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\begin{isabellebody}%
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\def\isabellecontext{Foundations}%
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\isacommand{theory}\isamarkupfalse%
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\ Foundations\isanewline
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\isakeyword{imports}\ Introduction\isanewline
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\isakeyword{begin}%
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\endisatagtheory
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\isamarkupsection{Code generation foundations \label{sec:program}%
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}
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\isamarkuptrue%
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\isamarkupsubsection{The \isa{Isabelle{\isacharslash}HOL} default setup%
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}
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\isamarkuptrue%
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\begin{isamarkuptext}%
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We have already seen how by default equations stemming from
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\hyperlink{command.definition}{\mbox{\isa{\isacommand{definition}}}}, \hyperlink{command.primrec}{\mbox{\isa{\isacommand{primrec}}}} and \hyperlink{command.fun}{\mbox{\isa{\isacommand{fun}}}}
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statements are used for code generation. This default behaviour
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can be changed, e.g.\ by providing different code equations.
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The customisations shown in this section are \emph{safe}
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as regards correctness: all programs that can be generated are partially
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correct.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isamarkupsubsection{Selecting code equations%
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}
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\isamarkuptrue%
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\begin{isamarkuptext}%
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Coming back to our introductory example, we
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could provide an alternative code equations for \isa{dequeue}
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explicitly:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{lemma}\isamarkupfalse%
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\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
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\ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\isanewline
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\ \ \ \ \ {\isacharparenleft}if\ xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ then\ {\isacharparenleft}None{\isacharcomma}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\isanewline
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\ \ \ \ \ \ \ else\ dequeue\ {\isacharparenleft}AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharparenleft}rev\ xs{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
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\ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\isanewline
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\ \ \ \ \ {\isacharparenleft}Some\ y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
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\ \ \isacommand{by}\isamarkupfalse%
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\ {\isacharparenleft}cases\ xs{\isacharcomma}\ simp{\isacharunderscore}all{\isacharparenright}\ {\isacharparenleft}cases\ {\isachardoublequoteopen}rev\ xs{\isachardoublequoteclose}{\isacharcomma}\ simp{\isacharunderscore}all{\isacharparenright}%
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\begin{isamarkuptext}%
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\noindent The annotation \isa{{\isacharbrackleft}code{\isacharbrackright}} is an \isa{Isar}
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\isa{attribute} which states that the given theorems should be
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considered as code equations for a \isa{fun} statement --
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the corresponding constant is determined syntactically. The resulting code:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\begin{isamarkuptext}%
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\isatypewriter%
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\noindent%
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\hspace*{0pt}dequeue ::~forall a.~Queue a -> (Maybe a,~Queue a);\\
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\hspace*{0pt}dequeue (AQueue xs (y :~ys)) = (Just y,~AQueue xs ys);\\
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\hspace*{0pt}dequeue (AQueue xs []) =\\
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\hspace*{0pt} ~(if null xs then (Nothing,~AQueue [] [])\\
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\hspace*{0pt} ~~~else dequeue (AQueue [] (reverse xs)));%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\begin{isamarkuptext}%
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\noindent You may note that the equality test \isa{xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}} has been
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replaced by the predicate \isa{null\ xs}. This is due to the default
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setup in the \qn{preprocessor} to be discussed further below (\secref{sec:preproc}).
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Changing the default constructor set of datatypes is also
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possible. See \secref{sec:datatypes} for an example.
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As told in \secref{sec:concept}, code generation is based
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on a structured collection of code theorems.
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This collection
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may be inspected using the \hyperlink{command.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}} command:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{code{\isacharunderscore}thms}\isamarkupfalse%
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\ dequeue%
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\begin{isamarkuptext}%
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\noindent prints a table with \emph{all} code equations
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for \isa{dequeue}, including
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\emph{all} code equations those equations depend
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on recursively.
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Similarly, the \hyperlink{command.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}} command shows a graph
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visualising dependencies between code equations.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isamarkupsubsection{\isa{class} and \isa{instantiation}%
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}
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\isamarkuptrue%
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\begin{isamarkuptext}%
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Concerning type classes and code generation, let us examine an example
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from abstract algebra:%
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\end{isamarkuptext}%
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\isamarkuptrue%
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\isacommand{class}\isamarkupfalse%
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\ semigroup\ {\isacharequal}\isanewline
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\ \ \isakeyword{fixes}\ mult\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ {\isacharparenleft}\isakeyword{infixl}\ {\isachardoublequoteopen}{\isasymotimes}{\isachardoublequoteclose}\ {\isadigit{7}}{\isadigit{0}}{\isacharparenright}\isanewline
