--- a/doc-src/TutorialI/Inductive/document/AB.tex Sun Nov 07 23:32:26 2010 +0100
+++ b/doc-src/TutorialI/Inductive/document/AB.tex Mon Nov 08 00:00:47 2010 +0100
@@ -42,14 +42,14 @@
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{datatype}\isamarkupfalse%
-\ alfa\ {\isacharequal}\ a\ {\isacharbar}\ b%
+\ alfa\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{7C}{\isacharbar}}\ b%
\begin{isamarkuptext}%
\noindent
For convenience we include the following easy lemmas as simplification rules:%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
-\ {\isacharbrackleft}simp{\isacharbrackright}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymnoteq}\ a{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharequal}\ b{\isacharparenright}\ {\isasymand}\ {\isacharparenleft}x\ {\isasymnoteq}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x\ {\isacharequal}\ a{\isacharparenright}{\isachardoublequoteclose}\isanewline
+\ {\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
%
\isadelimproof
%
@@ -57,7 +57,7 @@
%
\isatagproof
\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}case{\isacharunderscore}tac\ x{\isacharcomma}\ auto{\isacharparenright}%
+\ {\isaliteral{28}{\isacharparenleft}}case{\isaliteral{5F}{\isacharunderscore}}tac\ x{\isaliteral{2C}{\isacharcomma}}\ auto{\isaliteral{29}{\isacharparenright}}%
\endisatagproof
{\isafoldproof}%
%
@@ -74,21 +74,21 @@
of \isa{S}, \isa{A} and~\isa{B}:%
\end{isamarkuptext}%
\isamarkuptrue%
-\isacommand{inductive{\isacharunderscore}set}\isamarkupfalse%
+\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}set}\isamarkupfalse%
\isanewline
-\ \ S\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}alfa\ list\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ A\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}alfa\ list\ set{\isachardoublequoteclose}\ \isakeyword{and}\isanewline
-\ \ B\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}alfa\ list\ set{\isachardoublequoteclose}\isanewline
+\ \ S\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}alfa\ list\ set{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\isanewline
+\ \ A\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}alfa\ list\ set{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\isanewline
+\ \ B\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}alfa\ list\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
\isakeyword{where}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharbrackleft}{\isacharbrackright}\ {\isasymin}\ S{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}w\ {\isasymin}\ A\ {\isasymLongrightarrow}\ b{\isacharhash}w\ {\isasymin}\ S{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}w\ {\isasymin}\ B\ {\isasymLongrightarrow}\ a{\isacharhash}w\ {\isasymin}\ S{\isachardoublequoteclose}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ S{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ b{\isaliteral{23}{\isacharhash}}w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ S{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a{\isaliteral{23}{\isacharhash}}w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ S{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}w\ {\isasymin}\ S\ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ a{\isacharhash}w\ \ \ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}{\isasymlbrakk}\ v{\isasymin}A{\isacharsemicolon}\ w{\isasymin}A\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ b{\isacharhash}v{\isacharat}w\ {\isasymin}\ A{\isachardoublequoteclose}\isanewline
+{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ S\ \ \ \ \ \ \ \ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a{\isaliteral{23}{\isacharhash}}w\ \ \ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ v{\isaliteral{5C3C696E3E}{\isasymin}}A{\isaliteral{3B}{\isacharsemicolon}}\ w{\isaliteral{5C3C696E3E}{\isasymin}}A\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ b{\isaliteral{23}{\isacharhash}}v{\isaliteral{40}{\isacharat}}w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}w\ {\isasymin}\ S\ \ \ \ \ \ \ \ \ \ \ \ {\isasymLongrightarrow}\ b{\isacharhash}w\ \ \ {\isasymin}\ B{\isachardoublequoteclose}\isanewline
-{\isacharbar}\ {\isachardoublequoteopen}{\isasymlbrakk}\ v\ {\isasymin}\ B{\isacharsemicolon}\ w\ {\isasymin}\ B\ {\isasymrbrakk}\ {\isasymLongrightarrow}\ a{\isacharhash}v{\isacharat}w\ {\isasymin}\ B{\isachardoublequoteclose}%
+{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ S\ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ b{\isaliteral{23}{\isacharhash}}w\ \ \ {\isaliteral{5C3C696E3E}{\isasymin}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ v\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B{\isaliteral{3B}{\isacharsemicolon}}\ w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a{\isaliteral{23}{\isacharhash}}v{\isaliteral{40}{\isacharat}}w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B{\isaliteral{22}{\isachardoublequoteclose}}%
\begin{isamarkuptext}%
\noindent
First we show that all words in \isa{S} contain the same number of \isa{a}'s and \isa{b}'s. Since the definition of \isa{S} is by mutual
@@ -97,10 +97,10 @@
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
-\ correctness{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}w\ {\isasymin}\ S\ {\isasymlongrightarrow}\ size{\isacharbrackleft}x{\isasymleftarrow}w{\isachardot}\ x{\isacharequal}a{\isacharbrackright}\ {\isacharequal}\ size{\isacharbrackleft}x{\isasymleftarrow}w{\isachardot}\ x{\isacharequal}b{\isacharbrackright}{\isacharparenright}\ \ \ \ \ {\isasymand}\isanewline
