doc-src/TutorialI/Datatype/document/ABexpr.tex

 %
 \begin{isabellebody}%
 \def\isabellecontext{ABexpr}%
-\isamarkupfalse%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\isamarkuptrue%
 %
 \begin{isamarkuptext}%
 \index{datatypes!mutually recursive}%
 Sometimes it is necessary to define two datatypes that depend on each
 other. This is called \textbf{mutual recursion}. As an example consider a

 \item boolean expressions contain arithmetic expressions because of
   comparisons like ``$m<n$''.
 \end{itemize}
 In Isabelle this becomes%
 \end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
 \isacommand{datatype}\ {\isacharprime}a\ aexp\ {\isacharequal}\ IF\ \ \ {\isachardoublequote}{\isacharprime}a\ bexp{\isachardoublequote}\ {\isachardoublequote}{\isacharprime}a\ aexp{\isachardoublequote}\ {\isachardoublequote}{\isacharprime}a\ aexp{\isachardoublequote}\isanewline
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Sum\ \ {\isachardoublequote}{\isacharprime}a\ aexp{\isachardoublequote}\ {\isachardoublequote}{\isacharprime}a\ aexp{\isachardoublequote}\isanewline
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Diff\ {\isachardoublequote}{\isacharprime}a\ aexp{\isachardoublequote}\ {\isachardoublequote}{\isacharprime}a\ aexp{\isachardoublequote}\isanewline
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Var\ {\isacharprime}a\isanewline
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Num\ nat\isanewline
 \isakeyword{and}\ \ \ \ \ \ {\isacharprime}a\ bexp\ {\isacharequal}\ Less\ {\isachardoublequote}{\isacharprime}a\ aexp{\isachardoublequote}\ {\isachardoublequote}{\isacharprime}a\ aexp{\isachardoublequote}\isanewline
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ And\ \ {\isachardoublequote}{\isacharprime}a\ bexp{\isachardoublequote}\ {\isachardoublequote}{\isacharprime}a\ bexp{\isachardoublequote}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Neg\ \ {\isachardoublequote}{\isacharprime}a\ bexp{\isachardoublequote}\isamarkupfalse%
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isacharbar}\ Neg\ \ {\isachardoublequote}{\isacharprime}a\ bexp{\isachardoublequote}\isamarkuptrue%
 %
 \begin{isamarkuptext}%
 \noindent
 Type \isa{aexp} is similar to \isa{expr} in \S\ref{sec:ExprCompiler},
 except that we have added an \isa{IF} constructor,
 fixed the values to be of type \isa{nat} and declared the two binary
 operations \isa{Sum} and \isa{Diff}.  Boolean
 expressions can be arithmetic comparisons, conjunctions and negations.
 The semantics is given by two evaluation functions:%
 \end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
 \isacommand{consts}\ \ evala\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ aexp\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline
-\ \ \ \ \ \ \ \ evalb\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ bexp\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ bool{\isachardoublequote}\isamarkupfalse%
+\ \ \ \ \ \ \ \ evalb\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ bexp\ {\isasymRightarrow}\ {\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ nat{\isacharparenright}\ {\isasymRightarrow}\ bool{\isachardoublequote}\isamarkuptrue%
 %
 \begin{isamarkuptext}%
 \noindent
 Both take an expression and an environment (a mapping from variables \isa{{\isacharprime}a} to values
 \isa{nat}) and return its arithmetic/boolean
 value. Since the datatypes are mutually recursive, so are functions that
 operate on them. Hence they need to be defined in a single \isacommand{primrec}
 section:%
 \end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
 \isacommand{primrec}\isanewline
 \ \ {\isachardoublequote}evala\ {\isacharparenleft}IF\ b\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\isanewline