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\ \ \isakeyword{assumes}\ assoc{\isacharcolon}\ {\isachardoublequoteopen}{\isacharparenleft}x\ {\isasymotimes}\ y{\isacharparenright}\ {\isasymotimes}\ z\ {\isacharequal}\ x\ {\isasymotimes}\ {\isacharparenleft}y\ {\isasymotimes}\ z{\isacharparenright}{\isachardoublequoteclose}\isanewline
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\isanewline
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\isacommand{class}\isamarkupfalse%
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\ monoid\ {\isacharequal}\ semigroup\ {\isacharplus}\isanewline
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\ \ \isakeyword{fixes}\ neutral\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\ {\isacharparenleft}{\isachardoublequoteopen}{\isasymone}{\isachardoublequoteclose}{\isacharparenright}\isanewline
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\ \ \isakeyword{assumes}\ neutl{\isacharcolon}\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ x\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
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\ \ \ \ \isakeyword{and}\ neutr{\isacharcolon}\ {\isachardoublequoteopen}x\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ x{\isachardoublequoteclose}\isanewline
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\isanewline
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\isacommand{instantiation}\isamarkupfalse%
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\ nat\ {\isacharcolon}{\isacharcolon}\ monoid\isanewline
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\isakeyword{begin}\isanewline
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\isanewline
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\isacommand{primrec}\isamarkupfalse%
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\ mult{\isacharunderscore}nat\ \isakeyword{where}\isanewline
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\ \ \ \ {\isachardoublequoteopen}{\isadigit{0}}\ {\isasymotimes}\ n\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}nat{\isacharparenright}{\isachardoublequoteclose}\isanewline
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\ \ {\isacharbar}\ {\isachardoublequoteopen}Suc\ m\ {\isasymotimes}\ n\ {\isacharequal}\ n\ {\isacharplus}\ m\ {\isasymotimes}\ n{\isachardoublequoteclose}\isanewline
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\isanewline
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\isacommand{definition}\isamarkupfalse%
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\ neutral{\isacharunderscore}nat\ \isakeyword{where}\isanewline
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\ \ {\isachardoublequoteopen}{\isasymone}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isachardoublequoteclose}\isanewline
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\isanewline
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\isacommand{lemma}\isamarkupfalse%
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\ add{\isacharunderscore}mult{\isacharunderscore}distrib{\isacharcolon}\isanewline
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\ \ \isakeyword{fixes}\ n\ m\ q\ {\isacharcolon}{\isacharcolon}\ nat\isanewline
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\ \ \isakeyword{shows}\ {\isachardoublequoteopen}{\isacharparenleft}n\ {\isacharplus}\ m{\isacharparenright}\ {\isasymotimes}\ q\ {\isacharequal}\ n\ {\isasymotimes}\ q\ {\isacharplus}\ m\ {\isasymotimes}\ q{\isachardoublequoteclose}\isanewline
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\ \ \isacommand{by}\isamarkupfalse%
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\ {\isacharparenleft}induct\ n{\isacharparenright}\ simp{\isacharunderscore}all\isanewline
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\isanewline
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\isacommand{instance}\isamarkupfalse%
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\ \isacommand{proof}\isamarkupfalse%
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\isanewline
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\ \ \isacommand{fix}\isamarkupfalse%
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\ m\ n\ q\ {\isacharcolon}{\isacharcolon}\ nat\isanewline
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\ \ \isacommand{show}\isamarkupfalse%
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\ {\isachardoublequoteopen}m\ {\isasymotimes}\ n\ {\isasymotimes}\ q\ {\isacharequal}\ m\ {\isasymotimes}\ {\isacharparenleft}n\ {\isasymotimes}\ q{\isacharparenright}{\isachardoublequoteclose}\isanewline
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\ \ \ \ \isacommand{by}\isamarkupfalse%
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\ {\isacharparenleft}induct\ m{\isacharparenright}\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ add{\isacharunderscore}mult{\isacharunderscore}distrib{\isacharparenright}\isanewline
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\ \ \isacommand{show}\isamarkupfalse%
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\ {\isachardoublequoteopen}{\isasymone}\ {\isasymotimes}\ n\ {\isacharequal}\ n{\isachardoublequoteclose}\isanewline
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\ \ \ \ \isacommand{by}\isamarkupfalse%
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\ {\isacharparenleft}simp\ add{\isacharcolon}\ neutral{\isacharunderscore}nat{\isacharunderscore}def{\isacharparenright}\isanewline
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\ \ \isacommand{show}\isamarkupfalse%
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\ {\isachardoublequoteopen}m\ {\isasymotimes}\ {\isasymone}\ {\isacharequal}\ m{\isachardoublequoteclose}\isanewline
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\ \ \ \ \isacommand{by}\isamarkupfalse%
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\ {\isacharparenleft}induct\ m{\isacharparenright}\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ neutral{\isacharunderscore}nat{\isacharunderscore}def{\isacharparenright}\isanewline
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\isacommand{qed}\isamarkupfalse%
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\isanewline
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\isanewline
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\isacommand{end}\isamarkupfalse%
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\begin{isamarkuptext}%
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\noindent We define the natural operation of the natural numbers
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on monoids:%
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\end{isamarkuptext}%
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\isacommand{primrec}\isamarkupfalse%
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\ {\isacharparenleft}\isakeyword{in}\ monoid{\isacharparenright}\ pow\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
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\ \ \ \ {\isachardoublequoteopen}pow\ {\isadigit{0}}\ a\ {\isacharequal}\ {\isasymone}{\isachardoublequoteclose}\isanewline
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\ \ {\isacharbar}\ {\isachardoublequoteopen}pow\ {\isacharparenleft}Suc\ n{\isacharparenright}\ a\ {\isacharequal}\ a\ {\isasymotimes}\ pow\ n\ a{\isachardoublequoteclose}%
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\begin{isamarkuptext}%
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\noindent This we use to define the discrete exponentiation function:%
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\end{isamarkuptext}%
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\isacommand{definition}\isamarkupfalse%
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\ bexp\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
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\ \ {\isachardoublequoteopen}bexp\ n\ {\isacharequal}\ pow\ n\ {\isacharparenleft}Suc\ {\isacharparenleft}Suc\ {\isadigit{0}}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}%
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\begin{isamarkuptext}%
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\noindent The corresponding code in Haskell uses that language's native classes:%
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\end{isamarkuptext}%
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\begin{isamarkuptext}%
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\isatypewriter%
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\noindent%