-\ \ \ {\isacharparenleft}w\ {\isasymin}\ A\ {\isasymlongrightarrow}\ size{\isacharbrackleft}x{\isasymleftarrow}w{\isachardot}\ x{\isacharequal}a{\isacharbrackright}\ {\isacharequal}\ size{\isacharbrackleft}x{\isasymleftarrow}w{\isachardot}\ x{\isacharequal}b{\isacharbrackright}\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}\ {\isasymand}\isanewline
-\ \ \ {\isacharparenleft}w\ {\isasymin}\ B\ {\isasymlongrightarrow}\ size{\isacharbrackleft}x{\isasymleftarrow}w{\isachardot}\ x{\isacharequal}b{\isacharbrackright}\ {\isacharequal}\ size{\isacharbrackleft}x{\isasymleftarrow}w{\isachardot}\ x{\isacharequal}a{\isacharbrackright}\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}{\isachardoublequoteclose}%
+\ correctness{\isaliteral{3A}{\isacharcolon}}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ S\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}w{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}a{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}w{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}b{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\ \ \ \ \ {\isaliteral{5C3C616E643E}{\isasymand}}\isanewline
+\ \ \ {\isaliteral{28}{\isacharparenleft}}w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}w{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}a{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}w{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}b{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\isanewline
+\ \ \ {\isaliteral{28}{\isacharparenleft}}w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}w{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}b{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}w{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}a{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
\isadelimproof
%
\endisadelimproof
@@ -109,14 +109,14 @@
%
\begin{isamarkuptxt}%
\noindent
-These propositions are expressed with the help of the predefined \isa{filter} function on lists, which has the convenient syntax \isa{{\isacharbrackleft}x{\isasymleftarrow}xs{\isachardot}\ P\ x{\isacharbrackright}}, the list of all elements \isa{x} in \isa{xs} such that \isa{P\ x}
+These propositions are expressed with the help of the predefined \isa{filter} function on lists, which has the convenient syntax \isa{{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}xs{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{5D}{\isacharbrackright}}}, the list of all elements \isa{x} in \isa{xs} such that \isa{P\ x}
holds. Remember that on lists \isa{size} and \isa{length} are synonymous.
The proof itself is by rule induction and afterwards automatic:%
\end{isamarkuptxt}%
\isamarkuptrue%
\isacommand{by}\isamarkupfalse%
-\ {\isacharparenleft}rule\ S{\isacharunderscore}A{\isacharunderscore}B{\isachardot}induct{\isacharcomma}\ auto{\isacharparenright}%
+\ {\isaliteral{28}{\isacharparenleft}}rule\ S{\isaliteral{5F}{\isacharunderscore}}A{\isaliteral{5F}{\isacharunderscore}}B{\isaliteral{2E}{\isachardot}}induct{\isaliteral{2C}{\isacharcomma}}\ auto{\isaliteral{29}{\isacharparenright}}%
\endisatagproof
{\isafoldproof}%
%
@@ -141,13 +141,13 @@
word. We start with 0 and end (at the right end) with 2. Since each move to the
right increases or decreases the difference by 1, we must have passed through
1 on our way from 0 to 2. Formally, we appeal to the following discrete
-intermediate value theorem \isa{nat{\isadigit{0}}{\isacharunderscore}intermed{\isacharunderscore}int{\isacharunderscore}val}
+intermediate value theorem \isa{nat{\isadigit{0}}{\isaliteral{5F}{\isacharunderscore}}intermed{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5F}{\isacharunderscore}}val}
\begin{isabelle}%
-\ \ \ \ \ {\isasymlbrakk}{\isasymforall}i{\isacharless}n{\isachardot}\ {\isasymbar}f\ {\isacharparenleft}i\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}\ {\isacharminus}\ f\ i{\isasymbar}\ {\isasymle}\ {\isadigit{1}}{\isacharsemicolon}\ f\ {\isadigit{0}}\ {\isasymle}\ k{\isacharsemicolon}\ k\ {\isasymle}\ f\ n{\isasymrbrakk}\isanewline
-\isaindent{\ \ \ \ \ }{\isasymLongrightarrow}\ {\isasymexists}i{\isasymle}n{\isachardot}\ f\ i\ {\isacharequal}\ k%
+\ \ \ \ \ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}i{\isaliteral{3C}{\isacharless}}n{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6261723E}{\isasymbar}}f\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2D}{\isacharminus}}\ f\ i{\isaliteral{5C3C6261723E}{\isasymbar}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isadigit{1}}{\isaliteral{3B}{\isacharsemicolon}}\ f\ {\isadigit{0}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ k{\isaliteral{3B}{\isacharsemicolon}}\ k\ {\isaliteral{5C3C6C653E}{\isasymle}}\ f\ n{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
+\isaindent{\ \ \ \ \ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}i{\isaliteral{5C3C6C653E}{\isasymle}}n{\isaliteral{2E}{\isachardot}}\ f\ i\ {\isaliteral{3D}{\isacharequal}}\ k%
\end{isabelle}
-where \isa{f} is of type \isa{nat\ {\isasymRightarrow}\ int}, \isa{int} are the integers,
-\isa{{\isasymbar}{\isachardot}{\isasymbar}} is the absolute value function\footnote{See
+where \isa{f} is of type \isa{nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ int}, \isa{int} are the integers,
+\isa{{\isaliteral{5C3C6261723E}{\isasymbar}}{\isaliteral{2E}{\isachardot}}{\isaliteral{5C3C6261723E}{\isasymbar}}} is the absolute value function\footnote{See
Table~\ref{tab:ascii} in the Appendix for the correct \textsc{ascii}
syntax.}, and \isa{{\isadigit{1}}} is the integer 1 (see \S\ref{sec:numbers}).
