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\begin{isabellebody}%
\def\isabellecontext{Inductive{\isacharunderscore}Predicate}%
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\isadelimtheory
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\isatagtheory
\isacommand{theory}\isamarkupfalse%
\ Inductive{\isacharunderscore}Predicate\isanewline
\isakeyword{imports}\ Setup\isanewline
\isakeyword{begin}%
\endisatagtheory
{\isafoldtheory}%
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\endisadelimtheory
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\isamarkupsubsection{Inductive Predicates%
}
\isamarkuptrue%
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\begin{isamarkuptext}%
To execute inductive predicates, a special preprocessor, the predicate
 compiler, generates code equations from the introduction rules of the predicates.
The mechanisms of this compiler are described in \cite{Berghofer-Bulwahn-Haftmann:2009:TPHOL}.
Consider the simple predicate \isa{append} given by these two
introduction rules:%
\end{isamarkuptext}%
\isamarkuptrue%
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\begin{isamarkuptext}%
\isa{append\ {\isacharbrackleft}{\isacharbrackright}\ ys\ ys}\\
\noindent\isa{append\ xs\ ys\ zs\ {\isasymLongrightarrow}\ append\ {\isacharparenleft}x\ {\isacharhash}\ xs{\isacharparenright}\ ys\ {\isacharparenleft}x\ {\isacharhash}\ zs{\isacharparenright}}%
\end{isamarkuptext}%
\isamarkuptrue%
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\begin{isamarkuptext}%
\noindent To invoke the compiler, simply use \indexdef{}{command}{code\_pred}\hypertarget{command.code-pred}{\hyperlink{command.code-pred}{\mbox{\isa{\isacommand{code{\isacharunderscore}pred}}}}}:%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimquote
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\endisadelimquote
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\isatagquote
\isacommand{code{\isacharunderscore}pred}\isamarkupfalse%
\ append\ \isacommand{{\isachardot}}\isamarkupfalse%
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\begin{isamarkuptext}%
\noindent The \hyperlink{command.code-pred}{\mbox{\isa{\isacommand{code{\isacharunderscore}pred}}}} command takes the name
of the inductive predicate and then you put a period to discharge
a trivial correctness proof. 
The compiler infers possible modes 
for the predicate and produces the derived code equations.
Modes annotate which (parts of the) arguments are to be taken as input,
and which output. Modes are similar to types, but use the notation \isa{i}
for input and \isa{o} for output.
 
For \isa{append}, the compiler can infer the following modes:
\begin{itemize}
\item \isa{i\ {\isasymRightarrow}\ i\ {\isasymRightarrow}\ i\ {\isasymRightarrow}\ bool}
\item \isa{i\ {\isasymRightarrow}\ i\ {\isasymRightarrow}\ o\ {\isasymRightarrow}\ bool}
\item \isa{o\ {\isasymRightarrow}\ o\ {\isasymRightarrow}\ i\ {\isasymRightarrow}\ bool}
\end{itemize}
You can compute sets of predicates using \indexdef{}{command}{values}\hypertarget{command.values}{\hyperlink{command.values}{\mbox{\isa{\isacommand{values}}}}}:%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimquote
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\isatagquote
\isacommand{values}\isamarkupfalse%
\ {\isachardoublequoteopen}{\isacharbraceleft}zs{\isachardot}\ append\ {\isacharbrackleft}{\isacharparenleft}{\isadigit{1}}{\isacharcolon}{\isacharcolon}nat{\isacharparenright}{\isacharcomma}{\isadigit{2}}{\isacharcomma}{\isadigit{3}}{\isacharbrackright}\ {\isacharbrackleft}{\isadigit{4}}{\isacharcomma}{\isadigit{5}}{\isacharbrackright}\ zs{\isacharbraceright}{\isachardoublequoteclose}%
\endisatagquote
{\isafoldquote}%
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\begin{isamarkuptext}%
\noindent outputs \isa{{\isacharbraceleft}{\isacharbrackleft}{\isadigit{1}}{\isacharcomma}\ {\isadigit{2}}{\isacharcomma}\ {\isadigit{3}}{\isacharcomma}\ {\isadigit{4}}{\isacharcomma}\ {\isadigit{5}}{\isacharbrackright}{\isacharbraceright}}, and%
\end{isamarkuptext}%
\isamarkuptrue%