 \ \ \ \ \ {\isacharparenleft}if\ evalb\ b\ env\ then\ evala\ a{\isadigit{1}}\ env\ else\ evala\ a{\isadigit{2}}\ env{\isacharparenright}{\isachardoublequote}\isanewline
 \ \ {\isachardoublequote}evala\ {\isacharparenleft}Sum\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\ evala\ a{\isadigit{1}}\ env\ {\isacharplus}\ evala\ a{\isadigit{2}}\ env{\isachardoublequote}\isanewline
 \ \ {\isachardoublequote}evala\ {\isacharparenleft}Diff\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\ evala\ a{\isadigit{1}}\ env\ {\isacharminus}\ evala\ a{\isadigit{2}}\ env{\isachardoublequote}\isanewline
 \ \ {\isachardoublequote}evala\ {\isacharparenleft}Var\ v{\isacharparenright}\ env\ {\isacharequal}\ env\ v{\isachardoublequote}\isanewline
 \ \ {\isachardoublequote}evala\ {\isacharparenleft}Num\ n{\isacharparenright}\ env\ {\isacharequal}\ n{\isachardoublequote}\isanewline
 \isanewline
 \ \ {\isachardoublequote}evalb\ {\isacharparenleft}Less\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\ {\isacharparenleft}evala\ a{\isadigit{1}}\ env\ {\isacharless}\ evala\ a{\isadigit{2}}\ env{\isacharparenright}{\isachardoublequote}\isanewline
 \ \ {\isachardoublequote}evalb\ {\isacharparenleft}And\ b{\isadigit{1}}\ b{\isadigit{2}}{\isacharparenright}\ env\ {\isacharequal}\ {\isacharparenleft}evalb\ b{\isadigit{1}}\ env\ {\isasymand}\ evalb\ b{\isadigit{2}}\ env{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ {\isachardoublequote}evalb\ {\isacharparenleft}Neg\ b{\isacharparenright}\ env\ {\isacharequal}\ {\isacharparenleft}{\isasymnot}\ evalb\ b\ env{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
+\ \ {\isachardoublequote}evalb\ {\isacharparenleft}Neg\ b{\isacharparenright}\ env\ {\isacharequal}\ {\isacharparenleft}{\isasymnot}\ evalb\ b\ env{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
 %
 \begin{isamarkuptext}%
 \noindent
 In the same fashion we also define two functions that perform substitution:%
 \end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
 \isacommand{consts}\ substa\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ aexp{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ aexp\ {\isasymRightarrow}\ {\isacharprime}b\ aexp{\isachardoublequote}\isanewline
-\ \ \ \ \ \ \ substb\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ aexp{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ bexp\ {\isasymRightarrow}\ {\isacharprime}b\ bexp{\isachardoublequote}\isamarkupfalse%
+\ \ \ \ \ \ \ substb\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ aexp{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ bexp\ {\isasymRightarrow}\ {\isacharprime}b\ bexp{\isachardoublequote}\isamarkuptrue%
 %
 \begin{isamarkuptext}%
 \noindent
 The first argument is a function mapping variables to expressions, the
 substitution. It is applied to all variables in the second argument. As a
 result, the type of variables in the expression may change from \isa{{\isacharprime}a}
 to \isa{{\isacharprime}b}. Note that there are only arithmetic and no boolean variables.%
 \end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
 \isacommand{primrec}\isanewline
 \ \ {\isachardoublequote}substa\ s\ {\isacharparenleft}IF\ b\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\isanewline
 \ \ \ \ \ IF\ {\isacharparenleft}substb\ s\ b{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{2}}{\isacharparenright}{\isachardoublequote}\isanewline
 \ \ {\isachardoublequote}substa\ s\ {\isacharparenleft}Sum\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ Sum\ {\isacharparenleft}substa\ s\ a{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{2}}{\isacharparenright}{\isachardoublequote}\isanewline
 \ \ {\isachardoublequote}substa\ s\ {\isacharparenleft}Diff\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ Diff\ {\isacharparenleft}substa\ s\ a{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{2}}{\isacharparenright}{\isachardoublequote}\isanewline