@@ -160,9 +160,9 @@
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
-\ step{\isadigit{1}}{\isacharcolon}\ {\isachardoublequoteopen}{\isasymforall}i\ {\isacharless}\ size\ w{\isachardot}\isanewline
-\ \ {\isasymbar}{\isacharparenleft}int{\isacharparenleft}size{\isacharbrackleft}x{\isasymleftarrow}take\ {\isacharparenleft}i{\isacharplus}{\isadigit{1}}{\isacharparenright}\ w{\isachardot}\ P\ x{\isacharbrackright}{\isacharparenright}{\isacharminus}int{\isacharparenleft}size{\isacharbrackleft}x{\isasymleftarrow}take\ {\isacharparenleft}i{\isacharplus}{\isadigit{1}}{\isacharparenright}\ w{\isachardot}\ {\isasymnot}P\ x{\isacharbrackright}{\isacharparenright}{\isacharparenright}\isanewline
-\ \ \ {\isacharminus}\ {\isacharparenleft}int{\isacharparenleft}size{\isacharbrackleft}x{\isasymleftarrow}take\ i\ w{\isachardot}\ P\ x{\isacharbrackright}{\isacharparenright}{\isacharminus}int{\isacharparenleft}size{\isacharbrackleft}x{\isasymleftarrow}take\ i\ w{\isachardot}\ {\isasymnot}P\ x{\isacharbrackright}{\isacharparenright}{\isacharparenright}{\isasymbar}\ {\isasymle}\ {\isadigit{1}}{\isachardoublequoteclose}%
+\ step{\isadigit{1}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}i\ {\isaliteral{3C}{\isacharless}}\ size\ w{\isaliteral{2E}{\isachardot}}\isanewline
+\ \ {\isaliteral{5C3C6261723E}{\isasymbar}}{\isaliteral{28}{\isacharparenleft}}int{\isaliteral{28}{\isacharparenleft}}size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}take\ {\isaliteral{28}{\isacharparenleft}}i{\isaliteral{2B}{\isacharplus}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ w{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2D}{\isacharminus}}int{\isaliteral{28}{\isacharparenleft}}size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}take\ {\isaliteral{28}{\isacharparenleft}}i{\isaliteral{2B}{\isacharplus}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ w{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}P\ x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ {\isaliteral{2D}{\isacharminus}}\ {\isaliteral{28}{\isacharparenleft}}int{\isaliteral{28}{\isacharparenleft}}size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}take\ i\ w{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2D}{\isacharminus}}int{\isaliteral{28}{\isacharparenleft}}size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}take\ i\ w{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}P\ x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C6261723E}{\isasymbar}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}%
\isadelimproof
%
\endisadelimproof
@@ -172,7 +172,7 @@
\begin{isamarkuptxt}%
\noindent
The lemma is a bit hard to read because of the coercion function
-\isa{int\ {\isacharcolon}{\isacharcolon}\ nat\ {\isasymRightarrow}\ int}. It is required because \isa{size} returns
+\isa{int\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ int}. It is required because \isa{size} returns
a natural number, but subtraction on type~\isa{nat} will do the wrong thing.