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\isacommand{values}\isamarkupfalse%
\ {\isachardoublequoteopen}{\isacharbraceleft}{\isacharparenleft}xs{\isacharcomma}\ ys{\isacharparenright}{\isachardot}\ append\ xs\ ys\ {\isacharbrackleft}{\isacharparenleft}{\isadigit{2}}{\isacharcolon}{\isacharcolon}nat{\isacharparenright}{\isacharcomma}{\isadigit{3}}{\isacharbrackright}{\isacharbraceright}{\isachardoublequoteclose}%
\endisatagquote
{\isafoldquote}%
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\isadelimquote
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\begin{isamarkuptext}%
\noindent outputs \isa{{\isacharbraceleft}{\isacharparenleft}{\isacharbrackleft}{\isacharbrackright}{\isacharcomma}\ {\isacharbrackleft}{\isadigit{2}}{\isacharcomma}\ {\isadigit{3}}{\isacharbrackright}{\isacharparenright}{\isacharcomma}\ {\isacharparenleft}{\isacharbrackleft}{\isadigit{2}}{\isacharbrackright}{\isacharcomma}\ {\isacharbrackleft}{\isadigit{3}}{\isacharbrackright}{\isacharparenright}{\isacharcomma}\ {\isacharparenleft}{\isacharbrackleft}{\isadigit{2}}{\isacharcomma}\ {\isadigit{3}}{\isacharbrackright}{\isacharcomma}\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}{\isacharbraceright}}.%
\end{isamarkuptext}%
\isamarkuptrue%
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\begin{isamarkuptext}%
\noindent If you are only interested in the first elements of the set
comprehension (with respect to a depth-first search on the introduction rules), you can
pass an argument to
\hyperlink{command.values}{\mbox{\isa{\isacommand{values}}}} to specify the number of elements you want:%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimquote
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\isatagquote
\isacommand{values}\isamarkupfalse%
\ {\isadigit{1}}\ {\isachardoublequoteopen}{\isacharbraceleft}{\isacharparenleft}xs{\isacharcomma}\ ys{\isacharparenright}{\isachardot}\ append\ xs\ ys\ {\isacharbrackleft}{\isacharparenleft}{\isadigit{1}}{\isacharcolon}{\isacharcolon}nat{\isacharparenright}{\isacharcomma}{\isadigit{2}}{\isacharcomma}{\isadigit{3}}{\isacharcomma}{\isadigit{4}}{\isacharbrackright}{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\isacommand{values}\isamarkupfalse%
\ {\isadigit{3}}\ {\isachardoublequoteopen}{\isacharbraceleft}{\isacharparenleft}xs{\isacharcomma}\ ys{\isacharparenright}{\isachardot}\ append\ xs\ ys\ {\isacharbrackleft}{\isacharparenleft}{\isadigit{1}}{\isacharcolon}{\isacharcolon}nat{\isacharparenright}{\isacharcomma}{\isadigit{2}}{\isacharcomma}{\isadigit{3}}{\isacharcomma}{\isadigit{4}}{\isacharbrackright}{\isacharbraceright}{\isachardoublequoteclose}%
\endisatagquote
{\isafoldquote}%
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\endisadelimquote
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\begin{isamarkuptext}%
\noindent The \hyperlink{command.values}{\mbox{\isa{\isacommand{values}}}} command can only compute set
 comprehensions for which a mode has been inferred.

The code equations for a predicate are made available as theorems with
 the suffix \isa{equation}, and can be inspected with:%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimquote
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\endisadelimquote
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\isatagquote
\isacommand{thm}\isamarkupfalse%
\ append{\isachardot}equation%
\endisatagquote
{\isafoldquote}%
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\isadelimquote
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\endisadelimquote
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\begin{isamarkuptext}%
\noindent More advanced options are described in the following subsections.%
\end{isamarkuptext}%
\isamarkuptrue%
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\isamarkupsubsubsection{Alternative names for functions%
}
\isamarkuptrue%
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\begin{isamarkuptext}%
By default, the functions generated from a predicate are named after the predicate with the
mode mangled into the name (e.g., \isa{append{\isacharunderscore}i{\isacharunderscore}i{\isacharunderscore}o}).
