+%:%151=225%:%
+%:%152=226%:%
+%:%153=227%:%
+%:%154=228%:%
+%:%155=229%:%
+%:%156=230%:%
+%:%157=231%:%
+%:%158=232%:%
+%:%159=233%:%
+%:%160=234%:%
+%:%161=235%:%
+%:%162=236%:%
+%:%163=237%:%
+%:%164=238%:%
+%:%165=239%:%
+%:%166=240%:%
+%:%167=241%:%
+%:%168=242%:%
+%:%169=243%:%
+%:%170=244%:%
+%:%171=245%:%
+%:%172=246%:%
+%:%181=251%:%
+%:%193=257%:%
+%:%194=258%:%
+%:%195=259%:%
+%:%196=260%:%
 %:%196=261%:%
 %:%197=262%:%
 %:%198=263%:%
@@ -1815,40 +1849,40 @@
 %:%224=289%:%
 %:%225=290%:%
 %:%226=291%:%
-%:%226=292%:%
-%:%227=293%:%
-%:%228=294%:%
-%:%229=295%:%
-%:%230=296%:%
-%:%231=297%:%
-%:%232=298%:%
-%:%233=299%:%
-%:%234=300%:%
-%:%235=301%:%
-%:%236=302%:%
-%:%237=303%:%
-%:%238=304%:%
-%:%239=305%:%
-%:%240=306%:%
-%:%241=307%:%
-%:%242=308%:%
-%:%243=309%:%
-%:%244=310%:%
-%:%245=311%:%
-%:%246=312%:%
-%:%247=313%:%
-%:%248=314%:%
-%:%249=315%:%
-%:%250=316%:%
-%:%251=317%:%
-%:%252=318%:%
-%:%253=319%:%
-%:%254=320%:%
-%:%255=321%:%
-%:%256=322%:%
-%:%257=323%:%
-%:%258=324%:%
-%:%259=325%:%
+%:%227=292%:%
+%:%228=293%:%
+%:%229=294%:%
+%:%230=295%:%
+%:%231=296%:%
+%:%232=297%:%
+%:%233=298%:%
+%:%234=299%:%
+%:%235=300%:%
+%:%236=301%:%
+%:%237=302%:%
+%:%238=303%:%
+%:%239=304%:%
+%:%240=305%:%
+%:%241=306%:%
+%:%242=307%:%
+%:%243=308%:%
+%:%244=309%:%
+%:%245=310%:%
+%:%246=311%:%
+%:%247=312%:%
+%:%248=313%:%
+%:%249=314%:%
+%:%250=315%:%
+%:%251=316%:%
+%:%252=317%:%
+%:%253=318%:%
+%:%254=319%:%
+%:%255=320%:%
+%:%256=321%:%
+%:%257=322%:%
+%:%258=323%:%
+%:%259=324%:%
+%:%260=325%:%
 %:%260=326%:%
 %:%261=327%:%
 %:%262=328%:%
@@ -1921,89 +1955,89 @@
 %:%329=395%:%
 %:%330=396%:%
 %:%331=397%:%
-%:%340=404%:%
-%:%352=406%:%
-%:%353=407%:%
-%:%353=408%:%
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@@ -2060,41 +2094,41 @@
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@@ -2176,45 +2210,45 @@
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@@ -2238,116 +2272,116 @@
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@@ -2373,239 +2407,239 @@
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@@ -2688,52 +2722,52 @@
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@@ -2971,49 +3005,49 @@
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+%:%1382=1475%:%
+%:%1383=1476%:%
+%:%1384=1477%:%
+%:%1385=1478%:%
+%:%1386=1479%:%
+%:%1387=1480%:%
+%:%1388=1481%:%
+%:%1389=1482%:%
+%:%1390=1483%:%
+%:%1391=1484%:%
+%:%1392=1485%:%
+%:%1393=1486%:%
+%:%1394=1487%:%
+%:%1395=1488%:%
+%:%1396=1489%:%
+%:%1397=1490%:%
+%:%1398=1491%:%