Function \isa{take} is predefined and \isa{take\ i\ xs} is the prefix of
length \isa{i} of \isa{xs}; below we also need \isa{drop\ i\ xs}, which
@@ -184,9 +184,9 @@
\end{isamarkuptxt}%
\isamarkuptrue%
\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}induct{\isacharunderscore}tac\ w{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ w{\isaliteral{29}{\isacharparenright}}\isanewline
\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}auto\ simp\ add{\isacharcolon}\ abs{\isacharunderscore}if\ take{\isacharunderscore}Cons\ split{\isacharcolon}\ nat{\isachardot}split{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}auto\ simp\ add{\isaliteral{3A}{\isacharcolon}}\ abs{\isaliteral{5F}{\isacharunderscore}}if\ take{\isaliteral{5F}{\isacharunderscore}}Cons\ split{\isaliteral{3A}{\isacharcolon}}\ nat{\isaliteral{2E}{\isachardot}}split{\isaliteral{29}{\isacharparenright}}\isanewline
\isacommand{done}\isamarkupfalse%
%
\endisatagproof
@@ -201,9 +201,9 @@
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
-\ part{\isadigit{1}}{\isacharcolon}\isanewline
-\ {\isachardoublequoteopen}size{\isacharbrackleft}x{\isasymleftarrow}w{\isachardot}\ P\ x{\isacharbrackright}\ {\isacharequal}\ size{\isacharbrackleft}x{\isasymleftarrow}w{\isachardot}\ {\isasymnot}P\ x{\isacharbrackright}{\isacharplus}{\isadigit{2}}\ {\isasymLongrightarrow}\isanewline
-\ \ {\isasymexists}i{\isasymle}size\ w{\isachardot}\ size{\isacharbrackleft}x{\isasymleftarrow}take\ i\ w{\isachardot}\ P\ x{\isacharbrackright}\ {\isacharequal}\ size{\isacharbrackleft}x{\isasymleftarrow}take\ i\ w{\isachardot}\ {\isasymnot}P\ x{\isacharbrackright}{\isacharplus}{\isadigit{1}}{\isachardoublequoteclose}%
+\ part{\isadigit{1}}{\isaliteral{3A}{\isacharcolon}}\isanewline
+\ {\isaliteral{22}{\isachardoublequoteopen}}size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}w{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}w{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}P\ x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{2B}{\isacharplus}}{\isadigit{2}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
+\ \ {\isaliteral{5C3C6578697374733E}{\isasymexists}}i{\isaliteral{5C3C6C653E}{\isasymle}}size\ w{\isaliteral{2E}{\isachardot}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}take\ i\ w{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}take\ i\ w{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}P\ x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{2B}{\isacharplus}}{\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}%
\isadelimproof
%
\endisadelimproof
@@ -218,7 +218,7 @@
\end{isamarkuptxt}%
\isamarkuptrue%
\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}insert\ nat{\isadigit{0}}{\isacharunderscore}intermed{\isacharunderscore}int{\isacharunderscore}val{\isacharbrackleft}OF\ step{\isadigit{1}}{\isacharcomma}\ of\ {\isachardoublequoteopen}P{\isachardoublequoteclose}\ {\isachardoublequoteopen}w{\isachardoublequoteclose}\ {\isachardoublequoteopen}{\isadigit{1}}{\isachardoublequoteclose}{\isacharbrackright}{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}insert\ nat{\isadigit{0}}{\isaliteral{5F}{\isacharunderscore}}intermed{\isaliteral{5F}{\isacharunderscore}}int{\isaliteral{5F}{\isacharunderscore}}val{\isaliteral{5B}{\isacharbrackleft}}OF\ step{\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ of\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{22}{\isachardoublequoteopen}}w{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline
\isacommand{by}\isamarkupfalse%
\ force%
\endisatagproof
@@ -236,11 +236,11 @@
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
-\ part{\isadigit{2}}{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}{\isasymlbrakk}size{\isacharbrackleft}x{\isasymleftarrow}take\ i\ w\ {\isacharat}\ drop\ i\ w{\isachardot}\ P\ x{\isacharbrackright}\ {\isacharequal}\isanewline
-\ \ \ \ size{\isacharbrackleft}x{\isasymleftarrow}take\ i\ w\ {\isacharat}\ drop\ i\ w{\isachardot}\ {\isasymnot}P\ x{\isacharbrackright}{\isacharplus}{\isadigit{2}}{\isacharsemicolon}\isanewline
-\ \ \ \ size{\isacharbrackleft}x{\isasymleftarrow}take\ i\ w{\isachardot}\ P\ x{\isacharbrackright}\ {\isacharequal}\ size{\isacharbrackleft}x{\isasymleftarrow}take\ i\ w{\isachardot}\ {\isasymnot}P\ x{\isacharbrackright}{\isacharplus}{\isadigit{1}}{\isasymrbrakk}\isanewline
-\ \ \ {\isasymLongrightarrow}\ size{\isacharbrackleft}x{\isasymleftarrow}drop\ i\ w{\isachardot}\ P\ x{\isacharbrackright}\ {\isacharequal}\ size{\isacharbrackleft}x{\isasymleftarrow}drop\ i\ w{\isachardot}\ {\isasymnot}P\ x{\isacharbrackright}{\isacharplus}{\isadigit{1}}{\isachardoublequoteclose}\isanewline
+\ part{\isadigit{2}}{\isaliteral{3A}{\isacharcolon}}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}take\ i\ w\ {\isaliteral{40}{\isacharat}}\ drop\ i\ w{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline
+\ \ \ \ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}take\ i\ w\ {\isaliteral{40}{\isacharat}}\ drop\ i\ w{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}P\ x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{2B}{\isacharplus}}{\isadigit{2}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline
+\ \ \ \ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}take\ i\ w{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}take\ i\ w{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}P\ x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{2B}{\isacharplus}}{\isadigit{1}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
+\ \ \ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}drop\ i\ w{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}drop\ i\ w{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}P\ x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{2B}{\isacharplus}}{\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
%
\isadelimproof
%