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\hspace*{0pt}module Example where {\char123}\\
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\hspace*{0pt}\\
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\hspace*{0pt}data Nat = Zero{\char95}nat | Suc Nat;\\
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\hspace*{0pt}\\
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\hspace*{0pt}plus{\char95}nat ::~Nat -> Nat -> Nat;\\
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\hspace*{0pt}plus{\char95}nat (Suc m) n = plus{\char95}nat m (Suc n);\\
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\hspace*{0pt}plus{\char95}nat Zero{\char95}nat n = n;\\
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\hspace*{0pt}\\
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\hspace*{0pt}class Semigroup a where {\char123}\\
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\hspace*{0pt} ~mult ::~a -> a -> a;\\
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\hspace*{0pt}{\char125};\\
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\hspace*{0pt}\\
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\hspace*{0pt}class (Semigroup a) => Monoid a where {\char123}\\
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\hspace*{0pt} ~neutral ::~a;\\
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\hspace*{0pt}{\char125};\\
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\hspace*{0pt}\\
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\hspace*{0pt}pow ::~forall a.~(Monoid a) => Nat -> a -> a;\\
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\hspace*{0pt}pow Zero{\char95}nat a = neutral;\\
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\hspace*{0pt}pow (Suc n) a = mult a (pow n a);\\
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\hspace*{0pt}\\
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\hspace*{0pt}mult{\char95}nat ::~Nat -> Nat -> Nat;\\
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\hspace*{0pt}mult{\char95}nat Zero{\char95}nat n = Zero{\char95}nat;\\
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\hspace*{0pt}mult{\char95}nat (Suc m) n = plus{\char95}nat n (mult{\char95}nat m n);\\
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\hspace*{0pt}\\
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\hspace*{0pt}neutral{\char95}nat ::~Nat;\\
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\hspace*{0pt}neutral{\char95}nat = Suc Zero{\char95}nat;\\
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\hspace*{0pt}\\
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\hspace*{0pt}instance Semigroup Nat where {\char123}\\
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\hspace*{0pt} ~mult = mult{\char95}nat;\\
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\hspace*{0pt}{\char125};\\
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\hspace*{0pt}\\
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\hspace*{0pt}instance Monoid Nat where {\char123}\\
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\hspace*{0pt} ~neutral = neutral{\char95}nat;\\
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\hspace*{0pt}{\char125};\\
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\hspace*{0pt}\\
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\hspace*{0pt}bexp ::~Nat -> Nat;\\
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|
311 |
\hspace*{0pt}bexp n = pow n (Suc (Suc Zero{\char95}nat));\\
|
|
312 |
\hspace*{0pt}\\
|
|
313 |
\hspace*{0pt}{\char125}%
|
|
314 |
\end{isamarkuptext}%
|
|
315 |
\isamarkuptrue%
|
|
316 |
%
|
|
317 |
\endisatagquote
|
|
318 |
{\isafoldquote}%
|
|
319 |
%
|
|
320 |
\isadelimquote
|
|
321 |
%
|
|
322 |
\endisadelimquote
|
|
323 |
%
|
|
324 |
\begin{isamarkuptext}%
|
|
325 |
\noindent This is a convenient place to show how explicit dictionary construction
|
|
326 |
manifests in generated code (here, the same example in \isa{SML})
|
|
327 |
\cite{Haftmann-Nipkow:2010:code}:%
|
|
328 |
\end{isamarkuptext}%
|
|
329 |
\isamarkuptrue%
|
|
330 |
%
|
|
331 |
\isadelimquote
|
|
332 |
%
|
|
333 |
\endisadelimquote
|
|
334 |
%
|
|
335 |
\isatagquote
|
|
336 |
%
|
|
337 |
\begin{isamarkuptext}%
|
|
338 |
\isatypewriter%
|
|
339 |
\noindent%
|
|
340 |
\hspace*{0pt}structure Example :~sig\\
|
|
341 |
\hspace*{0pt} ~datatype nat = Zero{\char95}nat | Suc of nat\\
|
|
342 |
\hspace*{0pt} ~val plus{\char95}nat :~nat -> nat -> nat\\
|
|
343 |
\hspace*{0pt} ~type 'a semigroup\\
|
|
344 |
\hspace*{0pt} ~val mult :~'a semigroup -> 'a -> 'a -> 'a\\
|
|
345 |
\hspace*{0pt} ~type 'a monoid\\
|
|
346 |
\hspace*{0pt} ~val semigroup{\char95}monoid :~'a monoid -> 'a semigroup\\
|
|
347 |
\hspace*{0pt} ~val neutral :~'a monoid -> 'a\\
|
|
348 |
\hspace*{0pt} ~val pow :~'a monoid -> nat -> 'a -> 'a\\
|
|
349 |
\hspace*{0pt} ~val mult{\char95}nat :~nat -> nat -> nat\\
|
|
350 |
\hspace*{0pt} ~val neutral{\char95}nat :~nat\\
|
|
351 |
\hspace*{0pt} ~val semigroup{\char95}nat :~nat semigroup\\
|
|
352 |
\hspace*{0pt} ~val monoid{\char95}nat :~nat monoid\\
|
|
353 |
\hspace*{0pt} ~val bexp :~nat -> nat\\
|
|
354 |
\hspace*{0pt}end = struct\\
|
|
355 |
\hspace*{0pt}\\
|
|
356 |
\hspace*{0pt}datatype nat = Zero{\char95}nat | Suc of nat;\\
|
|
357 |
\hspace*{0pt}\\
|
|
358 |
\hspace*{0pt}fun plus{\char95}nat (Suc m) n = plus{\char95}nat m (Suc n)\\
|
|
359 |
\hspace*{0pt} ~| plus{\char95}nat Zero{\char95}nat n = n;\\
|
|
360 |
\hspace*{0pt}\\
|
|
361 |
\hspace*{0pt}type 'a semigroup = {\char123}mult :~'a -> 'a -> 'a{\char125};\\
|
|
362 |
\hspace*{0pt}val mult = {\char35}mult :~'a semigroup -> 'a -> 'a -> 'a;\\
|
|
363 |
\hspace*{0pt}\\
|
|
364 |
\hspace*{0pt}type 'a monoid = {\char123}semigroup{\char95}monoid :~'a semigroup,~neutral :~'a{\char125};\\
|
|
365 |
\hspace*{0pt}val semigroup{\char95}monoid = {\char35}semigroup{\char95}monoid :~'a monoid -> 'a semigroup;\\
|
|
366 |
\hspace*{0pt}val neutral = {\char35}neutral :~'a monoid -> 'a;\\
|
|
367 |
\hspace*{0pt}\\
|
|
368 |
\hspace*{0pt}fun pow A{\char95}~Zero{\char95}nat a = neutral A{\char95}\\
|
|
369 |
\hspace*{0pt} ~| pow A{\char95}~(Suc n) a = mult (semigroup{\char95}monoid A{\char95}) a (pow A{\char95}~n a);\\
|
|
370 |
\hspace*{0pt}\\
|
|
371 |
\hspace*{0pt}fun mult{\char95}nat Zero{\char95}nat n = Zero{\char95}nat\\
|
|
372 |
\hspace*{0pt} ~| mult{\char95}nat (Suc m) n = plus{\char95}nat n (mult{\char95}nat m n);\\
|
|
373 |
\hspace*{0pt}\\
|
|
374 |
\hspace*{0pt}val neutral{\char95}nat :~nat = Suc Zero{\char95}nat;\\
|
|
375 |
\hspace*{0pt}\\
|
|
376 |
\hspace*{0pt}val semigroup{\char95}nat = {\char123}mult = mult{\char95}nat{\char125}~:~nat semigroup;\\
|
|
377 |
\hspace*{0pt}\\
|
|
378 |
\hspace*{0pt}val monoid{\char95}nat = {\char123}semigroup{\char95}monoid = semigroup{\char95}nat,~neutral = neutral{\char95}nat{\char125}\\
|
|
379 |
\hspace*{0pt} ~:~nat monoid;\\
|
|
380 |
\hspace*{0pt}\\
|
|
381 |
\hspace*{0pt}fun bexp n = pow monoid{\char95}nat n (Suc (Suc Zero{\char95}nat));\\
|
|
382 |
\hspace*{0pt}\\
|
|
383 |
\hspace*{0pt}end;~(*struct Example*)%
|
|
384 |
\end{isamarkuptext}%
|
|
385 |
\isamarkuptrue%
|
|
386 |
%
|
|
387 |
\endisatagquote
|
|
388 |
{\isafoldquote}%
|
|
389 |
%
|
|
390 |
\isadelimquote
|
|
391 |
%
|
|
392 |
\endisadelimquote
|
|
393 |
%
|
|
394 |
\begin{isamarkuptext}%
|
|
395 |
\noindent Note the parameters with trailing underscore (\verb|A_|),
|
|
396 |
which are the dictionary parameters.%
|
|
397 |
\end{isamarkuptext}%
|
|
398 |
\isamarkuptrue%
|
|
399 |
%
|
|
400 |
\isamarkupsubsection{The preprocessor \label{sec:preproc}%
|
|
401 |
}
|
|
402 |
\isamarkuptrue%
|
|
403 |
%
|
|
404 |
\begin{isamarkuptext}%
|
|
405 |
Before selected function theorems are turned into abstract
|
|
406 |
code, a chain of definitional transformation steps is carried
|
|
407 |
out: \emph{preprocessing}. In essence, the preprocessor
|
|
408 |
consists of two components: a \emph{simpset} and \emph{function transformers}.
|
|
409 |
|
|
410 |
The \emph{simpset} can apply the full generality of the
|
|
411 |
Isabelle simplifier. Due to the interpretation of theorems as code
|
|
412 |
equations, rewrites are applied to the right hand side and the
|
|
413 |
arguments of the left hand side of an equation, but never to the
|
|
414 |
constant heading the left hand side. An important special case are
|
|
415 |
\emph{unfold theorems}, which may be declared and removed using
|
|
416 |
the \hyperlink{attribute.code-unfold}{\mbox{\isa{code{\isacharunderscore}unfold}}} or \emph{\hyperlink{attribute.code-unfold}{\mbox{\isa{code{\isacharunderscore}unfold}}} del}
|
|
417 |
attribute, respectively.