+%:%1399=1492%:%
+%:%1400=1493%:%
+%:%1401=1494%:%
+%:%1402=1495%:%
+%:%1403=1496%:%
+%:%1404=1497%:%
+%:%1405=1498%:%
+%:%1406=1499%:%
+%:%1407=1500%:%
+%:%1408=1501%:%
+%:%1417=1505%:%
+%:%1429=1512%:%
+%:%1430=1513%:%
+%:%1431=1514%:%
+%:%1432=1515%:%
+%:%1433=1516%:%
+%:%1434=1517%:%
+%:%1435=1518%:%
+%:%1444=1523%:%
 %:%1456=1527%:%
 %:%1457=1528%:%
 %:%1458=1529%:%
@@ -3070,118 +3104,118 @@
 %:%1509=1580%:%
 %:%1510=1581%:%
 %:%1511=1582%:%
-%:%1511=1698%:%
-%:%1512=1699%:%
-%:%1513=1700%:%
-%:%1514=1701%:%
-%:%1515=1702%:%
-%:%1515=1863%:%
-%:%1516=1864%:%
-%:%1517=1865%:%
-%:%1518=1866%:%
-%:%1519=1867%:%
-%:%1520=1868%:%
-%:%1521=1869%:%
-%:%1522=1870%:%
-%:%1523=1871%:%
-%:%1524=1872%:%
-%:%1525=1873%:%
-%:%1526=1874%:%
-%:%1527=1875%:%
-%:%1528=1876%:%
-%:%1529=1877%:%
-%:%1530=1878%:%
-%:%1531=1879%:%
-%:%1532=1880%:%
-%:%1533=1881%:%
-%:%1534=1882%:%
-%:%1534=1883%:%
-%:%1535=1884%:%
-%:%1535=1885%:%
-%:%1535=1886%:%
-%:%1536=1887%:%
-%:%1536=1888%:%
-%:%1537=1889%:%
-%:%1538=1890%:%
-%:%1539=1891%:%
-%:%1540=1892%:%
-%:%1541=1893%:%
-%:%1542=1894%:%
-%:%1543=1895%:%
-%:%1544=1896%:%
-%:%1545=1897%:%
-%:%1546=1898%:%
-%:%1547=1899%:%
-%:%1548=1900%:%
-%:%1549=1901%:%
-%:%1550=1902%:%
-%:%1551=1903%:%
-%:%1552=1904%:%
-%:%1553=1905%:%
-%:%1553=1906%:%
-%:%1554=1907%:%
-%:%1555=1908%:%
-%:%1556=1909%:%
-%:%1556=1910%:%
-%:%1556=1911%:%
-%:%1557=1912%:%
-%:%1558=1913%:%
-%:%1559=1914%:%
-%:%1559=1915%:%
-%:%1560=1916%:%
-%:%1560=1917%:%
-%:%1561=1918%:%
-%:%1562=1919%:%
-%:%1563=1920%:%
-%:%1564=1921%:%
-%:%1565=1922%:%
-%:%1566=1923%:%
-%:%1567=1924%:%
-%:%1568=1925%:%
-%:%1569=1926%:%
-%:%1570=1927%:%
-%:%1571=1928%:%
-%:%1572=1929%:%
-%:%1573=1930%:%
-%:%1574=1931%:%
-%:%1575=1932%:%
-%:%1576=1933%:%
-%:%1577=1934%:%
-%:%1586=1939%:%
-%:%1598=1943%:%
-%:%1599=1944%:%
-%:%1600=1945%:%
-%:%1601=1946%:%
-%:%1602=1947%:%
-%:%1603=1948%:%
-%:%1604=1949%:%
-%:%1605=1950%:%
-%:%1606=1951%:%
-%:%1607=1952%:%
-%:%1608=1953%:%
-%:%1609=1954%:%
-%:%1610=1955%:%
-%:%1611=1956%:%
-%:%1612=1957%:%
-%:%1613=1958%:%
-%:%1614=1959%:%
-%:%1615=1960%:%
-%:%1616=1961%:%
-%:%1617=1962%:%
-%:%1618=1963%:%
-%:%1619=1964%:%
-%:%1620=1965%:%
-%:%1621=1966%:%
-%:%1622=1967%:%
-%:%1623=1968%:%
-%:%1624=1969%:%
-%:%1625=1970%:%
-%:%1626=1971%:%
-%:%1627=1972%:%
-%:%1628=1973%:%
-%:%1629=1974%:%
-%:%1630=1975%:%
-%:%1631=1976%:%
+%:%1512=1583%:%
+%:%1513=1584%:%
+%:%1514=1585%:%
+%:%1515=1586%:%
+%:%1516=1587%:%
+%:%1517=1588%:%
+%:%1518=1589%:%
+%:%1519=1590%:%
+%:%1520=1591%:%
+%:%1521=1592%:%