 \ \ {\isachardoublequote}substa\ s\ {\isacharparenleft}Var\ v{\isacharparenright}\ {\isacharequal}\ s\ v{\isachardoublequote}\isanewline
 \ \ {\isachardoublequote}substa\ s\ {\isacharparenleft}Num\ n{\isacharparenright}\ {\isacharequal}\ Num\ n{\isachardoublequote}\isanewline
 \isanewline
 \ \ {\isachardoublequote}substb\ s\ {\isacharparenleft}Less\ a{\isadigit{1}}\ a{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ Less\ {\isacharparenleft}substa\ s\ a{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substa\ s\ a{\isadigit{2}}{\isacharparenright}{\isachardoublequote}\isanewline
 \ \ {\isachardoublequote}substb\ s\ {\isacharparenleft}And\ b{\isadigit{1}}\ b{\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ And\ {\isacharparenleft}substb\ s\ b{\isadigit{1}}{\isacharparenright}\ {\isacharparenleft}substb\ s\ b{\isadigit{2}}{\isacharparenright}{\isachardoublequote}\isanewline
-\ \ {\isachardoublequote}substb\ s\ {\isacharparenleft}Neg\ b{\isacharparenright}\ {\isacharequal}\ Neg\ {\isacharparenleft}substb\ s\ b{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
+\ \ {\isachardoublequote}substb\ s\ {\isacharparenleft}Neg\ b{\isacharparenright}\ {\isacharequal}\ Neg\ {\isacharparenleft}substb\ s\ b{\isacharparenright}{\isachardoublequote}\isamarkuptrue%
 %
 \begin{isamarkuptext}%
 Now we can prove a fundamental theorem about the interaction between
 evaluation and substitution: applying a substitution $s$ to an expression $a$
 and evaluating the result in an environment $env$ yields the same result as

 of $s(x)$ under $env$. If you try to prove this separately for arithmetic or
 boolean expressions (by induction), you find that you always need the other
 theorem in the induction step. Therefore you need to state and prove both
 theorems simultaneously:%
 \end{isamarkuptext}%
-\isamarkuptrue%
+\isamarkupfalse%
 \isacommand{lemma}\ {\isachardoublequote}evala\ {\isacharparenleft}substa\ s\ a{\isacharparenright}\ env\ {\isacharequal}\ evala\ a\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ evala\ {\isacharparenleft}s\ x{\isacharparenright}\ env{\isacharparenright}\ {\isasymand}\isanewline
 \ \ \ \ \ \ \ \ evalb\ {\isacharparenleft}substb\ s\ b{\isacharparenright}\ env\ {\isacharequal}\ evalb\ b\ {\isacharparenleft}{\isasymlambda}x{\isachardot}\ evala\ {\isacharparenleft}s\ x{\isacharparenright}\ env{\isacharparenright}{\isachardoublequote}\isanewline
-\isamarkupfalse%
-\isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ a\ \isakeyword{and}\ b{\isacharparenright}\isamarkupfalse%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isamarkupfalse%
+\isacommand{apply}{\isacharparenleft}induct{\isacharunderscore}tac\ a\ \isakeyword{and}\ b{\isacharparenright}\isamarkuptrue%
 %
 \begin{isamarkuptxt}%
 \noindent
 The resulting 8 goals (one for each constructor) are proved in one fell swoop:%
 \end{isamarkuptxt}%
+\isamarkupfalse%
+\isacommand{apply}\ simp{\isacharunderscore}all%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
 \isamarkuptrue%
-\isacommand{apply}\ simp{\isacharunderscore}all\isamarkupfalse%
-\isamarkupfalse%
 %
 \begin{isamarkuptext}%
 In general, given $n$ mutually recursive datatypes $\tau@1$, \dots, $\tau@n$,
 an inductive proof expects a goal of the form
 \[ P@1(x@1)\ \land \dots \land P@n(x@n) \]

   is really normal, i.e.\ no more \isa{And}s and \isa{Neg}s occur in
   it.  ({\em Hint:} proceed as in \S\ref{sec:boolex} and read the discussion
   of type annotations following lemma \isa{subst{\isacharunderscore}id} below).
 \end{exercise}%
 \end{isamarkuptext}%
-\isamarkuptrue%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
-\isamarkupfalse%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
 \end{isabellebody}%
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