@@ -248,7 +248,7 @@
%
\isatagproof
\isacommand{by}\isamarkupfalse%
-{\isacharparenleft}simp\ del{\isacharcolon}\ append{\isacharunderscore}take{\isacharunderscore}drop{\isacharunderscore}id{\isacharparenright}%
+{\isaliteral{28}{\isacharparenleft}}simp\ del{\isaliteral{3A}{\isacharcolon}}\ append{\isaliteral{5F}{\isacharunderscore}}take{\isaliteral{5F}{\isacharunderscore}}drop{\isaliteral{5F}{\isacharunderscore}}id{\isaliteral{29}{\isacharparenright}}%
\endisatagproof
{\isafoldproof}%
%
@@ -260,12 +260,12 @@
\noindent
In the proof we have disabled the normally useful lemma
\begin{isabelle}
-\isa{take\ n\ xs\ {\isacharat}\ drop\ n\ xs\ {\isacharequal}\ xs}
+\isa{take\ n\ xs\ {\isaliteral{40}{\isacharat}}\ drop\ n\ xs\ {\isaliteral{3D}{\isacharequal}}\ xs}
\rulename{append_take_drop_id}
\end{isabelle}
to allow the simplifier to apply the following lemma instead:
\begin{isabelle}%
-\ \ \ \ \ {\isacharbrackleft}x{\isasymin}xs{\isacharat}ys{\isachardot}\ P\ x{\isacharbrackright}\ {\isacharequal}\ {\isacharbrackleft}x{\isasymin}xs{\isachardot}\ P\ x{\isacharbrackright}\ {\isacharat}\ {\isacharbrackleft}x{\isasymin}ys{\isachardot}\ P\ x{\isacharbrackright}%
+\ \ \ \ \ {\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C696E3E}{\isasymin}}xs{\isaliteral{40}{\isacharat}}ys{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C696E3E}{\isasymin}}xs{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C696E3E}{\isasymin}}ys{\isaliteral{2E}{\isachardot}}\ P\ x{\isaliteral{5D}{\isacharbrackright}}%
\end{isabelle}
To dispose of trivial cases automatically, the rules of the inductive
@@ -273,7 +273,7 @@
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{declare}\isamarkupfalse%
-\ S{\isacharunderscore}A{\isacharunderscore}B{\isachardot}intros{\isacharbrackleft}simp{\isacharbrackright}%
+\ S{\isaliteral{5F}{\isacharunderscore}}A{\isaliteral{5F}{\isacharunderscore}}B{\isaliteral{2E}{\isachardot}}intros{\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}%
\begin{isamarkuptext}%
\noindent
This could have been done earlier but was not necessary so far.
@@ -284,10 +284,10 @@
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{theorem}\isamarkupfalse%
-\ completeness{\isacharcolon}\isanewline
-\ \ {\isachardoublequoteopen}{\isacharparenleft}size{\isacharbrackleft}x{\isasymleftarrow}w{\isachardot}\ x{\isacharequal}a{\isacharbrackright}\ {\isacharequal}\ size{\isacharbrackleft}x{\isasymleftarrow}w{\isachardot}\ x{\isacharequal}b{\isacharbrackright}\ \ \ \ \ {\isasymlongrightarrow}\ w\ {\isasymin}\ S{\isacharparenright}\ {\isasymand}\isanewline
-\ \ \ {\isacharparenleft}size{\isacharbrackleft}x{\isasymleftarrow}w{\isachardot}\ x{\isacharequal}a{\isacharbrackright}\ {\isacharequal}\ size{\isacharbrackleft}x{\isasymleftarrow}w{\isachardot}\ x{\isacharequal}b{\isacharbrackright}\ {\isacharplus}\ {\isadigit{1}}\ {\isasymlongrightarrow}\ w\ {\isasymin}\ A{\isacharparenright}\ {\isasymand}\isanewline
-\ \ \ {\isacharparenleft}size{\isacharbrackleft}x{\isasymleftarrow}w{\isachardot}\ x{\isacharequal}b{\isacharbrackright}\ {\isacharequal}\ size{\isacharbrackleft}x{\isasymleftarrow}w{\isachardot}\ x{\isacharequal}a{\isacharbrackright}\ {\isacharplus}\ {\isadigit{1}}\ {\isasymlongrightarrow}\ w\ {\isasymin}\ B{\isacharparenright}{\isachardoublequoteclose}%
+\ completeness{\isaliteral{3A}{\isacharcolon}}\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}w{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}a{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}w{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}b{\isaliteral{5D}{\isacharbrackright}}\ \ \ \ \ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ S{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\isanewline
+\ \ \ {\isaliteral{28}{\isacharparenleft}}size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}w{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}a{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}w{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}b{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\isanewline
+\ \ \ {\isaliteral{28}{\isacharparenleft}}size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}w{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}b{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ size{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}w{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}a{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ w\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
\isadelimproof
%
\endisadelimproof
@@ -299,16 +299,16 @@
The proof is by induction on \isa{w}. Structural induction would fail here
because, as we can see from the grammar, we need to make bigger steps than
merely appending a single letter at the front. Hence we induct on the length
-of \isa{w}, using the induction rule \isa{length{\isacharunderscore}induct}:%
+of \isa{w}, using the induction rule \isa{length{\isaliteral{5F}{\isacharunderscore}}induct}:%
\end{isamarkuptxt}%
\isamarkuptrue%
\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}induct{\isacharunderscore}tac\ w\ rule{\isacharcolon}\ length{\isacharunderscore}induct{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ w\ rule{\isaliteral{3A}{\isacharcolon}}\ length{\isaliteral{5F}{\isacharunderscore}}induct{\isaliteral{29}{\isacharparenright}}\isanewline
\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}rename{\isacharunderscore}tac\ w{\isacharparenright}%
+{\isaliteral{28}{\isacharparenleft}}rename{\isaliteral{5F}{\isacharunderscore}}tac\ w{\isaliteral{29}{\isacharparenright}}%
\begin{isamarkuptxt}%
\noindent
-The \isa{rule} parameter tells \isa{induct{\isacharunderscore}tac} explicitly which induction
+The \isa{rule} parameter tells \isa{induct{\isaliteral{5F}{\isacharunderscore}}tac} explicitly which induction
rule to use. For details see \S\ref{sec:complete-ind} below.