+%:%1522=1593%:%
+%:%1523=1594%:%
+%:%1524=1595%:%
+%:%1525=1596%:%
+%:%1526=1597%:%
+%:%1527=1598%:%
+%:%1528=1599%:%
+%:%1529=1600%:%
+%:%1530=1601%:%
+%:%1531=1602%:%
+%:%1532=1603%:%
+%:%1533=1604%:%
+%:%1534=1605%:%
+%:%1535=1606%:%
+%:%1536=1607%:%
+%:%1537=1608%:%
+%:%1538=1609%:%
+%:%1539=1610%:%
+%:%1540=1611%:%
+%:%1541=1612%:%
+%:%1542=1613%:%
+%:%1543=1614%:%
+%:%1544=1615%:%
+%:%1545=1616%:%
+%:%1545=1732%:%
+%:%1546=1733%:%
+%:%1547=1734%:%
+%:%1548=1735%:%
+%:%1549=1736%:%
+%:%1549=1897%:%
+%:%1550=1898%:%
+%:%1551=1899%:%
+%:%1552=1900%:%
+%:%1553=1901%:%
+%:%1554=1902%:%
+%:%1555=1903%:%
+%:%1556=1904%:%
+%:%1557=1905%:%
+%:%1558=1906%:%
+%:%1559=1907%:%
+%:%1560=1908%:%
+%:%1561=1909%:%
+%:%1562=1910%:%
+%:%1563=1911%:%
+%:%1564=1912%:%
+%:%1565=1913%:%
+%:%1566=1914%:%
+%:%1567=1915%:%
+%:%1568=1916%:%
+%:%1568=1917%:%
+%:%1569=1918%:%
+%:%1569=1919%:%
+%:%1569=1920%:%
+%:%1570=1921%:%
+%:%1570=1922%:%
+%:%1571=1923%:%
+%:%1572=1924%:%
+%:%1573=1925%:%
+%:%1574=1926%:%
+%:%1575=1927%:%
+%:%1576=1928%:%
+%:%1577=1929%:%
+%:%1578=1930%:%
+%:%1579=1931%:%
+%:%1580=1932%:%
+%:%1581=1933%:%
+%:%1582=1934%:%
+%:%1583=1935%:%
+%:%1584=1936%:%
+%:%1585=1937%:%
+%:%1586=1938%:%
+%:%1587=1939%:%
+%:%1587=1940%:%
+%:%1588=1941%:%
+%:%1589=1942%:%
+%:%1590=1943%:%
+%:%1590=1944%:%
+%:%1590=1945%:%
+%:%1591=1946%:%
+%:%1592=1947%:%
+%:%1593=1948%:%
+%:%1593=1949%:%
+%:%1594=1950%:%
+%:%1594=1951%:%
+%:%1595=1952%:%
+%:%1596=1953%:%
+%:%1597=1954%:%
+%:%1598=1955%:%
+%:%1599=1956%:%
+%:%1600=1957%:%
+%:%1601=1958%:%
+%:%1602=1959%:%
+%:%1603=1960%:%
+%:%1604=1961%:%
+%:%1605=1962%:%
+%:%1606=1963%:%
+%:%1607=1964%:%
+%:%1608=1965%:%
+%:%1609=1966%:%
+%:%1610=1967%:%
+%:%1611=1968%:%
+%:%1620=1973%:%
 %:%1632=1977%:%
 %:%1633=1978%:%
 %:%1634=1979%:%
@@ -3201,4 +3235,38 @@
 %:%1648=1993%:%
 %:%1649=1994%:%
 %:%1650=1995%:%
-%:%1663=2001%:%
\ No newline at end of file
+%:%1651=1996%:%
+%:%1652=1997%:%
+%:%1653=1998%:%
+%:%1654=1999%:%
+%:%1655=2000%:%
+%:%1656=2001%:%
+%:%1657=2002%:%
+%:%1658=2003%:%
+%:%1659=2004%:%
+%:%1660=2005%:%
+%:%1661=2006%:%
+%:%1662=2007%:%
+%:%1663=2008%:%
+%:%1664=2009%:%
+%:%1665=2010%:%
+%:%1666=2011%:%
+%:%1667=2012%:%
+%:%1668=2013%:%
+%:%1669=2014%:%
+%:%1670=2015%:%
+%:%1671=2016%:%
+%:%1672=2017%:%
+%:%1673=2018%:%
+%:%1674=2019%:%
+%:%1675=2020%:%
+%:%1676=2021%:%
+%:%1677=2022%:%
+%:%1678=2023%:%
+%:%1679=2024%:%
+%:%1680=2025%:%
+%:%1681=2026%:%
+%:%1682=2027%:%
+%:%1683=2028%:%
+%:%1684=2029%:%
+%:%1697=2035%:%
\ No newline at end of file