|
|
418 |
|
|
419 |
Some common applications:%
|
|
420 |
\end{isamarkuptext}%
|
|
421 |
\isamarkuptrue%
|
|
422 |
%
|
|
423 |
\begin{itemize}
|
|
424 |
%
|
|
425 |
\begin{isamarkuptext}%
|
|
426 |
\item replacing non-executable constructs by executable ones:%
|
|
427 |
\end{isamarkuptext}%
|
|
428 |
\isamarkuptrue%
|
|
429 |
%
|
|
430 |
\isadelimquote
|
|
431 |
%
|
|
432 |
\endisadelimquote
|
|
433 |
%
|
|
434 |
\isatagquote
|
|
435 |
\isacommand{lemma}\isamarkupfalse%
|
|
436 |
\ {\isacharbrackleft}code{\isacharunderscore}unfold{\isacharbrackright}{\isacharcolon}\isanewline
|
|
437 |
\ \ {\isachardoublequoteopen}x\ {\isasymin}\ set\ xs\ {\isasymlongleftrightarrow}\ List{\isachardot}member\ xs\ x{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
|
438 |
\ {\isacharparenleft}fact\ in{\isacharunderscore}set{\isacharunderscore}member{\isacharparenright}%
|
|
439 |
\endisatagquote
|
|
440 |
{\isafoldquote}%
|
|
441 |
%
|
|
442 |
\isadelimquote
|
|
443 |
%
|
|
444 |
\endisadelimquote
|
|
445 |
%
|
|
446 |
\begin{isamarkuptext}%
|
|
447 |
\item eliminating superfluous constants:%
|
|
448 |
\end{isamarkuptext}%
|
|
449 |
\isamarkuptrue%
|
|
450 |
%
|
|
451 |
\isadelimquote
|
|
452 |
%
|
|
453 |
\endisadelimquote
|
|
454 |
%
|
|
455 |
\isatagquote
|
|
456 |
\isacommand{lemma}\isamarkupfalse%
|
|
457 |
\ {\isacharbrackleft}code{\isacharunderscore}unfold{\isacharbrackright}{\isacharcolon}\isanewline
|
|
458 |
\ \ {\isachardoublequoteopen}{\isadigit{1}}\ {\isacharequal}\ Suc\ {\isadigit{0}}{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
|
459 |
\ {\isacharparenleft}fact\ One{\isacharunderscore}nat{\isacharunderscore}def{\isacharparenright}%
|
|
460 |
\endisatagquote
|
|
461 |
{\isafoldquote}%
|
|
462 |
%
|
|
463 |
\isadelimquote
|
|
464 |
%
|
|
465 |
\endisadelimquote
|
|
466 |
%
|
|
467 |
\begin{isamarkuptext}%
|
|
468 |
\item replacing executable but inconvenient constructs:%
|
|
469 |
\end{isamarkuptext}%
|
|
470 |
\isamarkuptrue%
|
|
471 |
%
|
|
472 |
\isadelimquote
|
|
473 |
%
|
|
474 |
\endisadelimquote
|
|
475 |
%
|
|
476 |
\isatagquote
|
|
477 |
\isacommand{lemma}\isamarkupfalse%
|
|
478 |
\ {\isacharbrackleft}code{\isacharunderscore}unfold{\isacharbrackright}{\isacharcolon}\isanewline
|
|
479 |
\ \ {\isachardoublequoteopen}xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymlongleftrightarrow}\ List{\isachardot}null\ xs{\isachardoublequoteclose}\ \isacommand{by}\isamarkupfalse%
|
|
480 |
\ {\isacharparenleft}fact\ eq{\isacharunderscore}Nil{\isacharunderscore}null{\isacharparenright}%
|
|
481 |
\endisatagquote
|
|
482 |
{\isafoldquote}%
|
|
483 |
%
|
|
484 |
\isadelimquote
|
|
485 |
%
|
|
486 |
\endisadelimquote
|
|
487 |
%
|
|
488 |
\end{itemize}
|
|
489 |
%
|
|
490 |
\begin{isamarkuptext}%
|
|
491 |
\noindent \emph{Function transformers} provide a very general interface,
|
|
492 |
transforming a list of function theorems to another
|
|
493 |
list of function theorems, provided that neither the heading
|
|
494 |
constant nor its type change. The \isa{{\isadigit{0}}} / \isa{Suc}
|
|
495 |
pattern elimination implemented in
|
|
496 |
theory \isa{Efficient{\isacharunderscore}Nat} (see \secref{eff_nat}) uses this
|
|
497 |
interface.
|
|
498 |
|
|
499 |
\noindent The current setup of the preprocessor may be inspected using
|
|
500 |
the \hyperlink{command.print-codeproc}{\mbox{\isa{\isacommand{print{\isacharunderscore}codeproc}}}} command.
|
|
501 |
\hyperlink{command.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}} provides a convenient
|
|
502 |
mechanism to inspect the impact of a preprocessor setup
|
|
503 |
on code equations.
|
|
504 |
|
|
505 |
\begin{warn}
|
|
506 |
|
|
507 |
Attribute \hyperlink{attribute.code-unfold}{\mbox{\isa{code{\isacharunderscore}unfold}}} also applies to the
|
|
508 |
preprocessor of the ancient \isa{SML\ code\ generator}; in case
|
|
509 |
this is not what you intend, use \hyperlink{attribute.code-inline}{\mbox{\isa{code{\isacharunderscore}inline}}} instead.
|
|
510 |
\end{warn}%
|
|
511 |
\end{isamarkuptext}%
|
|
512 |
\isamarkuptrue%
|
|
513 |
%
|
|
514 |
\isamarkupsubsection{Datatypes \label{sec:datatypes}%
|
|
515 |
}
|
|
516 |
\isamarkuptrue%
|
|
517 |
%
|
|
518 |
\begin{isamarkuptext}%
|
|
519 |
Conceptually, any datatype is spanned by a set of
|
|
520 |
\emph{constructors} of type \isa{{\isasymtau}\ {\isacharequal}\ {\isasymdots}\ {\isasymRightarrow}\ {\isasymkappa}\ {\isasymalpha}\isactrlisub {\isadigit{1}}\ {\isasymdots}\ {\isasymalpha}\isactrlisub n} where \isa{{\isacharbraceleft}{\isasymalpha}\isactrlisub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlisub n{\isacharbraceright}} is exactly the set of \emph{all} type variables in
|
|
521 |
\isa{{\isasymtau}}. The HOL datatype package by default registers any new
|
|
522 |
datatype in the table of datatypes, which may be inspected using the
|
|
523 |
\hyperlink{command.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} command.