In this case the result is that we may assume the lemma already
holds for all words shorter than \isa{w}. Because the induction step renames
@@ -319,18 +319,18 @@
\end{isamarkuptxt}%
\isamarkuptrue%
\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}case{\isacharunderscore}tac\ w{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}case{\isaliteral{5F}{\isacharunderscore}}tac\ w{\isaliteral{29}{\isacharparenright}}\isanewline
\ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}simp{\isacharunderscore}all{\isacharparenright}%
+{\isaliteral{28}{\isacharparenleft}}simp{\isaliteral{5F}{\isacharunderscore}}all{\isaliteral{29}{\isacharparenright}}%
\begin{isamarkuptxt}%
\noindent
Simplification disposes of the base case and leaves only a conjunction
of two step cases to be proved:
-if \isa{w\ {\isacharequal}\ a\ {\isacharhash}\ v} and \begin{isabelle}%
-\ \ \ \ \ length\ {\isacharparenleft}if\ x\ {\isacharequal}\ a\ then\ {\isacharbrackleft}x\ {\isasymin}\ v{\isacharbrackright}\ else\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\isanewline
-\isaindent{\ \ \ \ \ }length\ {\isacharparenleft}if\ x\ {\isacharequal}\ b\ then\ {\isacharbrackleft}x\ {\isasymin}\ v{\isacharbrackright}\ else\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharplus}\ {\isadigit{2}}%
+if \isa{w\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{23}{\isacharhash}}\ v} and \begin{isabelle}%
+\ \ \ \ \ length\ {\isaliteral{28}{\isacharparenleft}}if\ x\ {\isaliteral{3D}{\isacharequal}}\ a\ then\ {\isaliteral{5B}{\isacharbrackleft}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ v{\isaliteral{5D}{\isacharbrackright}}\ else\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline
+\isaindent{\ \ \ \ \ }length\ {\isaliteral{28}{\isacharparenleft}}if\ x\ {\isaliteral{3D}{\isacharequal}}\ b\ then\ {\isaliteral{5B}{\isacharbrackleft}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ v{\isaliteral{5D}{\isacharbrackright}}\ else\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}%
\end{isabelle} then
-\isa{b\ {\isacharhash}\ v\ {\isasymin}\ A}, and similarly for \isa{w\ {\isacharequal}\ b\ {\isacharhash}\ v}.
+\isa{b\ {\isaliteral{23}{\isacharhash}}\ v\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A}, and similarly for \isa{w\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{23}{\isacharhash}}\ v}.
We only consider the first case in detail.
After breaking the conjunction up into two cases, we can apply
@@ -338,77 +338,77 @@
\end{isamarkuptxt}%
\isamarkuptrue%
\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}rule\ conjI{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}rule\ conjI{\isaliteral{29}{\isacharparenright}}\isanewline
\ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}clarify{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}clarify{\isaliteral{29}{\isacharparenright}}\isanewline
\ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}frule\ part{\isadigit{1}}{\isacharbrackleft}of\ {\isachardoublequoteopen}{\isasymlambda}x{\isachardot}\ x{\isacharequal}a{\isachardoublequoteclose}{\isacharcomma}\ simplified{\isacharbrackright}{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}frule\ part{\isadigit{1}}{\isaliteral{5B}{\isacharbrackleft}}of\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}a{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{2C}{\isacharcomma}}\ simplified{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline
\ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}clarify{\isacharparenright}%
+{\isaliteral{28}{\isacharparenleft}}clarify{\isaliteral{29}{\isacharparenright}}%
\begin{isamarkuptxt}%
\noindent
-This yields an index \isa{i\ {\isasymle}\ length\ v} such that
+This yields an index \isa{i\ {\isaliteral{5C3C6C653E}{\isasymle}}\ length\ v} such that
\begin{isabelle}%
-\ \ \ \ \ length\ {\isacharbrackleft}x{\isasymleftarrow}take\ i\ v\ {\isachardot}\ x\ {\isacharequal}\ a{\isacharbrackright}\ {\isacharequal}\ length\ {\isacharbrackleft}x{\isasymleftarrow}take\ i\ v\ {\isachardot}\ x\ {\isacharequal}\ b{\isacharbrackright}\ {\isacharplus}\ {\isadigit{1}}%
+\ \ \ \ \ length\ {\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}take\ i\ v\ {\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ length\ {\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}take\ i\ v\ {\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}%
\end{isabelle}
With the help of \isa{part{\isadigit{2}}} it follows that
\begin{isabelle}%
-\ \ \ \ \ length\ {\isacharbrackleft}x{\isasymleftarrow}drop\ i\ v\ {\isachardot}\ x\ {\isacharequal}\ a{\isacharbrackright}\ {\isacharequal}\ length\ {\isacharbrackleft}x{\isasymleftarrow}drop\ i\ v\ {\isachardot}\ x\ {\isacharequal}\ b{\isacharbrackright}\ {\isacharplus}\ {\isadigit{1}}%