You can specify your own names as follows:%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimquote
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\isatagquote
\isacommand{code{\isacharunderscore}pred}\isamarkupfalse%
\ {\isacharparenleft}modes{\isacharcolon}\ i\ {\isacharequal}{\isachargreater}\ i\ {\isacharequal}{\isachargreater}\ o\ {\isacharequal}{\isachargreater}\ bool\ as\ concat{\isacharcomma}\isanewline
\ \ o\ {\isacharequal}{\isachargreater}\ o\ {\isacharequal}{\isachargreater}\ i\ {\isacharequal}{\isachargreater}\ bool\ as\ split{\isacharcomma}\isanewline
\ \ i\ {\isacharequal}{\isachargreater}\ o\ {\isacharequal}{\isachargreater}\ i\ {\isacharequal}{\isachargreater}\ bool\ as\ suffix{\isacharparenright}\ append\ \isacommand{{\isachardot}}\isamarkupfalse%
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\endisatagquote
{\isafoldquote}%
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\isamarkupsubsubsection{Alternative introduction rules%
}
\isamarkuptrue%
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\begin{isamarkuptext}%
Sometimes the introduction rules of an predicate are not executable because they contain
non-executable constants or specific modes could not be inferred.
It is also possible that the introduction rules yield a function that loops forever
due to the execution in a depth-first search manner.
Therefore, you can declare alternative introduction rules for predicates with the attribute
\hyperlink{attribute.code-pred-intro}{\mbox{\isa{code{\isacharunderscore}pred{\isacharunderscore}intro}}}.
For example, the transitive closure is defined by:%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimquote
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\begin{isamarkuptext}%
\isa{r\ a\ b\ {\isasymLongrightarrow}\ r\isactrlsup {\isacharplus}\isactrlsup {\isacharplus}\ a\ b}\\
\noindent \isa{{\isasymlbrakk}r\isactrlsup {\isacharplus}\isactrlsup {\isacharplus}\ a\ b{\isacharsemicolon}\ r\ b\ c{\isasymrbrakk}\ {\isasymLongrightarrow}\ r\isactrlsup {\isacharplus}\isactrlsup {\isacharplus}\ a\ c}%
\end{isamarkuptext}%
\isamarkuptrue%
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\endisatagquote
{\isafoldquote}%
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\isadelimquote
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\endisadelimquote
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\begin{isamarkuptext}%
\noindent These rules do not suit well for executing the transitive closure 
with the mode \isa{{\isacharparenleft}i\ {\isasymRightarrow}\ o\ {\isasymRightarrow}\ bool{\isacharparenright}\ {\isasymRightarrow}\ i\ {\isasymRightarrow}\ o\ {\isasymRightarrow}\ bool}, as the second rule will
cause an infinite loop in the recursive call.
This can be avoided using the following alternative rules which are
declared to the predicate compiler by the attribute \hyperlink{attribute.code-pred-intro}{\mbox{\isa{code{\isacharunderscore}pred{\isacharunderscore}intro}}}:%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimquote
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\endisadelimquote
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\isatagquote
\isacommand{lemma}\isamarkupfalse%
\ {\isacharbrackleft}code{\isacharunderscore}pred{\isacharunderscore}intro{\isacharbrackright}{\isacharcolon}\isanewline
\ \ {\isachardoublequoteopen}r\ a\ b\ {\isasymLongrightarrow}\ r\isactrlsup {\isacharplus}\isactrlsup {\isacharplus}\ a\ b{\isachardoublequoteclose}\isanewline
\ \ {\isachardoublequoteopen}r\ a\ b\ {\isasymLongrightarrow}\ r\isactrlsup {\isacharplus}\isactrlsup {\isacharplus}\ b\ c\ {\isasymLongrightarrow}\ r\isactrlsup {\isacharplus}\isactrlsup {\isacharplus}\ a\ c{\isachardoublequoteclose}\isanewline
\isacommand{by}\isamarkupfalse%
\ auto%
\endisatagquote
{\isafoldquote}%
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\isadelimquote
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\endisadelimquote
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\begin{isamarkuptext}%
\noindent After declaring all alternative rules for the transitive closure,
you invoke \hyperlink{command.code-pred}{\mbox{\isa{\isacommand{code{\isacharunderscore}pred}}}} as usual.