|
|
524 |
|
|
525 |
In some cases, it is appropriate to alter or extend this table. As
|
|
526 |
an example, we will develop an alternative representation of the
|
|
527 |
queue example given in \secref{sec:intro}. The amortised
|
|
528 |
representation is convenient for generating code but exposes its
|
|
529 |
\qt{implementation} details, which may be cumbersome when proving
|
|
530 |
theorems about it. Therefore, here is a simple, straightforward
|
|
531 |
representation of queues:%
|
|
532 |
\end{isamarkuptext}%
|
|
533 |
\isamarkuptrue%
|
|
534 |
%
|
|
535 |
\isadelimquote
|
|
536 |
%
|
|
537 |
\endisadelimquote
|
|
538 |
%
|
|
539 |
\isatagquote
|
|
540 |
\isacommand{datatype}\isamarkupfalse%
|
|
541 |
\ {\isacharprime}a\ queue\ {\isacharequal}\ Queue\ {\isachardoublequoteopen}{\isacharprime}a\ list{\isachardoublequoteclose}\isanewline
|
|
542 |
\isanewline
|
|
543 |
\isacommand{definition}\isamarkupfalse%
|
|
544 |
\ empty\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
|
|
545 |
\ \ {\isachardoublequoteopen}empty\ {\isacharequal}\ Queue\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
|
|
546 |
\isanewline
|
|
547 |
\isacommand{primrec}\isamarkupfalse%
|
|
548 |
\ enqueue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
|
|
549 |
\ \ {\isachardoublequoteopen}enqueue\ x\ {\isacharparenleft}Queue\ xs{\isacharparenright}\ {\isacharequal}\ Queue\ {\isacharparenleft}xs\ {\isacharat}\ {\isacharbrackleft}x{\isacharbrackright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
550 |
\isanewline
|
|
551 |
\isacommand{fun}\isamarkupfalse%
|
|
552 |
\ dequeue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ option\ {\isasymtimes}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
|
|
553 |
\ \ \ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}Queue\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}None{\isacharcomma}\ Queue\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
554 |
\ \ {\isacharbar}\ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}Queue\ {\isacharparenleft}x\ {\isacharhash}\ xs{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Some\ x{\isacharcomma}\ Queue\ xs{\isacharparenright}{\isachardoublequoteclose}%
|
|
555 |
\endisatagquote
|
|
556 |
{\isafoldquote}%
|
|
557 |
%
|
|
558 |
\isadelimquote
|
|
559 |
%
|
|
560 |
\endisadelimquote
|
|
561 |
%
|
|
562 |
\begin{isamarkuptext}%
|
|
563 |
\noindent This we can use directly for proving; for executing,
|
|
564 |
we provide an alternative characterisation:%
|
|
565 |
\end{isamarkuptext}%
|
|
566 |
\isamarkuptrue%
|
|
567 |
%
|
|
568 |
\isadelimquote
|
|
569 |
%
|
|
570 |
\endisadelimquote
|
|
571 |
%
|
|
572 |
\isatagquote
|
|
573 |
\isacommand{definition}\isamarkupfalse%
|
|
574 |
\ AQueue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
|
|
575 |
\ \ {\isachardoublequoteopen}AQueue\ xs\ ys\ {\isacharequal}\ Queue\ {\isacharparenleft}ys\ {\isacharat}\ rev\ xs{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
576 |
\isanewline
|
|
577 |
\isacommand{code{\isacharunderscore}datatype}\isamarkupfalse%
|
|
578 |
\ AQueue%
|
|
579 |
\endisatagquote
|
|
580 |
{\isafoldquote}%
|
|
581 |
%
|
|
582 |
\isadelimquote
|
|
583 |
%
|
|
584 |
\endisadelimquote
|
|
585 |
%
|
|
586 |
\begin{isamarkuptext}%
|
|
587 |
\noindent Here we define a \qt{constructor} \isa{AQueue} which
|
|
588 |
is defined in terms of \isa{Queue} and interprets its arguments
|
|
589 |
according to what the \emph{content} of an amortised queue is supposed
|
|
590 |
to be. Equipped with this, we are able to prove the following equations
|
|
591 |
for our primitive queue operations which \qt{implement} the simple
|
|
592 |
queues in an amortised fashion:%
|
|
593 |
\end{isamarkuptext}%
|
|
594 |
\isamarkuptrue%
|
|
595 |
%
|
|
596 |
\isadelimquote
|
|
597 |
%
|
|
598 |
\endisadelimquote
|
|
599 |
%
|
|
600 |
\isatagquote
|
|
601 |
\isacommand{lemma}\isamarkupfalse%
|
|
602 |
\ empty{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
|
|
603 |
\ \ {\isachardoublequoteopen}empty\ {\isacharequal}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isachardoublequoteclose}\isanewline
|
|
604 |
\ \ \isacommand{unfolding}\isamarkupfalse%
|
|
605 |
\ AQueue{\isacharunderscore}def\ empty{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
|
|
606 |
\ simp\isanewline
|
|
607 |
\isanewline
|
|
608 |
\isacommand{lemma}\isamarkupfalse%
|
|
609 |
\ enqueue{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
|
|
610 |
\ \ {\isachardoublequoteopen}enqueue\ x\ {\isacharparenleft}AQueue\ xs\ ys{\isacharparenright}\ {\isacharequal}\ AQueue\ {\isacharparenleft}x\ {\isacharhash}\ xs{\isacharparenright}\ ys{\isachardoublequoteclose}\isanewline
|
|
611 |
\ \ \isacommand{unfolding}\isamarkupfalse%
|
|
612 |
\ AQueue{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
|
|
613 |
\ simp\isanewline
|
|
614 |
\isanewline
|
|
615 |
\isacommand{lemma}\isamarkupfalse%
|
|
616 |
\ dequeue{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
|
|
617 |
\ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\isanewline
|
|
618 |
\ \ \ \ {\isacharparenleft}if\ xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ then\ {\isacharparenleft}None{\isacharcomma}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\isanewline
|
|
619 |
\ \ \ \ else\ dequeue\ {\isacharparenleft}AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharparenleft}rev\ xs{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
620 |
\ \ {\isachardoublequoteopen}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}Some\ y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
621 |
\ \ \isacommand{unfolding}\isamarkupfalse%
|
|
622 |
\ AQueue{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
|
|
623 |
\ simp{\isacharunderscore}all%
|
|
624 |
\endisatagquote
|
|
625 |
{\isafoldquote}%
|
|
626 |
%
|
|
627 |
\isadelimquote
|
|
628 |
%
|
|
629 |
\endisadelimquote
|
|
630 |
%
|
|
631 |
\begin{isamarkuptext}%
|
|
632 |
\noindent For completeness, we provide a substitute for the
|
|
633 |
\isa{case} combinator on queues:%
|