+\ \ \ \ \ length\ {\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}drop\ i\ v\ {\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ length\ {\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5C3C6C6566746172726F773E}{\isasymleftarrow}}drop\ i\ v\ {\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}%
\end{isabelle}%
\end{isamarkuptxt}%
\isamarkuptrue%
\ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}drule\ part{\isadigit{2}}{\isacharbrackleft}of\ {\isachardoublequoteopen}{\isasymlambda}x{\isachardot}\ x{\isacharequal}a{\isachardoublequoteclose}{\isacharcomma}\ simplified{\isacharbrackright}{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}drule\ part{\isadigit{2}}{\isaliteral{5B}{\isacharbrackleft}}of\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}a{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{2C}{\isacharcomma}}\ simplified{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline
\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}assumption{\isacharparenright}%
+{\isaliteral{28}{\isacharparenleft}}assumption{\isaliteral{29}{\isacharparenright}}%
\begin{isamarkuptxt}%
\noindent
-Now it is time to decompose \isa{v} in the conclusion \isa{b\ {\isacharhash}\ v\ {\isasymin}\ A}
-into \isa{take\ i\ v\ {\isacharat}\ drop\ i\ v},%
+Now it is time to decompose \isa{v} in the conclusion \isa{b\ {\isaliteral{23}{\isacharhash}}\ v\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A}
+into \isa{take\ i\ v\ {\isaliteral{40}{\isacharat}}\ drop\ i\ v},%
\end{isamarkuptxt}%
\isamarkuptrue%
\ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}rule{\isacharunderscore}tac\ n{\isadigit{1}}{\isacharequal}i\ \isakeyword{and}\ t{\isacharequal}v\ \isakeyword{in}\ subst{\isacharbrackleft}OF\ append{\isacharunderscore}take{\isacharunderscore}drop{\isacharunderscore}id{\isacharbrackright}{\isacharparenright}%
+{\isaliteral{28}{\isacharparenleft}}rule{\isaliteral{5F}{\isacharunderscore}}tac\ n{\isadigit{1}}{\isaliteral{3D}{\isacharequal}}i\ \isakeyword{and}\ t{\isaliteral{3D}{\isacharequal}}v\ \isakeyword{in}\ subst{\isaliteral{5B}{\isacharbrackleft}}OF\ append{\isaliteral{5F}{\isacharunderscore}}take{\isaliteral{5F}{\isacharunderscore}}drop{\isaliteral{5F}{\isacharunderscore}}id{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}%
\begin{isamarkuptxt}%
\noindent
(the variables \isa{n{\isadigit{1}}} and \isa{t} are the result of composing the
-theorems \isa{subst} and \isa{append{\isacharunderscore}take{\isacharunderscore}drop{\isacharunderscore}id})
+theorems \isa{subst} and \isa{append{\isaliteral{5F}{\isacharunderscore}}take{\isaliteral{5F}{\isacharunderscore}}drop{\isaliteral{5F}{\isacharunderscore}}id})
after which the appropriate rule of the grammar reduces the goal
-to the two subgoals \isa{take\ i\ v\ {\isasymin}\ A} and \isa{drop\ i\ v\ {\isasymin}\ A}:%
+to the two subgoals \isa{take\ i\ v\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A} and \isa{drop\ i\ v\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A}:%
\end{isamarkuptxt}%
\isamarkuptrue%
\ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}rule\ S{\isacharunderscore}A{\isacharunderscore}B{\isachardot}intros{\isacharparenright}%
+{\isaliteral{28}{\isacharparenleft}}rule\ S{\isaliteral{5F}{\isacharunderscore}}A{\isaliteral{5F}{\isacharunderscore}}B{\isaliteral{2E}{\isachardot}}intros{\isaliteral{29}{\isacharparenright}}%
\begin{isamarkuptxt}%
Both subgoals follow from the induction hypothesis because both \isa{take\ i\ v} and \isa{drop\ i\ v} are shorter than \isa{w}:%
\end{isamarkuptxt}%
\isamarkuptrue%
\ \ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}force\ simp\ add{\isacharcolon}\ min{\isacharunderscore}less{\isacharunderscore}iff{\isacharunderscore}disj{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}force\ simp\ add{\isaliteral{3A}{\isacharcolon}}\ min{\isaliteral{5F}{\isacharunderscore}}less{\isaliteral{5F}{\isacharunderscore}}iff{\isaliteral{5F}{\isacharunderscore}}disj{\isaliteral{29}{\isacharparenright}}\isanewline
\ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}force\ split\ add{\isacharcolon}\ nat{\isacharunderscore}diff{\isacharunderscore}split{\isacharparenright}%
+{\isaliteral{28}{\isacharparenleft}}force\ split\ add{\isaliteral{3A}{\isacharcolon}}\ nat{\isaliteral{5F}{\isacharunderscore}}diff{\isaliteral{5F}{\isacharunderscore}}split{\isaliteral{29}{\isacharparenright}}%
\begin{isamarkuptxt}%
-The case \isa{w\ {\isacharequal}\ b\ {\isacharhash}\ v} is proved analogously:%