As you have declared alternative rules for the predicate, you are urged to prove that these
introduction rules are complete, i.e., that you can derive an elimination rule for the
alternative rules:%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimquote
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\isatagquote
\isacommand{code{\isacharunderscore}pred}\isamarkupfalse%
\ tranclp\isanewline
\isacommand{proof}\isamarkupfalse%
\ {\isacharminus}\isanewline
\ \ \isacommand{case}\isamarkupfalse%
\ tranclp\isanewline
\ \ \isacommand{from}\isamarkupfalse%
\ this\ converse{\isacharunderscore}tranclpE{\isacharbrackleft}OF\ this{\isacharparenleft}{\isadigit{1}}{\isacharparenright}{\isacharbrackright}\ \isacommand{show}\isamarkupfalse%
\ thesis\ \isacommand{by}\isamarkupfalse%
\ metis\isanewline
\isacommand{qed}\isamarkupfalse%
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\endisatagquote
{\isafoldquote}%
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\isadelimquote
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\begin{isamarkuptext}%
\noindent Alternative rules can also be used for constants that have not
been defined inductively. For example, the lexicographic order which
is defined as:%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimquote
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\begin{isamarkuptext}%
\begin{isabelle}%
lexord\ r\ {\isacharequal}\isanewline
{\isacharbraceleft}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardot}\isanewline
\isaindent{\ }{\isasymexists}a\ v{\isachardot}\ y\ {\isacharequal}\ x\ {\isacharat}\ a\ {\isacharhash}\ v\ {\isasymor}\isanewline
\isaindent{\ {\isasymexists}a\ v{\isachardot}\ }{\isacharparenleft}{\isasymexists}u\ a\ b\ v\ w{\isachardot}\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymin}\ r\ {\isasymand}\ x\ {\isacharequal}\ u\ {\isacharat}\ a\ {\isacharhash}\ v\ {\isasymand}\ y\ {\isacharequal}\ u\ {\isacharat}\ b\ {\isacharhash}\ w{\isacharparenright}{\isacharbraceright}%
\end{isabelle}%
\end{isamarkuptext}%
\isamarkuptrue%
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\endisatagquote
{\isafoldquote}%
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\isadelimquote
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\endisadelimquote
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\begin{isamarkuptext}%
\noindent To make it executable, you can derive the following two rules and prove the
elimination rule:%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimproof
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\endisadelimproof
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\isatagproof
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\endisatagproof
{\isafoldproof}%
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\isadelimproof
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\endisadelimproof
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\isadelimquote
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\endisadelimquote
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\isatagquote
\isacommand{lemma}\isamarkupfalse%
\ {\isacharbrackleft}code{\isacharunderscore}pred{\isacharunderscore}intro{\isacharbrackright}{\isacharcolon}\isanewline
\ \ {\isachardoublequoteopen}append\ xs\ {\isacharparenleft}a\ {\isacharhash}\ v{\isacharparenright}\ ys\ {\isasymLongrightarrow}\ lexord\ r\ {\isacharparenleft}xs{\isacharcomma}\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
\isacommand{lemma}\isamarkupfalse%
\ {\isacharbrackleft}code{\isacharunderscore}pred{\isacharunderscore}intro{\isacharbrackright}{\isacharcolon}\isanewline
\ \ {\isachardoublequoteopen}append\ u\ {\isacharparenleft}a\ {\isacharhash}\ v{\isacharparenright}\ xs\ {\isasymLongrightarrow}\ append\ u\ {\isacharparenleft}b\ {\isacharhash}\ w{\isacharparenright}\ ys\ {\isasymLongrightarrow}\ r\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\isanewline
\ \ {\isasymLongrightarrow}\ lexord\ r\ {\isacharparenleft}xs{\isacharcomma}\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
\isacommand{code{\isacharunderscore}pred}\isamarkupfalse%
\ lexord%
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{\isafoldquote}%
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\isadelimquote
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\endisadelimquote
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\isamarkupsubsubsection{Options for values%
}
\isamarkuptrue%
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\begin{isamarkuptext}%
In the presence of higher-order predicates, multiple modes for some predicate could be inferred
that are not disambiguated by the pattern of the set comprehension.
To disambiguate the modes for the arguments of a predicate, you can state
the modes explicitly in the \hyperlink{command.values}{\mbox{\isa{\isacommand{values}}}} command. 
Consider the simple predicate \isa{succ}:%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{inductive}\isamarkupfalse%
\ succ\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
\isakeyword{where}\isanewline
\ \ {\isachardoublequoteopen}succ\ {\isadigit{0}}\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isachardoublequoteclose}\isanewline
{\isacharbar}\ {\isachardoublequoteopen}succ\ x\ y\ {\isasymLongrightarrow}\ succ\ {\isacharparenleft}Suc\ x{\isacharparenright}\ {\isacharparenleft}Suc\ y{\isacharparenright}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{code{\isacharunderscore}pred}\isamarkupfalse%
\ succ%
\isadelimproof
\ %
\endisadelimproof
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\isatagproof
\isacommand{{\isachardot}}\isamarkupfalse%
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\endisatagproof
{\isafoldproof}%
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\isadelimproof
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\endisadelimproof
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\begin{isamarkuptext}%
\noindent For this, the predicate compiler can infer modes \isa{o\ {\isasymRightarrow}\ o\ {\isasymRightarrow}\ bool}, \isa{i\ {\isasymRightarrow}\ o\ {\isasymRightarrow}\ bool},
  \isa{o\ {\isasymRightarrow}\ i\ {\isasymRightarrow}\ bool} and \isa{i\ {\isasymRightarrow}\ i\ {\isasymRightarrow}\ bool}.