|
634 |
\end{isamarkuptext}%
|
|
635 |
\isamarkuptrue%
|
|
636 |
%
|
|
637 |
\isadelimquote
|
|
638 |
%
|
|
639 |
\endisadelimquote
|
|
640 |
%
|
|
641 |
\isatagquote
|
|
642 |
\isacommand{lemma}\isamarkupfalse%
|
|
643 |
\ queue{\isacharunderscore}case{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
|
|
644 |
\ \ {\isachardoublequoteopen}queue{\isacharunderscore}case\ f\ {\isacharparenleft}AQueue\ xs\ ys{\isacharparenright}\ {\isacharequal}\ f\ {\isacharparenleft}ys\ {\isacharat}\ rev\ xs{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
645 |
\ \ \isacommand{unfolding}\isamarkupfalse%
|
|
646 |
\ AQueue{\isacharunderscore}def\ \isacommand{by}\isamarkupfalse%
|
|
647 |
\ simp%
|
|
648 |
\endisatagquote
|
|
649 |
{\isafoldquote}%
|
|
650 |
%
|
|
651 |
\isadelimquote
|
|
652 |
%
|
|
653 |
\endisadelimquote
|
|
654 |
%
|
|
655 |
\begin{isamarkuptext}%
|
|
656 |
\noindent The resulting code looks as expected:%
|
|
657 |
\end{isamarkuptext}%
|
|
658 |
\isamarkuptrue%
|
|
659 |
%
|
|
660 |
\isadelimquote
|
|
661 |
%
|
|
662 |
\endisadelimquote
|
|
663 |
%
|
|
664 |
\isatagquote
|
|
665 |
%
|
|
666 |
\begin{isamarkuptext}%
|
|
667 |
\isatypewriter%
|
|
668 |
\noindent%
|
|
669 |
\hspace*{0pt}structure Example :~sig\\
|
|
670 |
\hspace*{0pt} ~val foldl :~('a -> 'b -> 'a) -> 'a -> 'b list -> 'a\\
|
|
671 |
\hspace*{0pt} ~val rev :~'a list -> 'a list\\
|
|
672 |
\hspace*{0pt} ~val null :~'a list -> bool\\
|
|
673 |
\hspace*{0pt} ~datatype 'a queue = AQueue of 'a list * 'a list\\
|
|
674 |
\hspace*{0pt} ~val empty :~'a queue\\
|
|
675 |
\hspace*{0pt} ~val dequeue :~'a queue -> 'a option * 'a queue\\
|
|
676 |
\hspace*{0pt} ~val enqueue :~'a -> 'a queue -> 'a queue\\
|
|
677 |
\hspace*{0pt}end = struct\\
|
|
678 |
\hspace*{0pt}\\
|
|
679 |
\hspace*{0pt}fun foldl f a [] = a\\
|
|
680 |
\hspace*{0pt} ~| foldl f a (x ::~xs) = foldl f (f a x) xs;\\
|
|
681 |
\hspace*{0pt}\\
|
|
682 |
\hspace*{0pt}fun rev xs = foldl (fn xsa => fn x => x ::~xsa) [] xs;\\
|
|
683 |
\hspace*{0pt}\\
|
|
684 |
\hspace*{0pt}fun null [] = true\\
|
|
685 |
\hspace*{0pt} ~| null (x ::~xs) = false;\\
|
|
686 |
\hspace*{0pt}\\
|
|
687 |
\hspace*{0pt}datatype 'a queue = AQueue of 'a list * 'a list;\\
|
|
688 |
\hspace*{0pt}\\
|
|
689 |
\hspace*{0pt}val empty :~'a queue = AQueue ([],~[]);\\
|
|
690 |
\hspace*{0pt}\\
|
|
691 |
\hspace*{0pt}fun dequeue (AQueue (xs,~y ::~ys)) = (SOME y,~AQueue (xs,~ys))\\
|
|
692 |
\hspace*{0pt} ~| dequeue (AQueue (xs,~[])) =\\
|
|
693 |
\hspace*{0pt} ~~~(if null xs then (NONE,~AQueue ([],~[]))\\
|
|
694 |
\hspace*{0pt} ~~~~~else dequeue (AQueue ([],~rev xs)));\\
|
|
695 |
\hspace*{0pt}\\
|
|
696 |
\hspace*{0pt}fun enqueue x (AQueue (xs,~ys)) = AQueue (x ::~xs,~ys);\\
|
|
697 |
\hspace*{0pt}\\
|
|
698 |
\hspace*{0pt}end;~(*struct Example*)%
|
|
699 |
\end{isamarkuptext}%
|
|
700 |
\isamarkuptrue%
|
|
701 |
%
|
|
702 |
\endisatagquote
|
|
703 |
{\isafoldquote}%
|
|
704 |
%
|
|
705 |
\isadelimquote
|
|
706 |
%
|
|
707 |
\endisadelimquote
|
|
708 |
%
|
|
709 |
\begin{isamarkuptext}%
|
|
710 |
\noindent From this example, it can be glimpsed that using own
|
|
711 |
constructor sets is a little delicate since it changes the set of
|
|
712 |
valid patterns for values of that type. Without going into much
|
|
713 |
detail, here some practical hints:
|
|
714 |
|
|
715 |
\begin{itemize}
|
|
716 |
|
|
717 |
\item When changing the constructor set for datatypes, take care
|
|
718 |
to provide alternative equations for the \isa{case} combinator.
|
|
719 |
|
|
720 |
\item Values in the target language need not to be normalised --
|
|
721 |
different values in the target language may represent the same
|
|
722 |
value in the logic.
|
|
723 |
|
|
724 |
\item Usually, a good methodology to deal with the subtleties of
|
|
725 |
pattern matching is to see the type as an abstract type: provide
|
|
726 |
a set of operations which operate on the concrete representation
|
|
727 |
of the type, and derive further operations by combinations of
|
|
728 |
these primitive ones, without relying on a particular
|
|
729 |
representation.
|
|
730 |
|
|
731 |
\end{itemize}%
|
|
732 |
\end{isamarkuptext}%
|
|
733 |
\isamarkuptrue%
|
|
734 |
%
|
|
735 |
\isamarkupsubsection{Equality%
|
|
736 |
}
|
|
737 |
\isamarkuptrue%
|
|
738 |
%
|
|
739 |
\begin{isamarkuptext}%
|
|
740 |
Surely you have already noticed how equality is treated
|
|
741 |
by the code generator:%
|
|
742 |
\end{isamarkuptext}%
|
|
743 |
\isamarkuptrue%
|
|
744 |
%
|
|
745 |
\isadelimquote
|
|
746 |
%
|
|
747 |
\endisadelimquote
|
|
748 |
%
|
|
749 |
\isatagquote
|
|
750 |
\isacommand{primrec}\isamarkupfalse%
|
|
751 |
\ collect{\isacharunderscore}duplicates\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list\ {\isasymRightarrow}\ {\isacharprime}a\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
|
|
752 |
\ \ {\isachardoublequoteopen}collect{\isacharunderscore}duplicates\ xs\ ys\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharequal}\ xs{\isachardoublequoteclose}\isanewline
|
|
753 |
\ \ {\isacharbar}\ {\isachardoublequoteopen}collect{\isacharunderscore}duplicates\ xs\ ys\ {\isacharparenleft}z{\isacharhash}zs{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ z\ {\isasymin}\ set\ xs\isanewline
|
|
754 |
\ \ \ \ \ \ then\ if\ z\ {\isasymin}\ set\ ys\isanewline
|
|
755 |
\ \ \ \ \ \ \ \ then\ collect{\isacharunderscore}duplicates\ xs\ ys\ zs\isanewline
|
|
756 |
\ \ \ \ \ \ \ \ else\ collect{\isacharunderscore}duplicates\ xs\ {\isacharparenleft}z{\isacharhash}ys{\isacharparenright}\ zs\isanewline
|
|
757 |
\ \ \ \ \ \ else\ collect{\isacharunderscore}duplicates\ {\isacharparenleft}z{\isacharhash}xs{\isacharparenright}\ {\isacharparenleft}z{\isacharhash}ys{\isacharparenright}\ zs{\isacharparenright}{\isachardoublequoteclose}%
|
|
758 |
\endisatagquote
|
|
759 |
{\isafoldquote}%
|
|
760 |
%
|
|
761 |
\isadelimquote
|
|
762 |
%
|
|
763 |
\endisadelimquote
|
|
764 |
%
|
|
765 |
\begin{isamarkuptext}%
|
|
766 |
\noindent During preprocessing, the membership test is rewritten,
|
|
767 |