+The case \isa{w\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{23}{\isacharhash}}\ v} is proved analogously:%
\end{isamarkuptxt}%
\isamarkuptrue%
\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}clarify{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}clarify{\isaliteral{29}{\isacharparenright}}\isanewline
\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}frule\ part{\isadigit{1}}{\isacharbrackleft}of\ {\isachardoublequoteopen}{\isasymlambda}x{\isachardot}\ x{\isacharequal}b{\isachardoublequoteclose}{\isacharcomma}\ simplified{\isacharbrackright}{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}frule\ part{\isadigit{1}}{\isaliteral{5B}{\isacharbrackleft}}of\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}b{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{2C}{\isacharcomma}}\ simplified{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline
\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}clarify{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}clarify{\isaliteral{29}{\isacharparenright}}\isanewline
\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}drule\ part{\isadigit{2}}{\isacharbrackleft}of\ {\isachardoublequoteopen}{\isasymlambda}x{\isachardot}\ x{\isacharequal}b{\isachardoublequoteclose}{\isacharcomma}\ simplified{\isacharbrackright}{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}drule\ part{\isadigit{2}}{\isaliteral{5B}{\isacharbrackleft}}of\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ x{\isaliteral{3D}{\isacharequal}}b{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{2C}{\isacharcomma}}\ simplified{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline
\ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}assumption{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}assumption{\isaliteral{29}{\isacharparenright}}\isanewline
\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}rule{\isacharunderscore}tac\ n{\isadigit{1}}{\isacharequal}i\ \isakeyword{and}\ t{\isacharequal}v\ \isakeyword{in}\ subst{\isacharbrackleft}OF\ append{\isacharunderscore}take{\isacharunderscore}drop{\isacharunderscore}id{\isacharbrackright}{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}rule{\isaliteral{5F}{\isacharunderscore}}tac\ n{\isadigit{1}}{\isaliteral{3D}{\isacharequal}}i\ \isakeyword{and}\ t{\isaliteral{3D}{\isacharequal}}v\ \isakeyword{in}\ subst{\isaliteral{5B}{\isacharbrackleft}}OF\ append{\isaliteral{5F}{\isacharunderscore}}take{\isaliteral{5F}{\isacharunderscore}}drop{\isaliteral{5F}{\isacharunderscore}}id{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline
\isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}rule\ S{\isacharunderscore}A{\isacharunderscore}B{\isachardot}intros{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}rule\ S{\isaliteral{5F}{\isacharunderscore}}A{\isaliteral{5F}{\isacharunderscore}}B{\isaliteral{2E}{\isachardot}}intros{\isaliteral{29}{\isacharparenright}}\isanewline
\ \isacommand{apply}\isamarkupfalse%
-{\isacharparenleft}force\ simp\ add{\isacharcolon}\ min{\isacharunderscore}less{\isacharunderscore}iff{\isacharunderscore}disj{\isacharparenright}\isanewline
+{\isaliteral{28}{\isacharparenleft}}force\ simp\ add{\isaliteral{3A}{\isacharcolon}}\ min{\isaliteral{5F}{\isacharunderscore}}less{\isaliteral{5F}{\isacharunderscore}}iff{\isaliteral{5F}{\isacharunderscore}}disj{\isaliteral{29}{\isacharparenright}}\isanewline
\isacommand{by}\isamarkupfalse%
-{\isacharparenleft}force\ simp\ add{\isacharcolon}\ min{\isacharunderscore}less{\isacharunderscore}iff{\isacharunderscore}disj\ split\ add{\isacharcolon}\ nat{\isacharunderscore}diff{\isacharunderscore}split{\isacharparenright}%
+{\isaliteral{28}{\isacharparenleft}}force\ simp\ add{\isaliteral{3A}{\isacharcolon}}\ min{\isaliteral{5F}{\isacharunderscore}}less{\isaliteral{5F}{\isacharunderscore}}iff{\isaliteral{5F}{\isacharunderscore}}disj\ split\ add{\isaliteral{3A}{\isacharcolon}}\ nat{\isaliteral{5F}{\isacharunderscore}}diff{\isaliteral{5F}{\isacharunderscore}}split{\isaliteral{29}{\isacharparenright}}%
\endisatagproof
{\isafoldproof}%
%
@@ -429,7 +429,7 @@
length of a word. This deprives them of the beauty of rule induction,
and in the easy direction (correctness) their reasoning is more
detailed than our \isa{auto}. For the hard part (completeness), they
-consider just one of the cases that our \isa{simp{\isacharunderscore}all} disposes of
+consider just one of the cases that our \isa{simp{\isaliteral{5F}{\isacharunderscore}}all} disposes of
automatically. Then they conclude the proof by saying about the
remaining cases: ``We do this in a manner similar to our method of
proof for part (1); this part is left to the reader''. But this is