The invocation of \hyperlink{command.values}{\mbox{\isa{\isacommand{values}}}} \isa{{\isacharbraceleft}n{\isachardot}\ tranclp\ succ\ {\isadigit{1}}{\isadigit{0}}\ n{\isacharbraceright}} loops, as multiple
modes for the predicate \isa{succ} are possible and here the first mode \isa{o\ {\isasymRightarrow}\ o\ {\isasymRightarrow}\ bool}
is chosen. To choose another mode for the argument, you can declare the mode for the argument between
the \hyperlink{command.values}{\mbox{\isa{\isacommand{values}}}} and the number of elements.%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimquote
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\endisadelimquote
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\isatagquote
\isacommand{values}\isamarkupfalse%
\ {\isacharbrackleft}mode{\isacharcolon}\ i\ {\isacharequal}{\isachargreater}\ o\ {\isacharequal}{\isachargreater}\ bool{\isacharbrackright}\ {\isadigit{2}}{\isadigit{0}}\ {\isachardoublequoteopen}{\isacharbraceleft}n{\isachardot}\ tranclp\ succ\ {\isadigit{1}}{\isadigit{0}}\ n{\isacharbraceright}{\isachardoublequoteclose}\isanewline
\isacommand{values}\isamarkupfalse%
\ {\isacharbrackleft}mode{\isacharcolon}\ o\ {\isacharequal}{\isachargreater}\ i\ {\isacharequal}{\isachargreater}\ bool{\isacharbrackright}\ {\isadigit{1}}{\isadigit{0}}\ {\isachardoublequoteopen}{\isacharbraceleft}n{\isachardot}\ tranclp\ succ\ n\ {\isadigit{1}}{\isadigit{0}}{\isacharbraceright}{\isachardoublequoteclose}%
\endisatagquote
{\isafoldquote}%
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\isadelimquote
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\endisadelimquote
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\isamarkupsubsubsection{Embedding into functional code within Isabelle/HOL%
}
\isamarkuptrue%
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\begin{isamarkuptext}%
To embed the computation of an inductive predicate into functions that are defined in Isabelle/HOL,
you have a number of options:
\begin{itemize}
\item You want to use the first-order predicate with the mode
  where all arguments are input. Then you can use the predicate directly, e.g.
\begin{quote}
  \isa{valid{\isacharunderscore}suffix\ ys\ zs\ {\isacharequal}}\\
  \isa{{\isacharparenleft}if\ append\ {\isacharbrackleft}Suc\ {\isadigit{0}}{\isacharcomma}\ {\isadigit{2}}{\isacharbrackright}\ ys\ zs\ then\ Some\ ys\ else\ None{\isacharparenright}}
\end{quote}
\item If you know that the execution returns only one value (it is deterministic), then you can
  use the combinator \isa{Predicate{\isachardot}the}, e.g., a functional concatenation of lists
  is defined with
\begin{quote}
  \isa{functional{\isacharunderscore}concat\ xs\ ys\ {\isacharequal}\ Predicate{\isachardot}the\ {\isacharparenleft}append{\isacharunderscore}i{\isacharunderscore}i{\isacharunderscore}o\ xs\ ys{\isacharparenright}}
\end{quote}
  Note that if the evaluation does not return a unique value, it raises a run-time error
  \isa{not{\isacharunderscore}unique}.
\end{itemize}%
\end{isamarkuptext}%
\isamarkuptrue%
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\isamarkupsubsubsection{Further Examples%
}
\isamarkuptrue%
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\begin{isamarkuptext}%
Further examples for compiling inductive predicates can be found in
the \isa{HOL{\isacharslash}ex{\isacharslash}Predicate{\isacharunderscore}Compile{\isacharunderscore}ex} theory file.
There are also some examples in the Archive of Formal Proofs, notably
in the \isa{POPLmark{\isacharminus}deBruijn} and the \isa{FeatherweightJava} sessions.%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimtheory
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\endisadelimtheory
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\isatagtheory
\isacommand{end}\isamarkupfalse%
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\endisatagtheory
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\isadelimtheory
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\endisadelimtheory
\isanewline
\end{isabellebody}%
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