resulting in \isa{List{\isachardot}member}, which itself
|
|
768 |
performs an explicit equality check.%
|
|
769 |
\end{isamarkuptext}%
|
|
770 |
\isamarkuptrue%
|
|
771 |
%
|
|
772 |
\isadelimquote
|
|
773 |
%
|
|
774 |
\endisadelimquote
|
|
775 |
%
|
|
776 |
\isatagquote
|
|
777 |
%
|
|
778 |
\begin{isamarkuptext}%
|
|
779 |
\isatypewriter%
|
|
780 |
\noindent%
|
|
781 |
\hspace*{0pt}structure Example :~sig\\
|
|
782 |
\hspace*{0pt} ~type 'a eq\\
|
|
783 |
\hspace*{0pt} ~val eq :~'a eq -> 'a -> 'a -> bool\\
|
|
784 |
\hspace*{0pt} ~val eqa :~'a eq -> 'a -> 'a -> bool\\
|
|
785 |
\hspace*{0pt} ~val member :~'a eq -> 'a list -> 'a -> bool\\
|
|
786 |
\hspace*{0pt} ~val collect{\char95}duplicates :\\
|
|
787 |
\hspace*{0pt} ~~~'a eq -> 'a list -> 'a list -> 'a list -> 'a list\\
|
|
788 |
\hspace*{0pt}end = struct\\
|
|
789 |
\hspace*{0pt}\\
|
|
790 |
\hspace*{0pt}type 'a eq = {\char123}eq :~'a -> 'a -> bool{\char125};\\
|
|
791 |
\hspace*{0pt}val eq = {\char35}eq :~'a eq -> 'a -> 'a -> bool;\\
|
|
792 |
\hspace*{0pt}\\
|
|
793 |
\hspace*{0pt}fun eqa A{\char95}~a b = eq A{\char95}~a b;\\
|
|
794 |
\hspace*{0pt}\\
|
|
795 |
\hspace*{0pt}fun member A{\char95}~[] y = false\\
|
|
796 |
\hspace*{0pt} ~| member A{\char95}~(x ::~xs) y = eqa A{\char95}~x y orelse member A{\char95}~xs y;\\
|
|
797 |
\hspace*{0pt}\\
|
|
798 |
\hspace*{0pt}fun collect{\char95}duplicates A{\char95}~xs ys [] = xs\\
|
|
799 |
\hspace*{0pt} ~| collect{\char95}duplicates A{\char95}~xs ys (z ::~zs) =\\
|
|
800 |
\hspace*{0pt} ~~~(if member A{\char95}~xs z\\
|
|
801 |
\hspace*{0pt} ~~~~~then (if member A{\char95}~ys z then collect{\char95}duplicates A{\char95}~xs ys zs\\
|
|
802 |
\hspace*{0pt} ~~~~~~~~~~~~else collect{\char95}duplicates A{\char95}~xs (z ::~ys) zs)\\
|
|
803 |
\hspace*{0pt} ~~~~~else collect{\char95}duplicates A{\char95}~(z ::~xs) (z ::~ys) zs);\\
|
|
804 |
\hspace*{0pt}\\
|
|
805 |
\hspace*{0pt}end;~(*struct Example*)%
|
|
806 |
\end{isamarkuptext}%
|
|
807 |
\isamarkuptrue%
|
|
808 |
%
|
|
809 |
\endisatagquote
|
|
810 |
{\isafoldquote}%
|
|
811 |
%
|
|
812 |
\isadelimquote
|
|
813 |
%
|
|
814 |
\endisadelimquote
|
|
815 |
%
|
|
816 |
\begin{isamarkuptext}%
|
|
817 |
\noindent Obviously, polymorphic equality is implemented the Haskell
|
|
818 |
way using a type class. How is this achieved? HOL introduces
|
|
819 |
an explicit class \isa{eq} with a corresponding operation
|
|
820 |
\isa{eq{\isacharunderscore}class{\isachardot}eq} such that \isa{eq{\isacharunderscore}class{\isachardot}eq\ {\isacharequal}\ op\ {\isacharequal}}.
|
|
821 |
The preprocessing framework does the rest by propagating the
|
|
822 |
\isa{eq} constraints through all dependent code equations.
|
|
823 |
For datatypes, instances of \isa{eq} are implicitly derived
|
|
824 |
when possible. For other types, you may instantiate \isa{eq}
|
|
825 |
manually like any other type class.%
|
|
826 |
\end{isamarkuptext}%
|
|
827 |
\isamarkuptrue%
|
|
828 |
%
|
|
829 |
\isamarkupsubsection{Explicit partiality%
|
|
830 |
}
|
|
831 |
\isamarkuptrue%
|
|
832 |
%
|
|
833 |
\begin{isamarkuptext}%
|
|
834 |
Partiality usually enters the game by partial patterns, as
|
|
835 |
in the following example, again for amortised queues:%
|
|
836 |
\end{isamarkuptext}%
|
|
837 |
\isamarkuptrue%
|
|
838 |
%
|
|
839 |
\isadelimquote
|
|
840 |
%
|
|
841 |
\endisadelimquote
|
|
842 |
%
|
|
843 |
\isatagquote
|
|
844 |
\isacommand{definition}\isamarkupfalse%
|
|
845 |
\ strict{\isacharunderscore}dequeue\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
|
|
846 |
\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue\ q\ {\isacharequal}\ {\isacharparenleft}case\ dequeue\ q\isanewline
|
|
847 |
\ \ \ \ of\ {\isacharparenleft}Some\ x{\isacharcomma}\ q{\isacharprime}{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}x{\isacharcomma}\ q{\isacharprime}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
848 |
\isanewline
|
|
849 |
\isacommand{lemma}\isamarkupfalse%
|
|
850 |
\ strict{\isacharunderscore}dequeue{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
|
|
851 |
\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
852 |
\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\isanewline
|
|
853 |
\ \ \ \ {\isacharparenleft}case\ rev\ xs\ of\ y\ {\isacharhash}\ ys\ {\isasymRightarrow}\ {\isacharparenleft}y{\isacharcomma}\ AQueue\ {\isacharbrackleft}{\isacharbrackright}\ ys{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
854 |
\ \ \isacommand{by}\isamarkupfalse%
|
|
855 |
\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ strict{\isacharunderscore}dequeue{\isacharunderscore}def\ dequeue{\isacharunderscore}AQueue\ split{\isacharcolon}\ list{\isachardot}splits{\isacharparenright}%
|
|
856 |
\endisatagquote
|
|
857 |
{\isafoldquote}%
|
|
858 |
%
|
|
859 |
\isadelimquote
|
|
860 |
%
|
|
861 |
\endisadelimquote
|
|
862 |
%
|
|
863 |
\begin{isamarkuptext}%
|
|
864 |
\noindent In the corresponding code, there is no equation
|
|
865 |
for the pattern \isa{AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharbrackleft}{\isacharbrackright}}:%
|
|
866 |
\end{isamarkuptext}%
|
|
867 |
\isamarkuptrue%
|
|
868 |
%
|
|
869 |
\isadelimquote
|
|
870 |
%
|
|
871 |
\endisadelimquote
|
|
872 |
%
|
|
873 |
\isatagquote
|
|
874 |
%
|
|
875 |
\begin{isamarkuptext}%
|
|
876 |
\isatypewriter%
|
|
877 |
\noindent%
|
|
878 |
\hspace*{0pt}strict{\char95}dequeue ::~forall a.~Queue a -> (a,~Queue a);\\
|
|
879 |
\hspace*{0pt}strict{\char95}dequeue (AQueue xs []) =\\
|
|
880 |
\hspace*{0pt} ~let {\char123}\\
|
|
881 |
\hspace*{0pt} ~~~(y :~ys) = reverse xs;\\
|
|
882 |
\hspace*{0pt} ~{\char125}~in (y,~AQueue [] ys);\\
|
|
883 |
\hspace*{0pt}strict{\char95}dequeue (AQueue xs (y :~ys)) = (y,~AQueue xs ys);%
|
|
884 |
\end{isamarkuptext}%
|
|
885 |
\isamarkuptrue%
|
|
886 |
%
|
|
887 |
\endisatagquote
|
|
888 |
{\isafoldquote}%
|
|
889 |
%
|
|
890 |
\isadelimquote
|
|
891 |
%
|
|
892 |
\endisadelimquote
|
|
893 |
%
|
|
894 |
\begin{isamarkuptext}%
|
|
895 |
\noindent In some cases it is desirable to have this
|
|
896 |
pseudo-\qt{partiality} more explicitly, e.g.~as follows:%
|
|
897 |
\end{isamarkuptext}%
|
|
898 |
\isamarkuptrue%
|
|
899 |
%
|
|
900 |
\isadelimquote
|
|
901 |
%
|
|
902 |
\endisadelimquote
|
|
903 |
%
|
|
904 |
\isatagquote
|
|
905 |
\isacommand{axiomatization}\isamarkupfalse%
|
|
906 |
\ empty{\isacharunderscore}queue\ {\isacharcolon}{\isacharcolon}\ {\isacharprime}a\isanewline
|
|
907 |
\isanewline
|
|
908 |
\isacommand{definition}\isamarkupfalse%
|
|
909 |
\ strict{\isacharunderscore}dequeue{\isacharprime}\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}{\isacharprime}a\ queue\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymtimes}\ {\isacharprime}a\ queue{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
|
|
910 |
\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue{\isacharprime}\ q\ {\isacharequal}\ {\isacharparenleft}case\ dequeue\ q\ of\ {\isacharparenleft}Some\ x{\isacharcomma}\ q{\isacharprime}{\isacharparenright}\ {\isasymRightarrow}\ {\isacharparenleft}x{\isacharcomma}\ q{\isacharprime}{\isacharparenright}\ {\isacharbar}\ {\isacharunderscore}\ {\isasymRightarrow}\ empty{\isacharunderscore}queue{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
911 |
\isanewline
|
|
912 |
\isacommand{lemma}\isamarkupfalse%
|
|
913 |
\ strict{\isacharunderscore}dequeue{\isacharprime}{\isacharunderscore}AQueue\ {\isacharbrackleft}code{\isacharbrackright}{\isacharcolon}\isanewline
|
|
914 |
\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue{\isacharprime}\ {\isacharparenleft}AQueue\ xs\ {\isacharbrackleft}{\isacharbrackright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}if\ xs\ {\isacharequal}\ {\isacharbrackleft}{\isacharbrackright}\ then\ empty{\isacharunderscore}queue\isanewline
|
|
915 |
\ \ \ \ \ else\ strict{\isacharunderscore}dequeue{\isacharprime}\ {\isacharparenleft}AQueue\ {\isacharbrackleft}{\isacharbrackright}\ {\isacharparenleft}rev\ xs{\isacharparenright}{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
916 |
\ \ {\isachardoublequoteopen}strict{\isacharunderscore}dequeue{\isacharprime}\ {\isacharparenleft}AQueue\ xs\ {\isacharparenleft}y\ {\isacharhash}\ ys{\isacharparenright}{\isacharparenright}\ {\isacharequal}\isanewline
|
|
917 |
\ \ \ \ \ {\isacharparenleft}y{\isacharcomma}\ AQueue\ xs\ ys{\isacharparenright}{\isachardoublequoteclose}\isanewline
|
|
918 |
\ \ \isacommand{by}\isamarkupfalse%
|
|
919 |
\ {\isacharparenleft}simp{\isacharunderscore}all\ add{\isacharcolon}\ strict{\isacharunderscore}dequeue{\isacharprime}{\isacharunderscore}def\ dequeue{\isacharunderscore}AQueue\ split{\isacharcolon}\ list{\isachardot}splits{\isacharparenright}%
|
|
920 |
\endisatagquote
|
|
921 |
{\isafoldquote}%
|
|
922 |
%
|
|
923 |
\isadelimquote
|
|
924 |
%
|
|
925 |
\endisadelimquote
|
|
926 |
%
|
|
927 |
\begin{isamarkuptext}%
|
|
928 |
Observe that on the right hand side of the definition of \isa{strict{\isacharunderscore}dequeue{\isacharprime}}, the unspecified constant \isa{empty{\isacharunderscore}queue} occurs.
|
|
929 |
|
|
930 |
Normally, if constants without any code equations occur in a
|
|
931 |
program, the code generator complains (since in most cases this is
|
|
932 |
indeed an error). But such constants can also be thought
|
|
933 |
of as function definitions which always fail,
|
|
934 |
since there is never a successful pattern match on the left hand
|
|
935 |
side. In order to categorise a constant into that category
|
|
936 |
explicitly, use \hyperlink{command.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}}:%
|
|
937 |
\end{isamarkuptext}%
|
|
938 |
\isamarkuptrue%
|
|
939 |
%
|
|
940 |
\isadelimquote
|
|
941 |
%
|
|
942 |
\endisadelimquote
|
|
943 |
%
|
|
944 |
\isatagquote
|
|
945 |
\isacommand{code{\isacharunderscore}abort}\isamarkupfalse%
|
|
946 |
\ empty{\isacharunderscore}queue%
|
|
947 |
\endisatagquote
|
|
948 |
{\isafoldquote}%
|
|
949 |
%
|
|
950 |
\isadelimquote
|
|
951 |
%
|
|
952 |
\endisadelimquote
|
|
953 |
%
|
|
954 |
\begin{isamarkuptext}%
|
|
955 |
\noindent Then the code generator will just insert an error or
|
|
956 |
exception at the appropriate position:%
|
|
957 |
\end{isamarkuptext}%
|
|
958 |
\isamarkuptrue%
|
|
959 |
%
|
|
960 |
\isadelimquote
|
|
961 |
%
|
|
962 |
\endisadelimquote
|
|
963 |
%
|
|
964 |
\isatagquote
|
|
965 |
%
|
|
966 |
\begin{isamarkuptext}%
|
|
967 |
\isatypewriter%
|
|
968 |
\noindent%
|
|
969 |
\hspace*{0pt}empty{\char95}queue ::~forall a.~a;\\
|
|
970 |
\hspace*{0pt}empty{\char95}queue = error {\char34}empty{\char95}queue{\char34};\\
|
|
971 |
\hspace*{0pt}\\
|
|
972 |
\hspace*{0pt}strict{\char95}dequeue ::~forall a.~Queue a -> (a,~Queue a);\\
|
|
973 |
\hspace*{0pt}strict{\char95}dequeue (AQueue xs (y :~ys)) = (y,~AQueue xs ys);\\
|
|
974 |
\hspace*{0pt}strict{\char95}dequeue (AQueue xs []) =\\
|
|
975 |
\hspace*{0pt} ~(if null xs then empty{\char95}queue\\
|
|
976 |
\hspace*{0pt} ~~~else strict{\char95}dequeue (AQueue [] (reverse xs)));%
|
|
977 |
\end{isamarkuptext}%
|
|
978 |
\isamarkuptrue%
|
|
979 |
%
|
|
980 |
\endisatagquote
|
|
981 |
{\isafoldquote}%
|
|
982 |
%
|
|
983 |
\isadelimquote
|
|
984 |
%
|
|
985 |
\endisadelimquote
|
|
986 |
%
|
|
987 |
\begin{isamarkuptext}%
|
|
988 |
\noindent This feature however is rarely needed in practice.
|
|
989 |
Note also that the \isa{HOL} default setup already declares
|
|
990 |
\isa{undefined} as \hyperlink{command.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}}, which is most
|
|
991 |
likely to be used in such situations.%
|
|
992 |
\end{isamarkuptext}%
|
|
993 |
\isamarkuptrue%
|
|
994 |
%
|
|
995 |
\isadelimtheory
|
|
996 |
%
|
|
997 |
\endisadelimtheory
|
|
998 |
%
|
|
999 |
\isatagtheory
|
|
1000 |
\isacommand{end}\isamarkupfalse%
|
|
1001 |
%
|
|
1002 |
\endisatagtheory
|
|
1003 |
{\isafoldtheory}%
|
|
1004 |
%
|
|
1005 |
\isadelimtheory
|
|
1006 |
%
|
|
1007 |
\endisadelimtheory
|
|
1008 |
\isanewline
|
|
1009 |
\end{isabellebody}%
|
|
1010 |
%%% Local Variables:
|
|
1011 |
%%% mode: latex
|
|
1012 |
%%% TeX-master: "root"
|
|
1013 |
%%% End:
|